# Paths of Real Roots for Odd-Degree Polynomials with Parameters

Authors: Junxiang Xu
Journal: Abstract and Applied Analysis
 (2020)

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## Abstract

In this paper, we construct a connected path of real roots of odd-degree real polynomials depending on parameters continuously, which joins with all parameters. Moreover, most of the points on the path are odd-multiple real roots except for at most countable even-multiple real roots. Furthermore, some geometrical features of the path can be seen easily from the construction.


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## Body

## 1. Problem

The problem of real roots of approximating real polynomials (small perturbations of real polynomials) with parameters is not only an elementary problem in theory but also a very important one in application. On the one hand, it is very interesting to understand the topological structure of the set of real roots of (approximating) real polynomials with parameters. On the other hand, it can arise in the study of small perturbations of degenerate systems such as in [1–5]. For example, when we consider the persistence of degenerate elliptic lower dimensional invariant tori in Hamiltonian systems, we must solve real roots of approximating real polynomials with a small parameter; moreover, we require to know how these real roots depend on the small parameter, which is very important for us to deal with the problem of small divisors in the KAM theory. It is well known that the small divisor condition in the KAM theory is usually guaranteed by adjusting some parameters. For this purpose, it is necessary to know the dependence of the frequencies on the parameter. This problem is the main motivation of this paper. We will understand it more precisely in a forthcoming paper, in which we will prove the persistence of degenerate elliptic lower dimensional invariant tori in Hamiltonian systems by the results obtained in this paper.If an approximating polynomial depends on a small parameter, by some division theorems in [6–10], it can decompose into a product of a polynomial and a nonzero factor. Thus, in this paper, we mainly consider odd-degree real polynomials. We should first note that by the Leray-Schauder continuation theorem [11, 12], we can also have a path in the set of real roots of f joining with all parameters. However, we have no more information about the geometrical structure of the path. This is because the continuation theorem does not require too much smoothness assumption. In this paper, we construct a path in the set of all the real roots which joins with all parameters; moreover, it includes at most countable even-multiple real roots. From the proof, we can know more information about the geometrical structure of the path. This is what we are interested in.


## 2. Main Results

Consider a real polynomial of degree2d+1:
(1)fξ,x=x2d+1+∑j=12d+1cjξx2d+1−j,where the coefficients cj,1≤j≤2d+1 depend continuously on a parameter ξ∈a,b. It is easy to know that f has at least one real root for every ξ. Moreover, it is well known (see [13]) that there exist 2d+1 complex roots continuously depending on ξ, which are also called 2d+1 continuous bifurcations of roots. More precisely, we have the following result.Lemma 1 (see [12]).
Consider anm-degree polynomial
(2)fξ,x=xm+∑1≤j≤mcjξxm−j,where cj are continuous on a,b. Then, there exist m continuous complex-valued functions.
(3)x1ξ,x2ξ,⋯xmξξ∈a,b,such that the set
(4)ξ,xjξξ∈a,b,j=1,2,⋯m,consists of all roots of fξ,xforξ∈a,b.However, it is possible that there does not exist any continuous bifurcation of real roots forξ on a,b. But by our observation, we find that for a family of parameterized odd-degree real polynomial fξ,x, there exists a connected path of real roots on the ξ,x-plane, which joins with two odd-multiple real roots on the lines ξ=a and ξ=b, respectively. This shows that the set of real roots of fξ,x contains a path which can connect all parameters ξ∈a,b. Moreover, the path includes at most countable even-multiple real roots; in other words, almost everywhere, points on the path are odd-multiple real roots.To state our result, we first introduce some notations and definitions.Ifx0 is an m-multiple real root of fξ0,x, that is,
(5)fxkξ0,x0=0,k=0,1,⋯,m−1,fxmξ0,x0≠0,we call ξ0,x0 an m-multiple real root of f.Let(6)af,a,b=ξ,x∈a,b×ℝfξ,x=0,ξ∈a,b,be the set which consists of all real roots of f. Sometimes, we write A=Af,a,b for simplicity if fand a,b are fixed. Let
(7)Aodd=P∈APisanodd‐multiplerealroot,Aeven=P∈APisaneven‐multiplerealroot.Let(8)Aξ=Af,a,bξ=Af,a,b∩ξ×ℝ..Obviously,Aξ consists of all real roots of the polynomial fξ,x. Our result is given as follows.Theorem 2.
There exists a continuous mapping(9)γ:t∈0,1⟶γt=ξt,xt∈A⊂ℝ2,such that γ0∈Aa and γ1∈Ab are odd-multiple real roots. Moreover, let
(10)γ=γtt∈0,1,Tp=t∈0,1γt=P,then for P∈γ, we have
(11)Tp#≤themultiplicityofP,where TP# indicates the cardinality of the set TP. Furthermore, the sets P∈γ∣TP#≥2 and γ∩Aeven are at most countable.Remark 3.
Theorem2 describes some information of topological structure of the set of real roots of f. Obviously, the continuous curve γt∣t∈0,1 is usually not unique. However, it is possible that γ consists of all real roots of f, that is, γt∣t∈0,1=A. So the result in Theorem 2 is optimal. Moreover, (11) implies that
(12)t∈0,1ξt=η#≤2d+1,∀η∈a,b.
Thus, there exists an at most countable subsetI⊂0,1 such that γ:t∈0,1\I⟶γt∈A is an injective mapping.The result of Theorem2 is a global property of real roots of odd-degree real polynomials. For some local property of real roots of real polynomials, we refer to [14] and the references therein.


## 3. Proof of Theorem 2

In order to prove Theorem2, we give some important definitions.


### 3.1. Definitions

Below, we first define paths of odd-multiple real roots, crossing points, basis points, and reflecting points.Definition 4 (paths of odd-multiple real roots).
Let(13)γ:t∈0,1⟶γt=ξt,xt∈A⊂ℝ2,be a continuous mapping with P0=γ0=ξ0,x0 and P1=γ1=ξ1,x1. γ is called a path of odd-multiple real roots with endpoints P0 and P1 if the following hold:
(i)
Lett0∈0,1 and P=γt0=ξ∗,x∗, then one of the following holds true:
(1)
There existsδ0>0 such that
(14)ξt<ξ∗,∀t∈t0−δ0,t0andξt>ξ∗,∀t∈t0,t0+δ0,or
(15)ξt>ξ∗,∀t∈t0−δ0,t0andξt<ξ∗,∀t∈t0,t0+δ0Moreover, such t0 is unique for the point P if it exists.(2)
There existsδ0>0 such that
(16)ξt<ξ∗,∀0<t−t0<δ0,orξt>ξ∗,∀0<t−t0<δ0.(ii)
LetT~p=t∈0,1∣γt=P. If P∈γ is an odd-multiple real root such that there exists a unique t0∈T~p such that (1) holds, then such P is called a crossing point at t0. Moreover, for t0′∈T~p,ift0′≠t0, the result (2) holds at t0′; such P is called a basis point at t0′. If P is a crossing point at t0 and a basis point at t0′, we say that the basis point Patt0′ is based on the crossing point P at t0. If P∈γ is an odd-multiple real root such that the result (2) holds for all t0∈T~p, then P is a basis point, but not a crossing point; such P is called a pure basis point(iii)
IfP∈γ is an even-multiple real root, then only the result (2) holds. An even-multiple real root on γ is called a reflecting point(iv)
IfP0=P1 is an odd-multiple real root and it is not a crossing point, then there exists δ>0 such thatγt,0<t<δ and γt,1−δ<t<1 are on the same side of ξ=ξ0=ξ1(v)
IfP0=P1 is an odd-multiple real root and it is a crossing point, then there exists δ>0 such that γt,0<t<δ and γt,1−δ<t<1 are on different sides of ξ=ξ0=ξ1(vi)
IfP0=P1 is an even-multiple real root, then there exists δ>0 such that γt,0<t<δ and γt,1−δ<t<1 are on different sides of ξ=ξ0=ξ1(vii)
All basis points and reflecting points onγ are at most countable; thus, the points on γ are odd-multiple real roots almost everywhereA path of odd-multiple real roots is also called a path in brief if without any confusion (see Figure1).Figure 1
Path of odd-multiple real roots.By definition, an odd-multiple real root on a path can be both a crossing point and a basis point. Moreover, a path can have no pure basis point, and all odd-multiple real roots on the path can be crossing points except for its endpoints. This kind of special path is very important for the extension of paths in our problem.Definition 5.
Letγ=γt∣0≤t≤1 be a path with endpoints P0=γ0 and P1=γ1. Let γ∘=γt∣t∈0,1, which is called the interior point of γ. If for all P∈γ°∩Aodd, P is a crossing points of γ, γ is called a regular path. A regular path has no pure basis point.Leta,b⊂0,1. The set γa,b=γt∣t∈a,b is called a segment ofγ. By definition, it is easy to see that a segment of a path is still a path, but a segment of a regular path may not be regular. For P∈γ, P may be a crossing point on one segment and a pure basis point on another (also see Figure 1).An endpoint of a path is such a point that the path can be extended longer at this point.LetP1≠P0. There exists a small δ>0 such that for all 0<t≤δ,ξt is on the same side of ξ0. It is easy to see that we can always extend γ to cross P0. The same is true for P1.However, when the two endpoints of a path collide into one point, the path may not be extended again at the point; in this situation, the endpoints disappear. To avoid the endpoints losing their property as endpoints, we suggest the conditions (iv), (v), and (vi). More precisely, when two endpoints meet, they still keep the property of endpoints only after they collide into a pure basis point along the same side, into a crossing point along different sides, or into a reflecting point along different sides.Usually, we can extend a path at an endpoint, but the direction of extension about the parameterξ may be restricted. For this problem, we define directions of endpoints.Definition 6 (simple endpoints and directions of endpoints).
Let(17)γ:t∈0,1⟶γt=ξt,xt∈A⊂ℝ2,be a path of odd-multiple real roots with endpoints P0=ξ0,x0andP1=ξ1,x1.
We only considerP0 since P1 can be considered similarly. If P0 is neither a crossing point, nor a basis point, nor a reflecting point, we call it a simple endpoint. Moreover, if P0 is an odd(even)-multiple real root, we call it an odd(even)-multiple endpoint.
Below, we define the direction of endpoints in two cases ofP0≠P1 and P0=P1. First, note that there exists a sufficiently small δ0>0 such that either ξt>ξ0 for all t∈0,δ0 or ξt<ξ0 for all t∈0,δ0. That means that for sufficiently small t, γt are on the same side of ξ=ξ0.Case 1 (P0≠P1).
IfP0 is not a crossing point of the path γ and γt are on the left side (or right side) of ξ=ξ0 for all sufficiently small t>0, P0 is called a right (or left) endpoint. If P0 is a crossing point of γ and γt are on the left side (or right side) of ξ=ξ0 for all sufficiently small t>0, P0 is called a left (or right) endpoint.
IfP0 is an even-multiple real root and for all sufficiently small t>0, and γt are on the left side (or right side) of ξ=ξ0, P0 is called a left (or right) endpoint. For P1, we have the same statement and we omit the details.Case 2 (P0=P1).
In the condition (iv) of Definition4, if there exists δ>0 such that the set γt∣0<t<δ is on the right (left) side of ξ=ξ0=ξ1, we call P0 a left (right) endpoint and P1 a right (left) endpoint.
In the condition (v) of Definition4, if the set γt∣0<t<δ is on the right (left) side of ξ=ξ0=ξ1, we call P0 a right (left) endpoint and P1 a left (right) endpoint.
In the condition (vi) of Definition4, if the set γt∣0<t<δ is on the right (left) side of ξ=ξ0=ξ1, we call P0 a right (left) endpoint and P1 a left (right) endpoint.Remark 7.
A left (or right) endpoint means that the path can be extended to the left (or right) ofξ=ξ0 at the endpoint. In view of this point, in the cases of (iv)–(vi), we can define different directions of the endpoints. We choose the above definition of directions of endpoints for convenience in the extension of paths.Definition 8 (compatibility of paths).
Letγ1,γ2,⋯γn⊂A be paths of odd-multiple real roots and
(18)E=γj0,γj11≤j≤n,be the set of all endpoints of γj. Let G=γ1∪γ2∪⋯∪γn and G°=γ°1∪γ°2∪⋯∪γ°n.
If for allP∈G°∩Aodd, there exists a unique γj such that P is a crossing point of γj; moreover, if P∈γ°k with k≠j, P is a pure basis point of γk, then we say γj are compatible (at P). If P∈G°∩Aeven, P is always a reflecting point and we also say γj are compatible at P.
LetD be a domain of rectangles. If for all P∈D∩G°, γj are compatible at P, we say γj are compatible onD or G is regular on D.
Letγj be compatible paths and γ be a path of odd-multiple real roots with two endpoints P0=γ0andP1=γ1. If γ,γ1,⋯γn are compatible, we say γ is compatible with γj. Let γ^ be a path of odd-multiple real roots with two endpoints P^0=γ^=0 and P^0=γ^1. Let P^0=P1∉E. If γ+=γ∪γ^ is a path with two endpoints P0 and P^1 and it is also compatible with γj, we say γ+ is a compatible extension of γ with γj and γ^ is the prolonged segment. Obviously, if P1∉G°, then P1 must be a crossing point of γ+; if P1∈G°, then P1 must be a pure basis point of γ+.Remark 9.
Whenγ,γ1,⋯γn are compatible, formally, we can regard γ∪γ1∪⋯∪γn as a generalized regular path, because it may not be connected and can have many endpoints. In particular, if G=γ1∪⋯∪γn=∅, then γ is a regular path.
Letγ and γ^ be two compatible paths. Let P be an endpoint of both γ and γ^ with P=γ1=γ^0. If P∈Aodd and it has opposite endpoint directions for γ and γ^, or if P∈Aeven and it has the same endpoint direction, then γ+=γ∪γ^ is a compatible extension of γ.
Now supposeγ:t∈0,1⟶γt∈A and γ^:t∈1,2⟶γ^t∈A and P∗=γ1=γ^1, where we use γ^t+1 in place of the above γ^t given by standard definition without confusion.
Note that we can require a path to be defined on any intervala,b instead of 0,1 by rescaling the parameter of paths.
Ifγ1 and γ^1 are not crossing points, we can define γ+=γ∪γ^ as a regular path by
(19)γ+t=γt,0≤t≤1,γ+t=γ^t,1≤t≤2.
By definition, it is easy to see thatγ+ is well defined.
If bothγ1 and γ^1 are crossing points and they have different directions of endpoints, then γ+ as defined above may not satisfy the definition of paths. The above definition may result in γ+t crossing itself at P∗ more than 1 time. To avoid P∗ being crossed repeatedly, we can rearrange the parameter at P∗ so that γ+=γ∪γ^ can be defined as a regular path, but here, we omit the details.
Compatibility of paths is very important for extension of paths. It makes the extension of a path to not overlap any small segment of existing paths so that the extension cannot have any circulation.
When we extend a regular path, we require the path to keep the compatibility with the other existing paths. So at the first time when the path meets a new odd-multiple real root, we always let it go through the odd-multiple real root such that it becomes a crossing point of the path. When the path meets itself or some existing path at a crossing point, it must reflect at this point, and then it becomes a basis point. That means a new odd-multiple real root usually first becomes a crossing point and then a basis point. When the path meets an even-multiple real root, it is always extended by reflecting at this point.
We usually start a compatible extension of a regular path with a simple endpoint and require the prolonged segment to end also at a simple endpoint, so that we can easily see the extension to be compatible.Definition 10 (closed paths).
Letγ be a path of odd-multiple real roots with endpoints P0 and P1. We call γ a closed path of odd-multiple real roots, if P0=P1 with the following holding true:
(1)
IfP0 is a crossing point and there exists a sufficiently small δ0>0 such that γt∣t∈0,δ0 and γt∣t∈1−δ0,1 are on the same side of ξ=ξ0, then P0 is a crossing point of the closed path(2)
IfP0 is not a crossing point and there exists a sufficiently small δ0>0 such that γt∣t∈0,δ0 and γt∣t∈1−δ0,1are on different sides of ξ=ξ0, then P0 is a crossing point of the closed path(3)
IfP0 is not a crossing point and there exists a sufficiently small δ0>0 such that γt∣t∈0,δ0 and γt∣t∈1−δ0,1 are on the same side of ξ=ξ0, then P0 is a pure basis point of the closed path(4)
IfP0 is an even-multiple real root and there exists a sufficiently small δ0>0 such that γt∣t∈0,δ0 and γt∣t∈1−δ0,1 are on the same side of ξ=ξ0, then P0 is a reflecting point of the closed path
Whenγ is a closed path, we can regard γ as a continuous mapping: ℝ/ℤ=T⟶A, which is from the unit circle T to A. P0=P1 can be a crossing point or a pure basis point of the closed path (see Figure 2).
If a closed pathγ:t∈T=ℝ/ℤ⟶γt∈A⊂ℝ2 has no pure basis point, we call γ a closed regular path; see Figure 3.Figure 2
Closed path.Figure 3
Closed regular path.A closed regular path is a circulation and it cannot be extended longer.


### 3.2. Idea of Construction of Paths

Below, we give a more detailed description of the construction of paths.Letfξ,x be a polynomial of degree 2d+1 as in (1) and
(20)A2d+1=ξ,xxisa2d+1‐multiplerealrootoffξ,x,ξ∈a,b,B2d+1=ξξ,x∈A2d+1.Obviously,fx2dξ,x=0 determines a simple continuous curve
(21)l:x=−12d+1c1ξ,ξ∈a,b.Moreover,A2d+1⊂ℓ. Since L=a,b\B2d+1 is an open set of R, then L=∪n≥1In, where In=an,bn⊂a,b, which are disjointed each other.Letℓn be the segment of ℓ corresponding to ℓn. We want to find a path γn in Af,an,bn in place of ℓn.Moreover, we also find pathsγ± in Af,a±,b± for ξ∈a−,b− and ξ∈a+,b+, respectively, where a−=a,b−=minξ∈B2d+1 and a+=maxξ∈B2d+1,b+=b, if a or b does not belong to B2d+1. Let ℓ± be the two segments of l on a−,b− and a+,b+. Then, if we replace ln and ℓ± in l with γn and γ±, respectively, we obtain a path of odd-multiple real roots for ξ on a,b.The most important is to construct a pathγn on an,bn. Then, the path γ is obtained by grafting the simple continuous curve l with countable paths γn and ℓ±, where γn and ℓ± are composed of at most countable paths of polynomials of lower degree, maybe in a little complicated way. Moreover, each γn and ℓ± include at most countable reflecting points and basis points.To considerγn on an,bn, for shorting of notation, we drop the subscripts n. Then, suppose fa,x and fb,x have 2d+1-multiple real roots and fξ,x has no 2d+1-multiple real root for ξ∈a,b. Now we explain how to construct a path γ on a,b.The construction ofγ is based on extension of paths and induction for the degree of polynomials.First note that an odd-multiple real root is never isolated and usually it breaks into some lower-multiple real roots about the parameter. However, an even-multiple real root can be isolated, near to which there may be no other real roots. But, if there exists an odd-multiple real root near an even-multiple real root, there must be another odd-multiple real root near the even-multiple real root, that is, odd-multiple real roots always appear in pairs near an even-multiple real root. Thus, when a path arrives at an even-multiple real root, it can always reflect into some odd-multiple real roots at the even-multiple real root. So we define a path to be such a continuous curve that it is allowed to go through an odd-multiple real root and reflect when it meets an even-multiple real root.A path of odd-multiple real roots can be a little complicated near to a real root of higher-multiplicity; it can zigzag along the direction of thex-axes and have several recurrences at some parameters to approximate the real root of higher multiplicity. However, the times of the recurrence are no more than the multiplicity of the real root.Now, we first construct a regular path to join toPb. Let a<ξ0<b. Now, take an odd-multiple real root on Dξ0. By induction assumption, there exists a local regular path across the odd-multiple real root. Then, we extend the local regular path to the right. Since fξ,x has no 2d+1-multiple real root for ξ∈a,b, the real roots near any an odd-multiple real root can be all the real roots of a real polynomial of lower degree, and so we can extend it locally longer by induction assumption. We can extend the path until it arrives at Pb or ξ=ξ0. In the first situation, we have a desired path; otherwise, the path is extended to arrive at a new odd-multiple real root on Dξ0. By induction assumption, we can extend the path across ξ=ξ0 to the left a little and then have a regular path with two endpoints on the left of Dξ0.Then, take another odd-multiple real root onDξ0 and do the same way. If we cannot have a desired path, we have another compatible path with the previous path with two endpoints on the left of Dξ0. In this way, after at most d steps, only one odd-multiple real root is left on Dξ0. Take a path across this odd-multiple real root and extend it to the right, since all the odd-multiple real roots on Dξ0 are occupied by the previous compatible paths and the path cannot go back to Dξ0. Finally, it must go to Pb, and then we obtain a regular path joining to Pb.Then, we extend the path joining toPb to the left. If it is not extended to Pa, it must go back to Pb to form a closed path with Pb as a pure basis point. Then, take an odd-multiple real root on Dξ0 and do the same way as above to obtain a compatible path joining to Pb. In the same way, we extend the compatible path joining to Pb to the left. If we cannot have a desired path, we must have one more compatible closed path with Pb as a pure basis point. But we cannot have more than d number of these compatible closed paths, so after at most d iterations, the extension of the compatible path cannot go back to Pb; it has to go to Pa. Thus, the extension of the compatible path joins with the previous several compatible closed paths to form a regular path γ joining to Pa and Pb.We should also note that if two2d-multiplicity real roots are sufficiently close, the two corresponding simple real roots to the two 2d-multiplicity real roots can be connected by a saturation path of simple real roots. So a path cannot meet two 2d-multiplicity real roots which are very close; in other words, all 2d-multiplicity real roots on a path are discrete. Thus, the path γ can include at most countable 2d-multiple real roots. Moreover, γ can only have Pa or Pb as 2d+1-multiple basis points. This fact is very important for a path to contain at most countable basis points and reflecting points.


### 3.3. Lemmas

Below, we prove some lemmas which are required in the construction of paths.We first note that in Definition4, the conditions (1) of (i) and (vii) are very important; they guarantee that any small segment of a path cannot be overlapped one more times by the path itself. Moreover, a path cannot recur too many times about a fixed parameter.Lemma 11.
Letγt=ξt,xt∣t∈0,1 be a path of real roots. For any two intervals I=t1,t2⊂0,1 and I′=t1′,t2′⊂0,1,ifI∩I′=ϕ, then the set γI∩γI′ is at most countable. Moreover, Tξ0#≤2d+1, where Tξ0=t∈0,1∣ξt=ξ0.Proof.
IfγI∩γI′ is not countable, by the condition (vii) of the definition of paths, there exists a point P∈γt∣t∈I∩γt∣t∈I′ such that P is neither a basis point nor an even-multiple real root, then P is a pure crossing point. Again, there exist t0∈I and t0′∈I′ such that P=γt0=γt0′, but t0≠t0′, which contradicts with the fact that P can be crossed only once.
If there are several segments of the path crossing the parameterξ0, the condition (vii) implies that there exists ξn⟶ξ0 such that each segment crossing ξ=ξ0 does not meet the other segments on the line ξ=ξn. Otherwise, if for all ∣η−ξ0∣<δ0, at least two segments have an intersecting point Pη on ξ=η, these segments can intersect on Pηη−ξ0∣≤δ0, which has the cardinal number of the continuum and contradicts with the condition (vii).
Since the number of segments crossing the lineξ=ξ0 is no more than that of points of these segments on ξ=ξ0, which is no more than 2d+1, so the path oscillates for the parameter around ξ0 no more than 2d+1 times, which implies Tξ0#≤2d+1.This lemma means that a path cannot intersect itself everywhere on a nonempty segment, which guarantees a path to recur to a fixed parameterξ0 no more than 2d+1 times. Moreover, an m-multiple real root can be used by a path at most m-times. If the multiplicity is considered and an m-multiple real root is regarded as m different real roots, then a compatible extension can be thought to consist of new real roots and a path can be thought to consist of different real roots.For the construction of paths, we consider how a path can intersect with a lineξ=ξ0. For this purpose, we have the following lemma.Lemma 12.
Letγ be a path of odd-multiple real roots with two endpoints P1=ξ1,x1 and P2=ξ2,x2 with ξ1<ξ2. Then, the following hold true:
(1)
Ifξ1<ξ0<ξ2, then γ∩ξ0×ℝ has an odd number of crossing points(2)
Ifξ0<ξ1orξ0>ξ2, then γ∩ξ0×ℝ has an even number of crossing points.Proof.
(1)
By definition, we can moveP1 continuously to P2 along the path γ, and in the process, we go through odd-multiple real roots only one time.
Without loss of generality, supposeP1 and P2 are left and right endpoints, respectively. Then, the path starts with P1 to go to the right. Since ξ0∈ξ1,ξ2, it must intersect with ξ0×ℝ at some odd-multiple real root Q1. If it goes back across ξ0×ℝ at some odd-multiple real root Q2, then Q1≠Q2 since they can only be crossed only once. Then, it must cross ξ0×ℝ at another odd-multiple real root Q3; otherwise, it cannot arrive to P2. In this way, it is easy to see that the path must cross ξ0×ℝ an odd number of times.
(2)
This case can be considered similarly and we omit the detailLet(22)D=ξ,xa≤ξ≤b,c<x<d,be a rectangle domain of the ξ,x-plane. Let Dξ=ξ×c,d be a section of D at ξ.Let the boundary(23)∂D=lx−∪lx+∪lξ−∪lξ+,where
(24)lx−=a,b×c,lx+=a,b×d,lξ−=Da,lξ+=Db.Obviously, the rectangleD is open at the up-down sides lx±and closed at the left-right sides lξ±. Let D∘=a,b×c,d be the interior of D.Lemma 13.
Letγ be a path of odd-multiple real roots with endpoints P1 and P2 and γ∩lx±=ϕ. Suppose P1,P2∉D. Then, the number of crossing points in γ∩Dξ has the same odevity for all ξ∈a,b. In particular, if γ does not go through D, then the number of crossing points in γ∩Dξ is even; if γ goes through D, then the number of crossing points in γ∩Dξ is odd.Proof.
By definition of paths, it follows easily and we omit the details.By the implicit function theorem, for a simple real rootP0, there exists a path crossing P0, which only includes simple real roots. Below, we consider the maximal one among these paths.Lemma 14.
Suppose that bothfa,x and fb,x have a 2d+1-multiple real root. Fix ξ0∈a,b. Let
(25)fξ0,x0=0,fxξ0,x0≠0.
There exists a unique continuous functionx=xξ,ξ∈a0,b0⊂a,b such that
(26)x0=xξ0,fξ,xξ=0,fxξ,xξ≠0,∀ξ∈a0,b0.
Moreover,(27)fa0,xa0=fxa0,xa0=fb0,xb0=fxb0,xb0=0,fa0,xa0=fxa0,xa0=fb0,xb0=fxb0,xb0=0.
The continuous curveγ:ξ,xξ∣ξ∈a0,b0 is called a maximal (or saturation) path of the simple root P0=ξ0,x0.Proof.
By the implicit function theorem, there existsδ0>0 and a unique implicit function x=xξ,ξ∈ξ0−δ,ξ0+δ such that
(28)xξ0=x0,fξ,xξ=0,fxξ,xξ≠0,ξ∈ξ0−δ,ξ0+δ.
Below, we extend the continuous functionx=xξ from ξ0−δ,ξ0+δ to an interval as big as possible.
Let(29)J=a¯,b¯thereexistsauniquefunctionx=xξ,ξ∈a¯,b¯,suchthatxξ0=x0,fξ,xξ=0,fxξ,xξ≠0,ξ∈a¯,b¯.
Obviously,ξ0∈a¯,b¯. Let
(30)a0=inf=a¯,b¯∈Ja¯,b0=supa¯,b¯∈Jb¯.
Forξ∈a0,b0, we define x∗ξ=xξ,ξ∈a¯,b¯.
We prove thatx∗ξ is well defined on a0,b0. In fact, xξ,ξ∈a¯,b¯ is determined uniquely by ξ0,x0. It is easy to see that there exists α>0 such that
(31)fxξ,xξ≥α>0,ξ∈a¯,b¯.
By the implicit function theorem, there existsδ>0 such that for all η∈a¯,b¯, on the interval η−δ,η+δ, there uniquely determines an implicit function xξ such that
(32)fξ,xξ=0,ξ∈η−δ,η+δ.
This implies thatx=xξ can be obtained uniquely by extension of x=xξ from ξ∈ξ0−δ,ξ0+δ to [a¯,b¯].
Below, we provelimξ→a0+x∗ξ=xa0. Note that fξ,x∗ξ=0,fxξ,x∗ξ≠0. First, we prove the existence of the limit. If not so, there exist ξj1⟶a0+ and ξj2⟶a0+ such that x∗ξj1⟶x1 and x∗ξj2⟶x2 with x1<x2.
Takex1∗,x2∗ such that x1<x1∗<x2∗<x2. Without loss of generality, suppose
(33)x∗ξj1≤x1∗,x∗ξj2≥x2∗.
Sincex∗ξ is continuous, for all x∗ such that x1∗≤x∗≤x2∗, there exists ξj∗ between ξj1 and ξj2 such that x∗ξj∗=x∗. Also note ξj∗⟶a0+. By fξj∗,x∗ξj∗=0, it follows that
(34)fa0,x∗=0,∀x∗∈x1∗,x2∗.
Thus, allx∗∈x1∗,x2∗ are real roots of fa0,x. Since fa0,x=0 has at most 2d+1 real roots, this is a contradiction, which implies the existence of limξ→a0+x∗ξ.
Similarly,limξ→b0−x∗ξ also exists.
Obviously, we havefa0,x∗a0=0. If fxa0,x∗a0≠0, we can extend x∗ξatξ=a0 to a small neighbourhood of a0, which contradicts with the definition of a0.
The case ofξ=b0 can be discussed similarly.Remark 15.
Let∣fxξ0,x0∣≥α and fξ,x is continuous on D=a,b×−1,1 and fxξ,x≤M,∣fxxξ,x∣≤M,∀ξ,x∈a,b×−1,1. By the proof of the implicit theorem, it is easy to see that there exists constant δ>0, which is independent of ξ0,x0 and only depends on α, M, and the property of uniform continuity of f onD, such that xξ, the implicit function across ξ0,x0, is defined for ξ∈ξ0−δ,ξ0+δ⊂a0,b0. In general, the bigger the derivative fxξ0,x0, the longer the interval a0,b0. Moreover, we have that δ⟶0 as fxξ0,x0⟶0.


### 3.4. 3-Degree Polynomials

The proof of our result is strongly based on induction; for this purpose, we first consider 3-degree polynomials. Let(35)fξ,x=x3+c1ξx2+c2ξx+c3ξ, where cjj=1,2,3 are continuous on a,b.Let(36)Am=ξ,xξ,xisanm‐multiplerealrootoff,ξ∈a,b,Bm=ξξ,x∈Am,m=1,2,3.LetA=A1∪A2∪A3, which is the set of all real roots of f.Forξ∈B2, let Pξ1=ξ,xξ1 and Pξ2=ξ,xξ2 be, respectively, the simple real root and the 2-multiple real root of fξ,x. For ξ∈A3, let Pξ=ξ,x be the unique 3-multiple real root of fξ,x.Lemma 16.
AssumeA3=∅ and B2=a,b. Then, there exists a regular path in A joining with Pa1 and Pb1.Proof.
Denote byℓPa1 the maximal path across the simple root Pa1. By Lemma 14, it must end at some multiple real root. Since there is no multiple real root between ξ=a and ξ=b, on the right side, ℓPa1 goes through Pb1 or ends at Pb2=b,xb2.
IfℓPa1 does not reach Pb1, it must go to Pb2. Note that Pb2 is a 2-multiple real root, f has a simple real root ξ0,x0 such that a<ξ0<b and ξ0,x0∉ℓPa1; moreover, ξ0,x0 is sufficiently close to Pb2. Then, the maximal path across ξ0,x0 ends at Pb2 and Pa2. Denote the path by ℓPb2,Pa2.
On the other hand, since the maximal pathℓPb1 across Pb1 cannot go to Pa1, it must end at Pa2 on the left. Thus, ℓPa1∪ℓPb2,Pa2∪ℓPb1 forms a path joining with Pa1 and Pb1.
IfℓPa1 goes across Pb1, then it is just a path joining with Pa1 and Pb1.
Obviously, the path has no basis point and so it is a regular path. Thus, the lemma is proven.Lemma 17.
Assume thatA3=∅anda,b∈B2. Then, there exists a regular path between ξ=a and ξ=b, which joins with Pa1 and Pb1. Moreover, it includes at most finitely many 2-multiple real roots.Proof.
Obviously, ifB2=a,b, it is just the case in Lemma 16. Assume B2\a,b≠∅.
Letd0=infξ∈B2∣xξ2−xξ1∣.
We first claim that there existsξ0∈B2 such that d0=∣xξ02−xξ01∣>0.
In fact, we first haved0>0. If not so, there exist ξn∈B2 such that xξn1−xξn1⟶0. Without loss of generality, assume xξn2⟶x0,xξn1⟶x0,ξn⟶ξ0. Then, ξ0,x0 is a 3-multiple real root of f, which is a contradiction.
Now takeξn∈B2 such that xξn2−xξn1⟶d0. Since B2 is closed, there exists a subsequence ξnk⟶ξ0∈B2. Without loss of generality, assume ξn⟶ξ0. Since ∣xξ02−xξ01∣≥d0, the maximal path across Pξ01=ξ0,xξ01,lPξ01, exists for ∣ξ−ξ0∣≤δ0, where δ0 is a small constant. Let z1ξ,z2ξ,z3ξ be the three continuous bifurcations of roots of f. Then, lPξ01 must be on a continuous bifurcation. Without loss of generality, assume
(37)lPξ01⊂ξ,z1ξξ∈a,b.
Then,z1ξ0=xξ01. It follows that
(38)xξn1=z1ξn⟶z1ξ0=xξ01.
Noting thatz2ξ0=z3ξ0=xξ02 and
(39)xξn2=z2ξn=z3ξn⟶z2ξ0=z3ξ0=xξ02,we have xξn2−xξn1⟶xξ02−xξ01andthenxξ02−xξ01=d0.
Moreover, it follows easily that(40)fxξ,xξ1≥d02,∀ξ∈B2.
By Lemma14 and Remark 15, there exists δ>0, which is independent of ξ∈B2, such that for any η∈B2, a maximal path ℓη,xη1 is defined well for ∣ξ−η∣≤δ. More precisely, η−δ,η+δ is included in the definition interval of the maximal path.
Leta=ξ0<ξ1<⋯ξn=b such that ξj−ξj−1≤δ,j=1,2⋯n.
Ifξj−1,ξj∩B2≠∅, let
(41)aj=minξξ∈B2∩ξj−1,ξj,bj=maxξ∈B2∩ξj−1,ξj.
Obviously,ξj−1≤aj≤bj≤ξj and bj−aj≤δ. By Lemma 14 and using the same notation as in Lemma 16, the maximal path ℓPaj1 joins with Paj1 and Pbj1. Denote by ℓ¯j=ℓPaj1,Pbj1 the segment of ℓPaj1 joining with Paj1 and Pbj1. If aj=bj, we let lPaj1=Paj1. Obviously, there is no 2-multiple real root on the path ℓPaj1.
Ifξj−1,ξj∩B2=∅, we have no such path.
All these paths can be arranged aslpajk1k=1m with m≤n and
(42)a=aj1≤bj1≤⋯ajk≤bjk≤ajk+1≤bjk+1≤ajm≤bjm=b.
In particular, ifξ∈bjk,ajk+1,fξ,x has no 2-multiple real root. By Lemma 16, there exists a regular path
(43)ℓ~jk=ℓPbjk1,Pajk+11,such that ℓ~jk joins with Pbjk1 and Pajk+11.
Ifajk+1−bjk<δ, then ℓ~jk is a segment of the maximal path of the simple real root Pbjk, on which there is no 2-multiple real root. Only if ajk+1−bjk<δ can the 2-multiple real roots Pbjk2 and Pajk+12 be on ℓ~jk,k=1,2,⋯m−1.
It is easy to see that all the paths(44)ℓ¯j1,ℓ~j1,⋯ℓ¯jk,ℓ~jk⋯ℓ¯jm,ℓ~jm,can be continuously connected to form a regular path joining with Pa1 and Pb1. Moreover, it includes at most these 2-multiple real roots Pbjk2,Pajk+12k=1m.
By rescaling the parameter of the paths, we can define the path as a continuous mapping from0,1 to A, but we omit the details and refer to the proof of Lemma 26 below, where a more general case is considered.Lemma 18.
AssumeB3=a,b. Then, there exists a regular path ℓ in A joining with Pa and Pb; moreover, ℓ includes at most countable 2-multiple real roots.Proof.
Case 1.
Assume thatB2=∅. Let ξ0∈a,b and ξ0,x0∈A. Since fxξ0,x0≠0, by Lemma 14, the maximal path across ξ0,x0 must join with Pa and Pb.Case 2.
AssumeB2≠∅. Let
(45)a¯=infξ∈B2,b¯=supξ∈B2.
Ifa¯>a and b¯<b, then a¯,b¯∈B2. By Lemma 17, there exists a path joining with Pa¯1 and Pb¯1. Moreover, the maximal path across Pa¯1 reaches to ξ=a on the left and the maximal path across Pb¯1 reaches to ξ=b on the right. It is easy to see that three paths on a,a¯, a¯,b¯, and b¯,b form a regular path between ξ=a and ξ=b. Now we consider the case that a¯=a or b¯=b.
Leta¯=a. Take ξ0∈B2 and a sequence aj⊂B2 such that
(46)ξ0=a0>a1>a2>⋯>aj>aj+1⟶a.
Moreover,Paj1⟶Pa,Paj2⟶Pa. By Lemma 17, on aj−1,aj, we have a path ℓj joining with Paj−11 and Paj1. Then, all the paths ℓjj≥0 can define a path joining with Pa01 and Pa; for more details, we also refer to the proof of Lemma 26.
Ifb¯=b, in the same way, we can define a path connected with Pa01 and Pb. These two paths can be connected to form a regular path joining with Pa and Pb.Below, we consider the set of 3-multiple roots.Lemma 19.
For allξ,x∈A3, we have x=xξ=−c1ξ/3,ξ∈a,b. That means, all 3-multiple real roots of f are on the continuous curve
(47)l∗:ξ,xx=−c1ξ3,a≤ξ≤b.Proof.
Note that forξ,x∈A3, we have fxxξ,x=6x+2c1ξ=0; the result holds obviously.Now, we prove the following important lemma.Lemma 20.
There exists a regular pathℓ⊂A with two odd-multiple endpoints on Aa and Ab, respectively.Proof.
We first claim thatB3 is a closed set.
Letξn⟶ξ0,ξn∈B3. Then, there exists a subsequence xξnk of xξn, such that xξnk⟶x0. Since fξ,x,fxξ,x,fxxξ,x are continuous, by ξnk⟶ξ0 we have
(48)fξ0,x0=fxξ0,x0=fxxξ0,x0=0.
Thus,ξ0∈B3 and so B3 is closed.
Leta¯=minξ∣ξ∈B3,b¯=maxξ∣ξ∈B3. Then, we have a¯,b¯\B3=∪j≥1aj,bj where aj,bj∣j=1,2,⋯ are disjointed open intervals of each other.
By Lemma18, there exists a regular path ℓj between ξ=aj and ξ=bj such that Paj,Pbj∈ℓj.
Moreover, ifa<a¯, in the same way as the proof of Lemma 18, where we take ξ0=a and b=a¯, then there exists a regular path ℓl, which joins with Pa¯ and Pa∗, where Pa∗ is a simple root of fa,x. In the same way, if b¯<b, we have a regular path ℓr, which joins with Pb¯ and Pb∗, where Pb∗ is a simple root of fb,x. Now, we construct a path ℓ as follows:
(49)l=lj,ξ∈aj,bj,j≥1,l=l∗,ξ∈B3,l=ll,ξ∈a,a¯,l=lr,ξ∈b¯,b,where l∗ξ=ξ,−1/3c1ξ. Thus, ℓ is a path joining with Pa∗ and Pb∗.
Note that ifa=a¯ (or b=b¯), we have no path ℓl (or ℓr).
Below, we explain the continuity ofℓ. Let P0=ξ0,x0∈A3. By uniform continuity of continuous bifurcations of roots, it is easy to see that for any ϵ>0, there exists δ>0 such that if ∣ξ−ξ0∣≤δ and ξ,x is a real root of f, then x−x0≤ϵ. By this fact, the continuity of ℓ can be proven in the same way as in the extension of a continuous function on a closed set on ℝ to ℝ; here, we omit the details.
Moreover,ℓjj≥1,ℓl,ℓr include at most countable 2-multiple real roots; thus, the path ℓ includes at most countable 2-multiple real roots.
Furthermore, by the above construction, it is easy to see that the pathℓ has no basis point. Then, ℓ is a regular path in A, which connects from Aa to Ab.


### 3.5.2d+1-Degree Polynomials

In this subsection, we will prove our main results in the general case. To make an induction, we first give some definitions.Definition 21 (room of real roots and passage of odd-multiple real roots).
(1)
LetD=a,b×c,d be a rectangle in the ξ,x-plane and D°=a,b×c,d be the interior of D. Let I0=a0,b0⊂I=a,b and DI0=I0×c,d, where DI can be regarded as restriction of D for ξ to I0. Denote the section of D at ξ by Dξ=ξ×c,d⊂ℝ2. In particular, Da and Db are called the left side and the right side of D, respectively(2)
LetI=a,b and D be a rectangle as given above. A group f,m,D,I is called a room of real roots of f; it means that fξ,x is a real polynomial of degree m with respect to x and depends continuously in the parameter ξ on I; moreover, all real roots of fξ,x fall into D for all ξ∈I. If f has an m-multiple real root on D, f,m,D,I is called a minimal room of real roots of f(3)
If the left (right) side ofD contains an odd-multiple real root of f, the side is called a door of odd-multiple real roots. If a room of real roots of f has a door of odd-multiple real roots, it is called a passage of odd-multiple real roots. Otherwise, if the left (right) side of D contains no odd-multiple real root of f, the side is called a separating wall of odd-multiple real roots. If both the left side and the right side are separating walls of odd-multiple real roots, we call f,m,D,I an insulator of odd-multiple real roots. Let f,2d+1,D,I be a room of real roots. If fa,x and fb,x have 2d+1-multiple real roots, f,2d+1,D,I is called a one-way-passage of odd-multiple real roots(4)
IfD∘ contains no any real root of f, f,m,D,I is called a separating block of real roots(5)
Iff,m,D,I is a passage of odd-multiple real roots and f has an m-multiple real root in D, we call f,m,D,I a minimal passage of odd-multiple real roots. A minimal passage of odd-multiple real roots cannot be decomposed into smaller passagesA room of real roots is either a passage of odd-multiple real roots, or an insulator of odd-multiple real roots, or a separating block of real roots. Also, note that once a path enters into a one-way passage of odd-multiple real roots from one side, it can only go out of the one-way passage from the other side.Note that an odd-multiple real root is never isolated and an even-multiple real root can be isolated. Thus, there are a lot of odd-multiple real roots near to a door of odd-multiple real roots, so a passage is always neighbouring with some rooms of real roots. A separating block of real roots has no use in our problem, so it is usually ignored.Lemma 22 (decomposition of minimal rooms of real roots).
Letf,m,D,I be a room of real roots of f. Assume f has no any m-multiple root for all ξ∈I. Then, there exists δ>0, such that for any η∈I, if fη,x has a real root, there exist real polynomials fjξ,xj=1,2, with their degrees mj<m, which continuously depend on the parameter
(50)ξ∈I0=η−2δ,η+2δ∩I,and disjointed rectangles Dj⊂D,j=1,2, such that
(51)fξ,x=f1ξ,xf2ξ,x,ξ∈I0,and all real roots of fj fall into Dj,j=1,2. Moreover, f and fj have the same real roots on Dj,j=1,2.Remark 23.
In brief, the above statement can be written as(52)f,m,DI0,I0=f1,m1D1,I0×f2,m2,D2,I0.
Letf,m,D,I be a room of real roots. If fj,mj,Dj,I0 is a room of real roots, we call it a subroom of real roots of f,m,D,I. Similarly, we can define subpassage.
Also note thatfj,mj,Dj,I0 may not be a passage of odd-multiple real roots. In fact, fj,mj,Dj,I0 may be a separating block of real roots, which is usually ignored in our problem.Proof.
We first consider the case thatm=2d+1. Let xjξ,1≤j≤m be m continuous bifurcations of roots of f for ξ on I. Since f has no m-multiple real root, it follows that there exists α>0 such that
(53)maxi≠jxjξ−xiξ≥2d+1α,∀ξ∈I.
Letxj1η be a real root of fη,x. Then, there exist δ>0,c≥0,1≤r<m and two groups xj1,⋯,xjr and xjr+1,⋯,xjm such that
(54)xjkξ−xj1η<c,∀ξ∈I0,∀1≤k≤r<m,xjkξ−xj1η>c+α,∀ξ∈I0,∀r+1≤k≤m,where I0=η−2δ,η+2δ∩I. Since all xjξ are uniformly continuous in ξ on I, so δ can be independent of η∈I.
Note that forr+1≤k≤m,if∣Rexjkξ−xj1η∣<c+α, then xjkξ is a complex root.
Letx0=xj1η and M be a constant such that xjξ≤M,∀ξ∈I,j=1,2,⋯m.
Now we dividexjr+1,⋯xjm into two groups:
(55)xjkRexjk≤x0,r+1≤k≤m=x~j1≤j≤r~≤m−r,xjkRexjk>x0,r+1≤k≤m=x~jr~+1≤j≤m−r.
It is easy to see that for1≤j≤r~, if x~j is a real root, then x~j≤x0−c−α, and for r~+1≤j≤m−r, if x~j is a real root, then x~j>x0+c+α.
Let(56)g1ξ,x=x−xj1ξ⋯x−xjrξ,g2ξ,x=x−x~1ξ⋯x−x~r~ξ,g3ξ,x=x−x~r+1ξ⋯x−x~m−rξ.
Sincexj1η is a real number, it follows that xjkξ, x~jξ, and their conjugate complex roots belong to the same group; thus, all gjj=1,2,3 are real polynomials.
Ifr~≥1, let
(57)f1=g1g3,f2=g2,D1=I0×−M,x0+c,D2=I0×x0+c+α,M.
Ifr~=0, let
(58)f1=g1,f2=g3,D1=I0×x0−c,x0+c,D2=I0×−M,x0−c−α.The case thatm=2d can be considered in the same way by taking x0=xj1η to be a real root of fη,x, which is available by assumption.Iff has an m-multiple real root, the room of real roots f,m,D,I cannot be decomposed into smaller ones.The fact thatδ is independent of η guarantees a finite partition of a,b such that, on each small interval, we have the above decomposition. If some smaller room is not minimal, we can use the above lemma again until all small rooms are minimal. This result concerns with the following lemma.Lemma 24.
Considerf0,2d+1,D0,I0 and let f,m,D,I with I=a,b be a subroom of f0,2d+1,D0,I0 with m≤2d+1,D⊂D0,I⊂I0. Assume that f and f0 have same real roots on D; moreover, f has no any m-multiple root for all ξ∈a,b. Then, there exists finitely many minimal rooms of real roots of f,fj,mj,Dj,Ijj=1N with Ij=aj,bj, such that the following hold true:
(1)
f and fj have same real roots on Dj(2)
mj<m and fj has at least one mj-multiple real root on Dj(3)
Ifaj≠a and the left side Djaj contains a real root P of fj, then there is a room of real roots fi,mi,Di,Ii, which is neighboured with fj,mj,Dj,Ij such that P∈D°i and ai<aj<bi. For the right side, if bj≠b and Djbj contains a real root of fj, we have a similar result. Let δij=minai−bj,bi−bj, then we have δij>0. Since i,j≤N, the smallest distance of such δij is also positive(4)
Everyfj,mj,Dj,Ij is a minimal passage of odd-multiple real roots or an insulator of odd-multiple real roots; moreover, any real root of f on D is a real root of some fj on DjThe conclusion (3) is very important in extending of paths. By induction, a path can be always extended from an initial point in a passage of real rootsDj to arrive at an odd-multiple real root on a door of Dj. The new odd-multiple endpoint on the door of Dj is already in the interior of some Di. Then, we can do extension again in Di to arrive at another odd-multiple endpoint on a door of Di; in this process, the path must cross at least δij length of ξ. Since a path can only cross at most 2d+1b−a length about the parameter, the extension can repeat at most finitely many times before it cannot be extended again, which can only happen when the path has become a closed regular path or it arrives the boundary of the domain.Among the sequence of minimal roomsfj,mjDj,Ijj=1N in this lemma, there may be some minimal room which is completely included in a bigger one. In this case, we can drop away some smaller minimal rooms and the lemma still holds for the rest. So, without loss of generality, we can suppose Di\Dj≠∅ for all i≠j.Proof of Lemma 24.
By assumption,f has no any m-multiple real root. By Lemma 22, there exists δ>0 such that for η∈a,b, if fη,x has a real root, then there exist two real polynomials f1ξ,x and f2ξ,x with their degrees m1,m2<m, which continuously depend on the parameter ξ∈η−2δ,η+2δ∩a,b, and two disjointing rectangles D1 and D2 for ξ on η−2δ,η+2δ∩a,b, such that fξ,x=f1ξ,xf2ξ,x and all real roots of fi fall into Di, i=1,2.
Now, we choose a partition ofa,b:
(59)a=ξ0<ξ1<ξ2<⋯<ξn=b,such that ξj+1−ξj<δ. Then, let
(60)Ij1=ξj,ξj+2j=0,1,⋯n−2.
Iffη,x has a real root for some η∈ξj,ξj+2, Lemma 22 implies that on the small interval Ij1, we have the decomposition fj11,mj11,Dj11,Ij1 and fj21,mj21,Dj21,Ij1.
Iffη,x has no any real root for all η∈ξj,ξj+2, we drop away the room of real roots f,m,DIj1,Ij1. In this case, if there exists a real root P on Dξj or Dξj+2, P is an even-multiple real root and belongs to f,m,DIj−11,Ij−11 or f,m,DIj+11,Ij+11, so we do not lose any real root after dropping away f,mDIj1,Ij1.
Denote all these smaller rooms of real roots byfj1,mj1,Dj1,Ij1 for simplicity of notations. Note that some two intervals in Ij1 may be identical, but Dj1 are all different from each other. Let Ij1=aj1,bj1.
If somefj1 has no real root in the interior of Dj1, that is, (fj1,mj1,Dj1,Ij1) is a separating block of real roots, we drop it away. So without loss of generality, assume fj1 has at least one real root in the interior of Dj. Thus, fj,mj,Dj,Ij is a passage of odd-multiple real roots or an insulator of odd-multiple real roots. After dropping some separating blocks, for simplicity, we denote the subsequence by itself
(61)fj1,mj1,Dj1,Ij1j=1N1.
It is very important that for any side ofDj1, say Dj1aj1, if aj1≠a and P∈Dj1aj1 is a real root, then P must belong to the interior of some Di1i≠j. If bj1≠b and P∈Dj1bj1 is a real root, we have a similar result.
Now, if (f11,m11,D11,I11) is not minimal, in the same way as above, we decompose it into a sequence of smaller rooms of real roots
(62)f1k2,m1k2,D1k2,I1k2k≥1.
But we should note that when we make a partition ofI11=a11,b11, we require the points of the partition not to fall on M1, where
(63)M1=ξD11∩Dj1ξ≠ϕ,ξ∈aj1,bj1,1≤j≤N1.
SinceM1 is finite set, it is easy to have such partition of [a11,b11]. Such partition is very important, and it makes the sides of a neighbouring room of real roots to disjoint each other. Thus, any real root on a side must fall into the interior of a neighbouring room unless it falls into Da∪Db.
If (f11,m11,D11,I11) is minimal, we cannot decompose it and denote it by
(64)f112,m112,D112,I112=f11,m11,D11,I11.
Suppose we have decomposed(65)fjk1,mjk1,Djk1,Ijk1,j=1,2,⋯i−1,and obtained
(66)fjk2,mjk2,Djk2,Ijk2,j=1,2,⋯i−1.
Now, we consider (fi1,mi1,Di1,Ii1). In the same way, we make a partition of ai,bi as above; moreover, we require the points of the partition not to fall on Mi, where Mi=Mi1∪Mi2, where
(67)Mi1=ξDi1∩Dj1ξ≠ϕ,ξ∈aj1,bj1,j>i,Mi2=ξDi1∩Dj2ξ≠ϕ,ξ∈aj2,bj2,j=i,⋯i−1,k≥1.
If (fi1,mi1,Di1,Ii1) is minimal, we cannot decompose it and denote it by
(68)fi12,mi12,Di12,Ii12=fi1,mi1,Di1,Ii1.
In this way, we decompose all rooms (fj1,mj1,Dj1,Ij1) which are not minimal.
Then, we move away all separating blocks fromfjk2,mjk2,Djk2,Ijk2, and the rest is denoted by
(69)fj2,mj2,Dj2,Ij2j=1N2.
This step can iterate at mostm steps, and finally, we have a sequence of minimal rooms of real roots
(70)fjm,mjm,Djm,Ijmj=1Nm.
For simplicity, denote it by(71)fj,mj,Dj,Ijj=1N.According to the above decomposition, it is easy to see that if one side offj,mj,Dj,Ij does not fall on ξ=a or ξ=b, it may be divided into some parts such that each part is between the two sides of a neighbouring room fi,mi,Di,Ii, the overlapping region Dj∩Di can be regarded as a door way joining with fj,mj,Dj,Ij and fi,mi,Di,Ii. Any real root on the side of Dj is in the interior of some Di with i≠j.This is very important for our extension of paths. By induction assumption, a path of odd-multiple real roots can be extended inDj to a side of Dj. Since the new odd-multiple endpoint on the side of Dj falls into the interior of some Di, we can extend the path in Di again from the new endpoint to a side of Di. In this way, we can extend a path from the initial endpoint in some room to a neighbouring room and then go to the next neighbouring room. This process can go on unless it has become a closed regular path or arrived on ξ=a or ξ=b.We should note that once a path enters into an insulator of odd-multiple real roots, the whole path belongs to the insulator. Moreover, any extension of the path can only stay in the insulator forever and it can only be extended to be a closed path finally. So, only in some passages of odd-multiple real roots a path can be extended to be a desired path.A one-way passage of odd-multiple real roots can only allow one regular path to go through it. Once a segment of a path goes fromDa to Db, since there is a unique real root on Da and Db, respectively, according to the compatibility of paths, any segment of the path cannot enter into it again.We also note that a bigger room may be neighbouring with several smaller rooms. This case happens near to some multiple real roots, where some paths of lower-odd-multiple real roots bifurcate.The proof of Theorem2 is based on a special case that f has at least one 2d+1-multiple real root, which is considered in the following theorem:Theorem 25.
Consider a room of real rootsf,2d+1,D,I with I=a,b. If f has a 2d+1-multiple real root, then there exists a regular path of odd-multiple real roots in D, which joins with two odd-multiple endpoints on Da and Db, respectively.
The proof of Theorem25 is based on induction for degree of polynomials. So we first assume Theorem 25 already holds for polynomials of degree≤2d−1. Below, we prove that Theorem 25 also holds for polynomials of degree=2d+1. The proof is based on the following lemma.
Before giving the lemma, we first show some notations. Letm~≤2d+1 and f,m~,D,I be a room of real roots of f with I=a,b. Let Am~ be the set which consists of all m~-multiple real roots of f for all ξ∈I and Bm~=ξ∈I∣ξ,x∈Am~. For ξ∈Bm~, denote by Pξ the m~-multiple real root of fξ,x.Lemma 26.
Letf,m~,D,I be a subroom of real roots of f0,2d+1,D0,I0. Let γ0 be a regular path in f0,2d+1,D0,I0 with an odd-multiple endpoint P0∈D0\D and a simple odd-multiple endpoint P1∈D0. Let γ~j,⋯γ~k be paths in f0,2d+1,D0,I0, where k≤d.LetEj=γ~j0,γ~j1 be the endpoints of γ~j. Suppose that γ~j is a closed regular path or Ej⊂D0\D. Let γ~=γ~1∪⋯∪γ~ and Γ=γ0∪γ~. Suppose that {γ0,γ~1,⋯γ~k} are compatible on D.
(1)
Letm~=2d¯ and B2d¯=∅. Suppose that γ0 enters into D with P1∈D∘. Then γ0 can be extended in D to be a compatible path such that it has a simple odd-multiple endpoint P1+ on Da∪Db with outward direction. Here “outward direction” means, if P1+∈DaorDb, it is a left (or right) endpoint
Note that below the new endpoint in extension of paths always has outward direction in the above sense if without explanation.(2)
Letm~=2d¯ and P1∈D∘. Then, the following hold true:
(i)
LetB2d¯=b. Suppose that γ0 enters into D and arrives at P1=c,xc with a<c<b. Then, γ0 can be extended in D to be a compatible path such that it has a simple odd-multiple left endpoint on Da, or it reaches to Pb. Moreover, in the last situation, we can continue to extend γ0 with Pb as a reflecting point such that it has an odd-multiple left endpoint on Da; moreover, it is compatible with γ~(ii)
LetB2d¯≠∅. Then, γ0 can be extended to be a compatible path such that it has a simple odd-multiple endpoint on Da or Db with outward direction(3)
Letm~=2d¯+1 and B2d¯+1=∅. Suppose that γ0 enters into D with P1∈D∘. Then, γ0 can be extended to be a compatible path such that it has a simple odd-multiple endpoint on Da or Db with outward direction(4)
Letm~=2d¯+1 and B2d¯+1=b. Then, the following hold true:
(i)
Suppose thatγ0 enters into D with P1=Pc=c,xc∈D with a<c<b. Then, it can be extended to be a compatible path such that it has an odd-multiple left endpoint on Da, or it arrives at Pb. Moreover, in the last situation, if Pb is already a crossing point, γ0 can be extended to be a compatible path such that it has an odd-multiple left endpoint P1+ on Da(ii)
Cancel the existing pathsγ0 and γ~j in the assumption and consider f,m~,D,I to be an empty room of real roots. Then, there exists a regular path γ⊂D which joins with odd-multiple endpoints P0∈Da and P1=Pb with outward direction(5)
Letm~=2d¯+1 and B2d¯+1=a,b. Then, the following hold true:
(i)
Suppose that the pathγ0 reaches to D with P1∈D. Then, the path γ0 can be extended to be a compatible path such that it has a simple endpoint belonging to Pa,Pb with outward direction(ii)
In the same way as (4(ii)), there exists a regular pathγ⊂D, which joins with endpoints Pa and Pb with outward direction.(6)
Letm~=2d¯+1 and B2d¯+1≠∅. Then, the following hold true:
(i)
Letd¯≤d−1. Suppose that γ0 enters into D with P1∈D∘. Then, γ0 can be extended to be a compatible path such that it has a simple odd-multiple endpoint on Da∪Db with outward direction(ii)
Letd¯=d. Then, Theorem 25 holds trueRemark 27.
(i)
IfB2d¯=a, we have similar results to (2(i)) and (2(ii)). Moreover, if B2d¯+1=a, we have similar results to (4(i)) and (4(ii)). Here, we omit the details(ii)
We require endpoints ofγ~j to be outside of D to avoid γ0 being in a dilemma after it enters into D. Thus, when γ0 goes into D, it can always have some way to arrive at Da or DbProof.
We prove Lemma26 by induction. By the proof of Lemma 20, it is easy to see that Lemma 26 holds for d¯=1. Inductively suppose that Lemma 26 holds for all d¯≤m−1. Below, we prove it for d¯=m. Recall that we already make an induction assumption about d for Theorem 25. Here, we actually make an induction about both d and d¯ in Theorem 25 and Lemma 26.Proof of (1).
By assumption, we haveB2d¯=∅. Then, by Lemma 24, it follows that the room (f,2d¯,D,a,b) (d¯=m) can decompose into smaller minimal ones fj,mj,Dj,Ij with mj<2m.
Suppose that the pathγ0 has been extended to some room fj,mj,Dj,Ij such that P1∈D°j. By assumption it is easy to see that E=∪j=1kEj, the set of all the endpoints of {γ~j}, is outside of Dj. In addition, whatever mj=2m¯ or mj=2m¯+1, we have m¯≤m−1. By induction assumption of (2(ii)) and (6), we can extend γ0 in Dj to a side of Dj with a new odd-multiple left (or right) endpoint on Djaj (or Djbj).
Moreover, we can require the new odd-multiple endpoint to be simple and have outward direction about domainDj. In fact, we can always extend the path to reach to Djaj (or Djbj) such that the endpoint is neither a crossing point nor a reflecting point. If not so, the endpoint meets a crossing point or a reflecting point on Djaj (or Djbj), we can always continue the extension by reflecting at the crossing point or reflecting point such that the path enters into Dj again and then repeat the above process.
If the endpoint is already onDa∪Db, the result is proved. Otherwise, the odd-multiple endpoint is on a door of the passage fj,mj,Dj,Ij, which must be neighboured with another room of real roots, say, fi,mi,Di,Ii. Then, the path enters into Di and the odd-multiple endpoint on the door of fj,mj,Dj,Ij belongs to the interior of Di; thus, we can do extension in fi,mi,Di,Ii in the same way as in fj,mj,Dj,Ij.
If the extension does not go toDa∪Db, the process can repeat in the next neighbouring room. Each extension must go across the parameter of a positive length δ>0. It is easy to see that the total length of parameter of γ is no more than 2d¯b−a; thus, after at most finitely many extensions, the path must reach to Da∪Db with an odd-multiple endpoint with outward direction.
Thus, we haven segments of paths γ1,⋯,γn such that
(72)P0=γ10,P1=γ11,⋯,γn−11=Pn−1=γn0,Pn=γn1,where P0 is the initial endpoint of the path.
Note that eachγj is obtained when we do extension in some passage. Moreover, each γj is compatible with the existing paths since the extension is done according to the compatibility of paths by induction assumption. We should also note that once a path enters into an insulator, the path must totally fall into this insulator. However, since P0∉D, this situation cannot happen in our extension. Anyway, by induction assumption, the extension is well defined.
LetIj=i/n+1,j+1/n+1,j=0,1,⋯n and 0,1=∪j=0nIj. By rescaling the parameter t of paths, we suppose that the original path γ0t is defined for t on I0, and γjt is defined for t on Ij,j≥1. Then, we obtain an extension γ=∪j=0nγj with
(73)γt=γjt,t∈Ij,j=0,1⋯n.
By the above discussion, every extension follows the compatibility of paths, so all the paths are compatible. Thus, conclusion (1) is proven.Proof of (2).
Now, we turn to proving conclusion (2).Case of (2(i)).
(A) We first prove that onceγ0 enters into D, it can always be extended to arrive at Da or Pb; moreover, it is compatible with γ~=γ~∣j=1,2⋯k.
Take a partition(74)a=a0<c<a1<⋯<an<⋯bsuchthatan⟶b.
Sincef has no 2d¯-multiple real root for ξ∈a0,a1, by induction assumption and using the result (1) to the room of real roots f,2d¯,Da0,a1a0,a1, we can extend γ0 such that it goes to Da∪Da1. If it arrives at Da, the result is proved; otherwise, it goes to Da1. Denote the segment of extension by ℓ1. Note that here we can require the endpoint to be a simple one as discussed above; moreover, ℓ1 is compatible with Γ=γ~∪γ0.
Letγ1=γ0∪ℓ1, then γ1 is compatible with γ~. Now, we extend the path γ1 in Da0,a2 with a simple odd-multiple endpoint on Da1 as a new initial endpoint. In the same way as above, if it does not arrive at Da, it goes to Da2. Denote the segment of the extension by ℓ2. Moreover, ℓ2 is compatible with Γ1=γ~∪γ1. Let γ2=γ1∪ℓ2, then γ2 is compatible with γ~.
Then, we repeat the above process of extension. If the path is already extended toξ=an, we consider a0,an+1. In the same way, if it does not arrive at Da, it must arrive at ξ=an+1.
In this process, if the path does not arrive atDa, it must arrive at all ξ=an and have a sequence of segments of paths ln, which is obtained when extending on a0,an and given by a continuous mapping ℓn:t∈0,1⟶ℓnt with Pn=ℓn0∈Dan and Pn+1=ℓn1∈Dan+1, where Pn are all simple odd-multiple right endpoints of ℓn−1. Let
(75)0,1=∪n=1∞1−1n,1−1n+1∪1.
By rescaling the curve parametert, we can let γ0t,t∈0,1/2 and ℓn:t∈1−1/n+1,1−1/n+2⟶ℓnt with Pn=ℓn1−1/n+1andPn+1=ℓn1−1/n+2.
Define(76)γ^:γ^t=ℓnt,t∈1−1n+1,1−1n+2,∀n≥1,γ^1=Pb.
Then,γ^t,t∈1/2,1 is a continuous mapping with γ^0=P1 and γ^1=Pb. So γ=γ0∪γ^ is a continuous curve for t∈0,1 joining with P0 and Pb.
We should note that in the above extension,Pb may be already a reflecting point of γ~j or γ0. In this case, we can regard γ~j as two segments of paths with Pb as their endpoints. Obviously, all their endpoints are outside of Da0,an for all n≥1. Moreover, Γ is still compatible when γ~j is divided into two pieces of paths. Then, the assumption of (1) holds, so the above extension is reasonable.
Below, we show that the extensionγ is well defined and satisfies the definition of paths; thus, γ is a desired path.
By Lemma11, since the points on a path can recur to some parameter at most 2d¯ times, so ℓn can intersect with ℓ1,⋯,ℓn−1 at most 2d¯ times. So for any ℓm, there exists N>m such that for all n>N,ℓn∩ℓm=∅.
Letγn=γ0∪∪k=1n−1ℓk. By induction, all ℓn and γn follow the compatibility of paths.
It follows that allγn are compatible with γ~.
Letγ=γ0∪∪n=1∞ℓn∪Pb. Thus, for any P∈γ, if P≠Pb, there exists sufficiently large N>0 such that P∉∪n≥Nℓn and P∈γN=γ0∪∪n≤N−1ℓn. By induction γN is a well-defined compatible path, so P satisfies the definition of path as a point on γN. Since P∉∪n>Nℓn, then P also satisfies the definition of paths as a point on γ.
Below, we prove thatγt is continuous at t=1. Note that γ1=Pb. Let γt=ξt,xt. For any ϵ>0, there exists N>0 such that for all n>N, ℓn do not go across ξ=b−ϵ. Note that ℓn=ξnt,xnt,t∈1−1/n+1,1−1/n+2. Then, for all n>N,ξnt∈b−ϵ,b. Let δ=1/N+1. Then, if 0<1−t≤δ, there exists n>N such that t∈1−1/n+1,1−1/n+2,soξt=ξntandb−ξt=b−ξnt≤ϵ. Thus, ξt is continuous at t=1. Moreover, since Pb is a 2d¯-multiple real root, by the 2d¯ continuous bifurcations of roots of the polynomial f, one of which the real root ξt,xt must belong to; it follows easily that if t<1 and t⟶1, we have ξt⟶b and so xt⟶xb. Thus, γt⟶Pb,t⟶1, which implies that γt is continuous at t=1. Thus, γ is a compatible path with γ~, and Pb is a left even-multiple endpoint.
(B) Ifγ0 is already extended to arrive at Pb, we consider to extend it back from Pb to Da. We prove that there exists a path γ^, which joins also from an odd-multiple real root P1+∈Da to Pb, such that γ=γ0∪γ^ is a compatible path with γ~, which joins from P0 to Pb along γ0 and then goes back from Pb to P1+ along γ^.
Without loss of generality, suppose thatγ0 has arrived at Pb. Note that γ0 may meet Pb several times, that is, Pb is already a reflecting point of γ0. We first note that Lemma 13 implies that Γ=γ0∪γ~ has occupied odd number of crossing points on Da.
Sincef has even number of odd-multiple real roots on Da, we can take an odd multiple real root a,x1 on Da. By induction assumption of Theorem 25, there exists a small rectangle with a,x1 as the center and a small segment of path in the small rectangle, which goes across a,x1 from a left endpoint P0 on the left side to a right endpoint P1 on the right side. Then, P0∉D. Now, we extend the path from P1 to the right in the same way as in (A) of case (2(i)). Denote the path by γ1.
Ifγ1 is extended to Pb, let γ^=γ1 and then γ0∪γ^0 is a desired path. Otherwise, γ1 goes back to ξ=a without touching with Pb. We can require γ1 to be extended finally to an odd-multiple real root (a,x1′) with x1≠x1′. In fact, if the extension goes back to a,x1, it can also be extended to the right by reflecting at a,x1 and then repeat the above process. So, if it goes back to Da, we can always suppose it touches a different odd-multiple real root (a,x1′)on Da; moreover, we require it to cross Da a little with a simple endpoint of γ1 on the left of Da.
Denote the path joining witha,x1 and (a,x1′) still by γ1. Then, it has two endpoints of on the left of Da and occupies two odd-multiple real roots on Da. Note that Γ occupies odd number of odd-multiple real roots. Then Γ and γ1 have occupied odd number of odd-multiple real roots on Da.
In the same way as above, take another odd-multiple real roota,x2 on Da. By induction assumption of Theorem 25, there exists a small rectangle with a,x2 as the center and a small segment of path in the small rectangle, which goes across a,x2 from a left endpoint P0′ on the left side to a right endpoint P1′ on the right side. Then, P0′∉D. Now, we extend the path from P1′ to the right in the same way as in (A) of case (2(i)) to obtain a compatible path γ2.
We should note thatγ1 plays a role to restrict number of odd-multiple real roots on Da which can be taken as a crossing point of γ2. More precisely, since γ1 occupied at least two odd-multiple real roots on Da, a crossing point of γ2 on Da can be chosen in a smaller range.
Below, we analyze several cases of extension ofγ2.
(i)
γ2 does not recur to Da.
Ifγ2 does not go back to Da, it goes through all ξ=an, and then γ2 is extended from a,x2 to Pb in the same way as (A) of case (2(i)).
We should note thatγ2 is allowed to meet with γ0∪γ1 and γ~ according to the compatibility of paths.
Ifγ2 does not attach the path γ1, then γ2 is a desired path by taking off γ1.
Assume thatγ2 attaches γ1. Since γ1 is on the left of ξ=an1 with n1 being sufficiently big, so γ2 has no touching point with γ1 on the right of ξ=an1. Take the last crossing point of γ2 at ξ=an1 and denote it by an1,xn1.
Then, we drop awayγ1 and the segment of γ2 which joins from a,x2 to an1,xn1. The purpose of dropping γ1 away is to avoid influence of γ1 on γ2 such that γ2 can only have some touch with γ0 and γ~; thus, the next extension of γ2 can be a compatible path.
Then, we extend the segment ofγ2 between Pb and an1,xn1 to the left. Still denote by γ2 the segment of γ2 without confusion. If γ2 is extended to Da, then γ2 is a desired path. Otherwise, the extension goes through all ξ=an to arrive at Pb, then we have a closed path which reflects at Pb and denote it by γ¯1. Moreover, γ¯1 is compatible with γ0 and γ~.
(ii)
γ2 recurs to Da.
Ifγ2 goes back to ξ=a, we assume that it does not touch γ1 at ξ=a (if so, we can still extend it by reflecting at the touching point and finally γ2 can reach to an odd-multiple real root on Da, which does not belong to γ1). Moreover, in the same way as γ1, we require γ2 to cross Da a little with a simple endpoint on the left of Da so that its two endpoints are on the left of Da. Then, γ1∪γ2 occupy at least 4 odd-multiple real roots on Da and all their endpoints are on the left of Da. Moreover, γ0,γ1,γ2,γ~ are compatible.
At this stage, we can makeγ3. Using γ1∪γ2 in place of γ1 and γ3 in place of the above γ2, we can have a similar discussion as above.
Finally, if we have not obtained a desired path, either we obtain another closed pathγ¯2 withPb being a reflecting point, or γ1∪γ2∪γ3 occupies at least 6 odd-multiple real roots on Da; moreover, γ0,γ1,γ2,γ3,γ~ are compatible.
We1 should note thatγ¯2 is obtained by extension of a segment of the path after dropping γ1∪γ2. So γ0,γ¯1 and γ¯2 also follow the compatibility of paths with γ~, and so γ0,γ1,γ2,γ~ are compatible.
If we do not obtain a closed pathγ¯2, we can consider to find γ4 in the same way as above. Thus, if we do not find a desired path, we either find a closed path γ¯2 as above or a path γ4 as γj,1≤j≤3.
During this process, if we have not obtained a desired path, one of two situations happen: we find one more closed path withPb being a reflecting point or two more odd-multiple real roots on Da are costed by a new path. Moreover, all the paths are compatible.
Also note that we can have at mostγ1,⋯γd¯−1, which has occupied at least 2d¯−2 odd-multiple real roots on Da. At this stage, if we still have not obtained a desired path, we must obtain a closed path γ¯2 as above.
(iii)
Iteration of the above process
In this way, if the number of closed paths which reflect atPb stops increasing, then more and more odd-multiple real roots on Da are costed and after some n−1 steps, finally, only one odd-multiple real root on Da is left. Then, in n-th step, the new path γn cannot recur to Da, and it then must go to Pb. If we do not obtain a desired path, the number of closed paths which reflect at Pb finally increases by one.
In the same way, if, at some step, the number of closed paths reflecting atPb increases to d¯−1, in the next step, the number of closed paths which reflect at Pb cannot increase again and the new constructed path, denoted by γ^, has to go back to ξ=a. Finally, we obtain some closed paths γ¯1,⋯,γ¯mm≤d¯−1, which all reflect at Pb, and a path γ^, which joins with Pb and an odd-multiple real root on Da, such that all path γ0,γ¯1,⋯,γ¯m,γ^,γ~ are compatible. Hence, γ=γ0∪γ¯1∪⋯∪γ¯m∪γ^ is a desired path.
Note that these closed pathsγ¯1,⋯,γ¯m play as a role of bridge to join all the pathsγ0,γ¯1,⋯γ¯m,γ^ together to form a compatible path with γ~. In other words, these closed paths have no influence on the continuity of γ, but they may be important for γ to become a compatible path (see Figure 4).Case of (2(ii)).
Note thatB2d¯≠∅. Let a¯=infξ∣ξ∈B2d¯ and b¯=supξ∣ξ∈B2d¯. Obviously, a≤a¯<b¯<b and a¯,b¯∈B2d¯. Suppose P1=ξ1,x1 and a<ξ1<a¯ or b¯<ξ1<b. Then fξ,x has no 2d¯-multiple real root for ξ∈a,a¯ or ξ∈b¯,b. Then, this case is reduced to that of (2(i)).
Note thatP0∉D, then the case that ξ1∈a¯,b¯ cannot happen.
We should note that in a minimal room of real roots (f,2d¯,D,I), a path in D can meet at most two 2d¯-multiple real roots. This fact is very important for a path to include at most countable even-multiple real roots.Proof of (3).
The result (3) is about the case of odd-degree, the proof is in the same as that of the result (1) by using all induction assumptions and the result (2(ii)), which is just proven; here, we omit the details.Proof of (4).
Case of (4(i))
In the same way as the proof of (2(i)), the first part of the result can follow by induction assumption and the results (2) and (3); here, we omit the details. See Figures5–7.Figure 4
Simple extension at even-multiple real roots.Figure 5
Simple extension at odd-multiple real roots.Figure 6
Complicated extension at odd-multiple real roots.Figure 7
Complicated extension at odd-multiple real roots.Now, we consider the second part of the result and assume thatγ0 is already extended to Pb with P1=Pb. Below, we consider the two cases:
(i)
Pb is a crossing point of ΓBy assumption,P0∉D. Note that the path is extended in D to Pb. Since Pb is a crossing point of γ0 or γ~j, then γ0 cannot be extended to go through Pb and P1=Pb can only be a left endpoint according the compatibility of paths.IfPb is a crossing point of γ0, then all γ~j cannot go through D. If Pb is a crossing point of γ~j, then Pb is not a crossing point of γ0 or γ~j with i≠j. Anyway, Γ occupies even number of crossing points on Da. Since fa,x has odd number of odd-multiple real roots on Da, we can take an odd-multiple real root on Da and a small rectangle with this odd-multiple real root as the center. Then, take a small segment of path crossing this odd-multiple real root in the small rectangle and then extend it to the right in the same way as in the proof of the result (2(i)). If the path reaches to Pb, then γ0 can be extended from Pb to Da along the path; thus, the result is proven. Otherwise, the path does not reach to Pb and it goes back to Da. In the same way as in the proof of (2(i)), we take a new odd-multiple real root on Da and repeat the above process. If we do not obtain a desired path from Pb to Da, we have a closed path with Pb as a basis point, which is similar to the closed paths γ¯j in the proof of (2(i)), where Pb is a reflecting point of γ¯j. Finally, we obtain a desired path from Pb to Da.Here, we should note that the proof is essentially in the same way as that of the result (2(i)). In fact,Pb runs out of 1-multiplicity by a crossing point of γ~j and remains an even-multiplicity which can be used. So the crossing point Pb plays a role as an even-multiple real root, which allows any path arriving at Pb to be able to reflect at Pb.
(ii)
Pb is not a crossing point of ΓAlso note thatP0∉D and P0≠6=Pb. If the path is extended to the right to arrive at Pb, then Pb is a simple right endpoint and we need not do any extension in D. In fact, Pb will become a crossing point in the next extension of γ0 in a neighbouring room of real roots.Case of (4(ii)).
This case is a special situation of (4(i)) with absence of {γ~j}. By induction assumption, we take an odd-multiple real root on Da and extend a small segment of a path crossing this odd-multiple real root to the right. In the same way as the proof of (4(i)), it is easy to find a desired path; here, we omit the details.Proof of (5).
We first prove that, ifP1∉Pa,Pb, the path γ0 can be extended such that it arrives at Pa or Pb. Since P0∉D, then Pa or Pb is a crossing point of γ0 and P1∈D∘.
Note that oncePa is a crossing point of γ~j, then Pb is also a crossing point of γ~j; in this case, γ0 cannot enter into D. So D∩γ~=∅.
LetP1=ξ1,x1. Take a partition
(77)a<an<an−1<⋯<a0<ξ1<b0<⋯<bn−1<bn<b,such that an⟶a and bn⟶b. Since f has no 2d¯+1-multiple real root for ξ∈a0,b0, by induction assumption and the result (3), we can extend γ0 such that it arrives at ξ=a0 or ξ=b0. Denote the segment of extension by ℓ1.
Now, we considerγ1=γ0∪ℓ1 with an odd-multiple endpoint on ξ=a0 or ξ=b0 as a new initial endpoint and extend the path γ1 in a1,b1 such that it arrives to ξ=a1 or ξ=b1. Denote the segment of the extension by ℓ2.
Then, we can repeat the extension by the result (3) and in the same way as in the proof of (2(i)). If the path is already extended toξ=an or ξ=bn, we consider an+1,bn+1. Then, it can be extended to reach to ξ=an+1 or ξ=bn+1.
We should note that by Lemma24, there are at most finitely many 2d¯-degree minimal rooms of real roots in the decompositions of rooms on an,bn; thus, the extension of the path on an,bn includes at most finitely many 2d¯-multiple real roots.
Note that the path cannot oscillate toξ=a andξ=b more than 2d¯+1 times; thus, the path must go along one direction finally. If the path is extended to cross all ξ=an, then, in the same way as in the proof of the result (2(i)), it is extended to Pa finally. If the path is extended to cross all ξ=bn, it is extended to Pb.Case of (5(i)).
(i)
Pa and Pb are crossing points of Γ
Note thatP0∉D. If Pa is a crossing of γ~j, then Pb is also a crossing of γ~j, since its endpoints are outside of D. In this case, γ0 cannot enter into D. Since P1∈D, then P1=Pa or Pb. In this situation, P1 can only be a basis point in the extension. If Pa is a crossing of γ0, then Pa cannot be a crossing of γ~j, so Pb is also a crossing of γ0. Then, P1=Pa or Pb can also be a basis point in the extension. Whatever happens, the endpoint of γ0, P1∈Da∪Db always has outward direction; thus, the result holds.
(ii)
Only one ofPa and Pb is a crossing point of Γ
Without loss of generality, we assume thatPa is a crossing point and Pb is not. In this case, neither Pa nor Pb is a crossing point of γ~j,∀j. Actually, D∩γ~=∅, where γ~=∪jγ~j.
SinceP0∉D, then γ0 must enter into D from Pa. Since we can ignore the presence of γ~, this case is actually the case of (5(ii)); see the proof of the case of (5(ii)) below. Then, we can extend the path to Pb such that Pb is a right endpoint.
(iii)
NeitherPa nor Pb is a crossing point of Γ
In the same way as above, we haveD∩γ~=∅.Note thatP0∉D. Then, P1=Pa or Pb. Moreover, γ0∩Da,b=∅. Thus, this case is also reduced to that of the case (5(ii)).Case of (5(ii)).
Ifd¯<d−1, this result is from the induction assumption of Theorem 25.
Letd¯=d. First, take a section Dξ0, where a<ξ0<b. By the result (4), we have a regular path γ0 which joins with P0=Pa and an odd-multiple left endpoint P1∈Dξ0. Then, by the above discussion, it can be extended to Pb or Pa. In the first case, we have a desired path. Otherwise, it arrives at Pa, the path forms a closed path γ¯1 with Pa as a basis point. In the same way, by the result in the case (4(i)), we have a path γ0 which joins with P0=Paand an odd-multiple left endpoint P1∈Dξ0. Then, by the above discussion, it can be extended to Pb or Pa. By repeating the above process, either we obtain a desired path or we obtain a closed path γ¯2 with Pa as a basis point, which is compatible with γ¯1.In this way, if we do not obtain a desired path, then we obtain one more closed pathγ¯1 with Pa as a basis point, which is compatible with the previous ones γ¯1,⋯γ¯j−1. After at most d steps, when we have d number of such closed paths γ¯1,⋯γ¯d, which are compatible, we can have a path γ0, which joins with P0=Pa and an odd-multiple left endpoint P1∈Dξ0. Then, we extend γ0 from P1 to the left and it must go to Pb since there is no way for it to go back to Pa. Moreover, γ0 is compatible with γ¯1,⋯γ¯d. Let γ=γ0∪γ¯1∪⋯∪γ¯d, then γ is a desired path.Proof of (6).
Case of (6(i))
First note thatd¯≤d−1 and d¯=m, then it follows that m≤d−1. By induction assumption, Theorem 25 already holds.
By assumptionB2m+1 is a nonempty closed set. Let
(78)a¯=infξξ∈B2m+1,b¯=supξξ∈B2m+1.
Then,a¯,b¯∈B2m+1 and a≤a¯≤b¯≤b. Let
(79)a¯,b¯\B2m+1=∪j≥1aj,bj.
Then,fx,ξ has no 2m+1-multiple real root for ξ∈a,a¯∪b¯,b∪∪j≥1aj,bj.
NoteD=Da,a¯∪Db¯,b∪Da¯,b¯, where Da¯,b¯ is one-way passage of real roots. Below, we consider several cases according to distribution of the endpoints P0 and P1 about D.
By the induction assumption of Theorem25, there exists a regular path γ^ in Da¯,b¯ such that γ^ joins with Pa¯ and Pb¯. The path γ^ is useful in the proof of this part.
(i)
Pa¯ and Pb¯ are crossing points of Γ
By the discussion, we can suppose that there existsγ~j going through Pa¯ and Pb¯ or γ~j∩Da¯,b¯=∅,∀j. In the first case, γ0 cannot enter into D, so P1∈Da,a¯∪Db¯,b. Then, by the result of (4(i)), we can extend γ0 to Da or Db. In the second case, γ0 must enter into D and go through Da¯,b¯ with P1∈Da,a¯∪Db¯,b, then we can extend γ0 to Da or Db.
(ii)
Only one ofPa¯ and Pb¯ is a crossing point of Γ
Without loss of generality, supposePa¯ is a crossing point and Pb¯ is not. Since P0∉Da¯,b¯, then γ0 must enter into D from Pa¯. Then, we replace the segment of γ0 in Da¯,b¯ with γ~ to extend the path to Pb¯. Then, we extend the path to cross Pb¯ to reach to Db.
(iii)
NeitherPa¯ nor Pb¯ is a crossing point of Γ
SinceP0∉D, then P1∉D∘a¯,b¯ and γ0∩Da¯,b¯=∅. So P1 belongs to Da,a¯ or Db¯,b. If P1∈Da,a¯, the path γ0 can be extended to Pa¯ or Da. If it is extended to Pa¯, by joining the path γ^ with γ0, γ0 can be extended to Pb¯ and then to Db. If P1 belongs to Db¯,b, we have similar discussion.Case of (6(ii)) and Proof of Theorem 25.
Now, we are in a position to prove Theorem25. Recall
(80)A2d+1=ξ,xxisa2d+1‐multiplerealrootoffξ,x,B2d+1=ξξ,x∈A2d+1.
Then,B2d+1 is a closed set.
Let(81)a∗=minξξ∈B2d+1,b∗=maxξξ∈B2d+1.
Then,a∗,b∗∈B2d+1 and a≤a∗≤b∗≤b. Without loss of generality, suppose a<a∗ and b∗<b. By Lemma 26, there exist a path γl joining with (a∗,x∗) and Aa and another path γr joining with (b∗,y∗) and Ab. Let
(82)a∗,b∗\B2d+1=∪n≥1an,bn.
Letxn and yn be the 2d+1-multiple real root of fan,x=0 and fbn,x=0. By the result (5(ii)) of Lemma 26, there exists a path γn:γnt,t∈0,1, joining with an,xn and bn,yn.
By rescaling the parameter of the curveγn and suppose γn:γnt,t∈an,bn. Similarly, suppose γl:γlt,t∈a,a∗ and γr:γrt,t∈b∗,b.
Moreover, similar to Lemma19, it follows that A2d+1 is on the continuous curve x=−c1ξ/2d+1 determined by f2dξ,x=0,soxn=−c1an/2d+1,yn=−c1bn/2d+1.
Note thata,b=∪n≥1an,bn∪B2d+1∪a,a∗∪b∗,b. Then, define
(83)γt=γnt,an≤t≤bnn≥1,γt=t,−c1t2d+1,t∈B2d+1,γt=γlt,a≤t≤a∗,γt=γrt,b∗≤t≤b.
Then,γ:t∈a,b⟶A is a path joining with odd-multiple real roots on Da and Db, respectively. Of course, by rescaling of parameter we can write as γ:t∈0,1⟶A.
Below we show thatγ has at most countable reflecting points and basis points.
From the proof of the results (2(ii)) and (4(ii)), the pathγ includes at most countable number of 2d-multiple real roots as reflecting points on each interval an,bn. Moreover, only an,xn,bn,yn can be basis points of 2d+1-multiple real roots, so the path includes at most countable number of 2d+1-multiple real roots as basis points. Furthermore, the other reflecting points and basis points are lower-multiple real roots and they belong to some segments of the path in some lower-degree passages of odd-multiple real roots. By induction assumption, all reflecting points and basis points on these segments of the path are at most countable. Moreover, these lower-degree passages are also at most countable. Thus, all reflecting points and basis points on the path are also at most countable. For the idea of the proof, we refer to Figure 8.Proof of Theorem 2.
Now, we can prove our main result. By Theorem25, we only need to consider the case that f has no 2d+1-multiple real root in D. We reduce the case to the special one as in Theorem 25.
Leta0<a and b<b0, then a,b⊂a0,b0. It is easy to see that there exists a real polynomial of f0ξ,x of degree 2d+1, which depends continuously on ξ∈a0,b0, such that f0ξ,x=f0ξ,x,ξ∈a,b and both f0a0,x and f0b0,x has a 2d+1-multiple real root. Then, we use Theorem 25 or the result (5(ii)) of Lemma 26 to f0,2d+1,D0,I0, where I0=a0,b0. There exists a regular path γ0t,t∈0,1, such that γ00∈D0a0 and γ01∈D0b0.
Let(84)t0=maxt∈0,1∣γ0t∈Daisacrossingpointofγ0,t1=mint∈0,1∣γ0t∈Dbisacrossingpointofγ0.
Thus, it is easy to see thatγ0t0∈Da and γ0t1∈Db are odd-multiple real roots of f. Moreover, γt=γ0t∈D,t∈t0,t1 is a path on D with two odd-multiple real roots γ0t0 and γ0t1 on Da and Db, respectively.
RecallTP=t∈0,1γt=P. By construction of γ, the estimate (11) in Theorem 2 holds obviously. Moreover, for P∈γ, if TP#≥2, then P is a basis point or reflecting point; all these points are at most countable. Thus, Theorem 2 is proven.Remark 28.
From the above proof, we do not know if the pathγ in Theorem 2 is regular; however, it has no influence on our application.Figure 8
Passage of odd-multiple real roots.


## 3.1. Definitions

Below, we first define paths of odd-multiple real roots, crossing points, basis points, and reflecting points.Definition 4 (paths of odd-multiple real roots).
Let(13)γ:t∈0,1⟶γt=ξt,xt∈A⊂ℝ2,be a continuous mapping with P0=γ0=ξ0,x0 and P1=γ1=ξ1,x1. γ is called a path of odd-multiple real roots with endpoints P0 and P1 if the following hold:
(i)
Lett0∈0,1 and P=γt0=ξ∗,x∗, then one of the following holds true:
(1)
There existsδ0>0 such that
(14)ξt<ξ∗,∀t∈t0−δ0,t0andξt>ξ∗,∀t∈t0,t0+δ0,or
(15)ξt>ξ∗,∀t∈t0−δ0,t0andξt<ξ∗,∀t∈t0,t0+δ0Moreover, such t0 is unique for the point P if it exists.(2)
There existsδ0>0 such that
(16)ξt<ξ∗,∀0<t−t0<δ0,orξt>ξ∗,∀0<t−t0<δ0.(ii)
LetT~p=t∈0,1∣γt=P. If P∈γ is an odd-multiple real root such that there exists a unique t0∈T~p such that (1) holds, then such P is called a crossing point at t0. Moreover, for t0′∈T~p,ift0′≠t0, the result (2) holds at t0′; such P is called a basis point at t0′. If P is a crossing point at t0 and a basis point at t0′, we say that the basis point Patt0′ is based on the crossing point P at t0. If P∈γ is an odd-multiple real root such that the result (2) holds for all t0∈T~p, then P is a basis point, but not a crossing point; such P is called a pure basis point(iii)
IfP∈γ is an even-multiple real root, then only the result (2) holds. An even-multiple real root on γ is called a reflecting point(iv)
IfP0=P1 is an odd-multiple real root and it is not a crossing point, then there exists δ>0 such thatγt,0<t<δ and γt,1−δ<t<1 are on the same side of ξ=ξ0=ξ1(v)
IfP0=P1 is an odd-multiple real root and it is a crossing point, then there exists δ>0 such that γt,0<t<δ and γt,1−δ<t<1 are on different sides of ξ=ξ0=ξ1(vi)
IfP0=P1 is an even-multiple real root, then there exists δ>0 such that γt,0<t<δ and γt,1−δ<t<1 are on different sides of ξ=ξ0=ξ1(vii)
All basis points and reflecting points onγ are at most countable; thus, the points on γ are odd-multiple real roots almost everywhereA path of odd-multiple real roots is also called a path in brief if without any confusion (see Figure1).Figure 1
Path of odd-multiple real roots.By definition, an odd-multiple real root on a path can be both a crossing point and a basis point. Moreover, a path can have no pure basis point, and all odd-multiple real roots on the path can be crossing points except for its endpoints. This kind of special path is very important for the extension of paths in our problem.Definition 5.
Letγ=γt∣0≤t≤1 be a path with endpoints P0=γ0 and P1=γ1. Let γ∘=γt∣t∈0,1, which is called the interior point of γ. If for all P∈γ°∩Aodd, P is a crossing points of γ, γ is called a regular path. A regular path has no pure basis point.Leta,b⊂0,1. The set γa,b=γt∣t∈a,b is called a segment ofγ. By definition, it is easy to see that a segment of a path is still a path, but a segment of a regular path may not be regular. For P∈γ, P may be a crossing point on one segment and a pure basis point on another (also see Figure 1).An endpoint of a path is such a point that the path can be extended longer at this point.LetP1≠P0. There exists a small δ>0 such that for all 0<t≤δ,ξt is on the same side of ξ0. It is easy to see that we can always extend γ to cross P0. The same is true for P1.However, when the two endpoints of a path collide into one point, the path may not be extended again at the point; in this situation, the endpoints disappear. To avoid the endpoints losing their property as endpoints, we suggest the conditions (iv), (v), and (vi). More precisely, when two endpoints meet, they still keep the property of endpoints only after they collide into a pure basis point along the same side, into a crossing point along different sides, or into a reflecting point along different sides.Usually, we can extend a path at an endpoint, but the direction of extension about the parameterξ may be restricted. For this problem, we define directions of endpoints.Definition 6 (simple endpoints and directions of endpoints).
Let(17)γ:t∈0,1⟶γt=ξt,xt∈A⊂ℝ2,be a path of odd-multiple real roots with endpoints P0=ξ0,x0andP1=ξ1,x1.
We only considerP0 since P1 can be considered similarly. If P0 is neither a crossing point, nor a basis point, nor a reflecting point, we call it a simple endpoint. Moreover, if P0 is an odd(even)-multiple real root, we call it an odd(even)-multiple endpoint.
Below, we define the direction of endpoints in two cases ofP0≠P1 and P0=P1. First, note that there exists a sufficiently small δ0>0 such that either ξt>ξ0 for all t∈0,δ0 or ξt<ξ0 for all t∈0,δ0. That means that for sufficiently small t, γt are on the same side of ξ=ξ0.Case 1 (P0≠P1).
IfP0 is not a crossing point of the path γ and γt are on the left side (or right side) of ξ=ξ0 for all sufficiently small t>0, P0 is called a right (or left) endpoint. If P0 is a crossing point of γ and γt are on the left side (or right side) of ξ=ξ0 for all sufficiently small t>0, P0 is called a left (or right) endpoint.
IfP0 is an even-multiple real root and for all sufficiently small t>0, and γt are on the left side (or right side) of ξ=ξ0, P0 is called a left (or right) endpoint. For P1, we have the same statement and we omit the details.Case 2 (P0=P1).
In the condition (iv) of Definition4, if there exists δ>0 such that the set γt∣0<t<δ is on the right (left) side of ξ=ξ0=ξ1, we call P0 a left (right) endpoint and P1 a right (left) endpoint.
In the condition (v) of Definition4, if the set γt∣0<t<δ is on the right (left) side of ξ=ξ0=ξ1, we call P0 a right (left) endpoint and P1 a left (right) endpoint.
In the condition (vi) of Definition4, if the set γt∣0<t<δ is on the right (left) side of ξ=ξ0=ξ1, we call P0 a right (left) endpoint and P1 a left (right) endpoint.Remark 7.
A left (or right) endpoint means that the path can be extended to the left (or right) ofξ=ξ0 at the endpoint. In view of this point, in the cases of (iv)–(vi), we can define different directions of the endpoints. We choose the above definition of directions of endpoints for convenience in the extension of paths.Definition 8 (compatibility of paths).
Letγ1,γ2,⋯γn⊂A be paths of odd-multiple real roots and
(18)E=γj0,γj11≤j≤n,be the set of all endpoints of γj. Let G=γ1∪γ2∪⋯∪γn and G°=γ°1∪γ°2∪⋯∪γ°n.
If for allP∈G°∩Aodd, there exists a unique γj such that P is a crossing point of γj; moreover, if P∈γ°k with k≠j, P is a pure basis point of γk, then we say γj are compatible (at P). If P∈G°∩Aeven, P is always a reflecting point and we also say γj are compatible at P.
LetD be a domain of rectangles. If for all P∈D∩G°, γj are compatible at P, we say γj are compatible onD or G is regular on D.
Letγj be compatible paths and γ be a path of odd-multiple real roots with two endpoints P0=γ0andP1=γ1. If γ,γ1,⋯γn are compatible, we say γ is compatible with γj. Let γ^ be a path of odd-multiple real roots with two endpoints P^0=γ^=0 and P^0=γ^1. Let P^0=P1∉E. If γ+=γ∪γ^ is a path with two endpoints P0 and P^1 and it is also compatible with γj, we say γ+ is a compatible extension of γ with γj and γ^ is the prolonged segment. Obviously, if P1∉G°, then P1 must be a crossing point of γ+; if P1∈G°, then P1 must be a pure basis point of γ+.Remark 9.
Whenγ,γ1,⋯γn are compatible, formally, we can regard γ∪γ1∪⋯∪γn as a generalized regular path, because it may not be connected and can have many endpoints. In particular, if G=γ1∪⋯∪γn=∅, then γ is a regular path.
Letγ and γ^ be two compatible paths. Let P be an endpoint of both γ and γ^ with P=γ1=γ^0. If P∈Aodd and it has opposite endpoint directions for γ and γ^, or if P∈Aeven and it has the same endpoint direction, then γ+=γ∪γ^ is a compatible extension of γ.
Now supposeγ:t∈0,1⟶γt∈A and γ^:t∈1,2⟶γ^t∈A and P∗=γ1=γ^1, where we use γ^t+1 in place of the above γ^t given by standard definition without confusion.
Note that we can require a path to be defined on any intervala,b instead of 0,1 by rescaling the parameter of paths.
Ifγ1 and γ^1 are not crossing points, we can define γ+=γ∪γ^ as a regular path by
(19)γ+t=γt,0≤t≤1,γ+t=γ^t,1≤t≤2.
By definition, it is easy to see thatγ+ is well defined.
If bothγ1 and γ^1 are crossing points and they have different directions of endpoints, then γ+ as defined above may not satisfy the definition of paths. The above definition may result in γ+t crossing itself at P∗ more than 1 time. To avoid P∗ being crossed repeatedly, we can rearrange the parameter at P∗ so that γ+=γ∪γ^ can be defined as a regular path, but here, we omit the details.
Compatibility of paths is very important for extension of paths. It makes the extension of a path to not overlap any small segment of existing paths so that the extension cannot have any circulation.
When we extend a regular path, we require the path to keep the compatibility with the other existing paths. So at the first time when the path meets a new odd-multiple real root, we always let it go through the odd-multiple real root such that it becomes a crossing point of the path. When the path meets itself or some existing path at a crossing point, it must reflect at this point, and then it becomes a basis point. That means a new odd-multiple real root usually first becomes a crossing point and then a basis point. When the path meets an even-multiple real root, it is always extended by reflecting at this point.
We usually start a compatible extension of a regular path with a simple endpoint and require the prolonged segment to end also at a simple endpoint, so that we can easily see the extension to be compatible.Definition 10 (closed paths).
Letγ be a path of odd-multiple real roots with endpoints P0 and P1. We call γ a closed path of odd-multiple real roots, if P0=P1 with the following holding true:
(1)
IfP0 is a crossing point and there exists a sufficiently small δ0>0 such that γt∣t∈0,δ0 and γt∣t∈1−δ0,1 are on the same side of ξ=ξ0, then P0 is a crossing point of the closed path(2)
IfP0 is not a crossing point and there exists a sufficiently small δ0>0 such that γt∣t∈0,δ0 and γt∣t∈1−δ0,1are on different sides of ξ=ξ0, then P0 is a crossing point of the closed path(3)
IfP0 is not a crossing point and there exists a sufficiently small δ0>0 such that γt∣t∈0,δ0 and γt∣t∈1−δ0,1 are on the same side of ξ=ξ0, then P0 is a pure basis point of the closed path(4)
IfP0 is an even-multiple real root and there exists a sufficiently small δ0>0 such that γt∣t∈0,δ0 and γt∣t∈1−δ0,1 are on the same side of ξ=ξ0, then P0 is a reflecting point of the closed path
Whenγ is a closed path, we can regard γ as a continuous mapping: ℝ/ℤ=T⟶A, which is from the unit circle T to A. P0=P1 can be a crossing point or a pure basis point of the closed path (see Figure 2).
If a closed pathγ:t∈T=ℝ/ℤ⟶γt∈A⊂ℝ2 has no pure basis point, we call γ a closed regular path; see Figure 3.Figure 2
Closed path.Figure 3
Closed regular path.A closed regular path is a circulation and it cannot be extended longer.


## 3.2. Idea of Construction of Paths

Below, we give a more detailed description of the construction of paths.Letfξ,x be a polynomial of degree 2d+1 as in (1) and
(20)A2d+1=ξ,xxisa2d+1‐multiplerealrootoffξ,x,ξ∈a,b,B2d+1=ξξ,x∈A2d+1.Obviously,fx2dξ,x=0 determines a simple continuous curve
(21)l:x=−12d+1c1ξ,ξ∈a,b.Moreover,A2d+1⊂ℓ. Since L=a,b\B2d+1 is an open set of R, then L=∪n≥1In, where In=an,bn⊂a,b, which are disjointed each other.Letℓn be the segment of ℓ corresponding to ℓn. We want to find a path γn in Af,an,bn in place of ℓn.Moreover, we also find pathsγ± in Af,a±,b± for ξ∈a−,b− and ξ∈a+,b+, respectively, where a−=a,b−=minξ∈B2d+1 and a+=maxξ∈B2d+1,b+=b, if a or b does not belong to B2d+1. Let ℓ± be the two segments of l on a−,b− and a+,b+. Then, if we replace ln and ℓ± in l with γn and γ±, respectively, we obtain a path of odd-multiple real roots for ξ on a,b.The most important is to construct a pathγn on an,bn. Then, the path γ is obtained by grafting the simple continuous curve l with countable paths γn and ℓ±, where γn and ℓ± are composed of at most countable paths of polynomials of lower degree, maybe in a little complicated way. Moreover, each γn and ℓ± include at most countable reflecting points and basis points.To considerγn on an,bn, for shorting of notation, we drop the subscripts n. Then, suppose fa,x and fb,x have 2d+1-multiple real roots and fξ,x has no 2d+1-multiple real root for ξ∈a,b. Now we explain how to construct a path γ on a,b.The construction ofγ is based on extension of paths and induction for the degree of polynomials.First note that an odd-multiple real root is never isolated and usually it breaks into some lower-multiple real roots about the parameter. However, an even-multiple real root can be isolated, near to which there may be no other real roots. But, if there exists an odd-multiple real root near an even-multiple real root, there must be another odd-multiple real root near the even-multiple real root, that is, odd-multiple real roots always appear in pairs near an even-multiple real root. Thus, when a path arrives at an even-multiple real root, it can always reflect into some odd-multiple real roots at the even-multiple real root. So we define a path to be such a continuous curve that it is allowed to go through an odd-multiple real root and reflect when it meets an even-multiple real root.A path of odd-multiple real roots can be a little complicated near to a real root of higher-multiplicity; it can zigzag along the direction of thex-axes and have several recurrences at some parameters to approximate the real root of higher multiplicity. However, the times of the recurrence are no more than the multiplicity of the real root.Now, we first construct a regular path to join toPb. Let a<ξ0<b. Now, take an odd-multiple real root on Dξ0. By induction assumption, there exists a local regular path across the odd-multiple real root. Then, we extend the local regular path to the right. Since fξ,x has no 2d+1-multiple real root for ξ∈a,b, the real roots near any an odd-multiple real root can be all the real roots of a real polynomial of lower degree, and so we can extend it locally longer by induction assumption. We can extend the path until it arrives at Pb or ξ=ξ0. In the first situation, we have a desired path; otherwise, the path is extended to arrive at a new odd-multiple real root on Dξ0. By induction assumption, we can extend the path across ξ=ξ0 to the left a little and then have a regular path with two endpoints on the left of Dξ0.Then, take another odd-multiple real root onDξ0 and do the same way. If we cannot have a desired path, we have another compatible path with the previous path with two endpoints on the left of Dξ0. In this way, after at most d steps, only one odd-multiple real root is left on Dξ0. Take a path across this odd-multiple real root and extend it to the right, since all the odd-multiple real roots on Dξ0 are occupied by the previous compatible paths and the path cannot go back to Dξ0. Finally, it must go to Pb, and then we obtain a regular path joining to Pb.Then, we extend the path joining toPb to the left. If it is not extended to Pa, it must go back to Pb to form a closed path with Pb as a pure basis point. Then, take an odd-multiple real root on Dξ0 and do the same way as above to obtain a compatible path joining to Pb. In the same way, we extend the compatible path joining to Pb to the left. If we cannot have a desired path, we must have one more compatible closed path with Pb as a pure basis point. But we cannot have more than d number of these compatible closed paths, so after at most d iterations, the extension of the compatible path cannot go back to Pb; it has to go to Pa. Thus, the extension of the compatible path joins with the previous several compatible closed paths to form a regular path γ joining to Pa and Pb.We should also note that if two2d-multiplicity real roots are sufficiently close, the two corresponding simple real roots to the two 2d-multiplicity real roots can be connected by a saturation path of simple real roots. So a path cannot meet two 2d-multiplicity real roots which are very close; in other words, all 2d-multiplicity real roots on a path are discrete. Thus, the path γ can include at most countable 2d-multiple real roots. Moreover, γ can only have Pa or Pb as 2d+1-multiple basis points. This fact is very important for a path to contain at most countable basis points and reflecting points.


## 3.3. Lemmas

Below, we prove some lemmas which are required in the construction of paths.We first note that in Definition4, the conditions (1) of (i) and (vii) are very important; they guarantee that any small segment of a path cannot be overlapped one more times by the path itself. Moreover, a path cannot recur too many times about a fixed parameter.Lemma 11.
Letγt=ξt,xt∣t∈0,1 be a path of real roots. For any two intervals I=t1,t2⊂0,1 and I′=t1′,t2′⊂0,1,ifI∩I′=ϕ, then the set γI∩γI′ is at most countable. Moreover, Tξ0#≤2d+1, where Tξ0=t∈0,1∣ξt=ξ0.Proof.
IfγI∩γI′ is not countable, by the condition (vii) of the definition of paths, there exists a point P∈γt∣t∈I∩γt∣t∈I′ such that P is neither a basis point nor an even-multiple real root, then P is a pure crossing point. Again, there exist t0∈I and t0′∈I′ such that P=γt0=γt0′, but t0≠t0′, which contradicts with the fact that P can be crossed only once.
If there are several segments of the path crossing the parameterξ0, the condition (vii) implies that there exists ξn⟶ξ0 such that each segment crossing ξ=ξ0 does not meet the other segments on the line ξ=ξn. Otherwise, if for all ∣η−ξ0∣<δ0, at least two segments have an intersecting point Pη on ξ=η, these segments can intersect on Pηη−ξ0∣≤δ0, which has the cardinal number of the continuum and contradicts with the condition (vii).
Since the number of segments crossing the lineξ=ξ0 is no more than that of points of these segments on ξ=ξ0, which is no more than 2d+1, so the path oscillates for the parameter around ξ0 no more than 2d+1 times, which implies Tξ0#≤2d+1.This lemma means that a path cannot intersect itself everywhere on a nonempty segment, which guarantees a path to recur to a fixed parameterξ0 no more than 2d+1 times. Moreover, an m-multiple real root can be used by a path at most m-times. If the multiplicity is considered and an m-multiple real root is regarded as m different real roots, then a compatible extension can be thought to consist of new real roots and a path can be thought to consist of different real roots.For the construction of paths, we consider how a path can intersect with a lineξ=ξ0. For this purpose, we have the following lemma.Lemma 12.
Letγ be a path of odd-multiple real roots with two endpoints P1=ξ1,x1 and P2=ξ2,x2 with ξ1<ξ2. Then, the following hold true:
(1)
Ifξ1<ξ0<ξ2, then γ∩ξ0×ℝ has an odd number of crossing points(2)
Ifξ0<ξ1orξ0>ξ2, then γ∩ξ0×ℝ has an even number of crossing points.Proof.
(1)
By definition, we can moveP1 continuously to P2 along the path γ, and in the process, we go through odd-multiple real roots only one time.
Without loss of generality, supposeP1 and P2 are left and right endpoints, respectively. Then, the path starts with P1 to go to the right. Since ξ0∈ξ1,ξ2, it must intersect with ξ0×ℝ at some odd-multiple real root Q1. If it goes back across ξ0×ℝ at some odd-multiple real root Q2, then Q1≠Q2 since they can only be crossed only once. Then, it must cross ξ0×ℝ at another odd-multiple real root Q3; otherwise, it cannot arrive to P2. In this way, it is easy to see that the path must cross ξ0×ℝ an odd number of times.
(2)
This case can be considered similarly and we omit the detailLet(22)D=ξ,xa≤ξ≤b,c<x<d,be a rectangle domain of the ξ,x-plane. Let Dξ=ξ×c,d be a section of D at ξ.Let the boundary(23)∂D=lx−∪lx+∪lξ−∪lξ+,where
(24)lx−=a,b×c,lx+=a,b×d,lξ−=Da,lξ+=Db.Obviously, the rectangleD is open at the up-down sides lx±and closed at the left-right sides lξ±. Let D∘=a,b×c,d be the interior of D.Lemma 13.
Letγ be a path of odd-multiple real roots with endpoints P1 and P2 and γ∩lx±=ϕ. Suppose P1,P2∉D. Then, the number of crossing points in γ∩Dξ has the same odevity for all ξ∈a,b. In particular, if γ does not go through D, then the number of crossing points in γ∩Dξ is even; if γ goes through D, then the number of crossing points in γ∩Dξ is odd.Proof.
By definition of paths, it follows easily and we omit the details.By the implicit function theorem, for a simple real rootP0, there exists a path crossing P0, which only includes simple real roots. Below, we consider the maximal one among these paths.Lemma 14.
Suppose that bothfa,x and fb,x have a 2d+1-multiple real root. Fix ξ0∈a,b. Let
(25)fξ0,x0=0,fxξ0,x0≠0.
There exists a unique continuous functionx=xξ,ξ∈a0,b0⊂a,b such that
(26)x0=xξ0,fξ,xξ=0,fxξ,xξ≠0,∀ξ∈a0,b0.
Moreover,(27)fa0,xa0=fxa0,xa0=fb0,xb0=fxb0,xb0=0,fa0,xa0=fxa0,xa0=fb0,xb0=fxb0,xb0=0.
The continuous curveγ:ξ,xξ∣ξ∈a0,b0 is called a maximal (or saturation) path of the simple root P0=ξ0,x0.Proof.
By the implicit function theorem, there existsδ0>0 and a unique implicit function x=xξ,ξ∈ξ0−δ,ξ0+δ such that
(28)xξ0=x0,fξ,xξ=0,fxξ,xξ≠0,ξ∈ξ0−δ,ξ0+δ.
Below, we extend the continuous functionx=xξ from ξ0−δ,ξ0+δ to an interval as big as possible.
Let(29)J=a¯,b¯thereexistsauniquefunctionx=xξ,ξ∈a¯,b¯,suchthatxξ0=x0,fξ,xξ=0,fxξ,xξ≠0,ξ∈a¯,b¯.
Obviously,ξ0∈a¯,b¯. Let
(30)a0=inf=a¯,b¯∈Ja¯,b0=supa¯,b¯∈Jb¯.
Forξ∈a0,b0, we define x∗ξ=xξ,ξ∈a¯,b¯.
We prove thatx∗ξ is well defined on a0,b0. In fact, xξ,ξ∈a¯,b¯ is determined uniquely by ξ0,x0. It is easy to see that there exists α>0 such that
(31)fxξ,xξ≥α>0,ξ∈a¯,b¯.
By the implicit function theorem, there existsδ>0 such that for all η∈a¯,b¯, on the interval η−δ,η+δ, there uniquely determines an implicit function xξ such that
(32)fξ,xξ=0,ξ∈η−δ,η+δ.
This implies thatx=xξ can be obtained uniquely by extension of x=xξ from ξ∈ξ0−δ,ξ0+δ to [a¯,b¯].
Below, we provelimξ→a0+x∗ξ=xa0. Note that fξ,x∗ξ=0,fxξ,x∗ξ≠0. First, we prove the existence of the limit. If not so, there exist ξj1⟶a0+ and ξj2⟶a0+ such that x∗ξj1⟶x1 and x∗ξj2⟶x2 with x1<x2.
Takex1∗,x2∗ such that x1<x1∗<x2∗<x2. Without loss of generality, suppose
(33)x∗ξj1≤x1∗,x∗ξj2≥x2∗.
Sincex∗ξ is continuous, for all x∗ such that x1∗≤x∗≤x2∗, there exists ξj∗ between ξj1 and ξj2 such that x∗ξj∗=x∗. Also note ξj∗⟶a0+. By fξj∗,x∗ξj∗=0, it follows that
(34)fa0,x∗=0,∀x∗∈x1∗,x2∗.
Thus, allx∗∈x1∗,x2∗ are real roots of fa0,x. Since fa0,x=0 has at most 2d+1 real roots, this is a contradiction, which implies the existence of limξ→a0+x∗ξ.
Similarly,limξ→b0−x∗ξ also exists.
Obviously, we havefa0,x∗a0=0. If fxa0,x∗a0≠0, we can extend x∗ξatξ=a0 to a small neighbourhood of a0, which contradicts with the definition of a0.
The case ofξ=b0 can be discussed similarly.Remark 15.
Let∣fxξ0,x0∣≥α and fξ,x is continuous on D=a,b×−1,1 and fxξ,x≤M,∣fxxξ,x∣≤M,∀ξ,x∈a,b×−1,1. By the proof of the implicit theorem, it is easy to see that there exists constant δ>0, which is independent of ξ0,x0 and only depends on α, M, and the property of uniform continuity of f onD, such that xξ, the implicit function across ξ0,x0, is defined for ξ∈ξ0−δ,ξ0+δ⊂a0,b0. In general, the bigger the derivative fxξ0,x0, the longer the interval a0,b0. Moreover, we have that δ⟶0 as fxξ0,x0⟶0.


## 3.4. 3-Degree Polynomials

The proof of our result is strongly based on induction; for this purpose, we first consider 3-degree polynomials. Let(35)fξ,x=x3+c1ξx2+c2ξx+c3ξ, where cjj=1,2,3 are continuous on a,b.Let(36)Am=ξ,xξ,xisanm‐multiplerealrootoff,ξ∈a,b,Bm=ξξ,x∈Am,m=1,2,3.LetA=A1∪A2∪A3, which is the set of all real roots of f.Forξ∈B2, let Pξ1=ξ,xξ1 and Pξ2=ξ,xξ2 be, respectively, the simple real root and the 2-multiple real root of fξ,x. For ξ∈A3, let Pξ=ξ,x be the unique 3-multiple real root of fξ,x.Lemma 16.
AssumeA3=∅ and B2=a,b. Then, there exists a regular path in A joining with Pa1 and Pb1.Proof.
Denote byℓPa1 the maximal path across the simple root Pa1. By Lemma 14, it must end at some multiple real root. Since there is no multiple real root between ξ=a and ξ=b, on the right side, ℓPa1 goes through Pb1 or ends at Pb2=b,xb2.
IfℓPa1 does not reach Pb1, it must go to Pb2. Note that Pb2 is a 2-multiple real root, f has a simple real root ξ0,x0 such that a<ξ0<b and ξ0,x0∉ℓPa1; moreover, ξ0,x0 is sufficiently close to Pb2. Then, the maximal path across ξ0,x0 ends at Pb2 and Pa2. Denote the path by ℓPb2,Pa2.
On the other hand, since the maximal pathℓPb1 across Pb1 cannot go to Pa1, it must end at Pa2 on the left. Thus, ℓPa1∪ℓPb2,Pa2∪ℓPb1 forms a path joining with Pa1 and Pb1.
IfℓPa1 goes across Pb1, then it is just a path joining with Pa1 and Pb1.
Obviously, the path has no basis point and so it is a regular path. Thus, the lemma is proven.Lemma 17.
Assume thatA3=∅anda,b∈B2. Then, there exists a regular path between ξ=a and ξ=b, which joins with Pa1 and Pb1. Moreover, it includes at most finitely many 2-multiple real roots.Proof.
Obviously, ifB2=a,b, it is just the case in Lemma 16. Assume B2\a,b≠∅.
Letd0=infξ∈B2∣xξ2−xξ1∣.
We first claim that there existsξ0∈B2 such that d0=∣xξ02−xξ01∣>0.
In fact, we first haved0>0. If not so, there exist ξn∈B2 such that xξn1−xξn1⟶0. Without loss of generality, assume xξn2⟶x0,xξn1⟶x0,ξn⟶ξ0. Then, ξ0,x0 is a 3-multiple real root of f, which is a contradiction.
Now takeξn∈B2 such that xξn2−xξn1⟶d0. Since B2 is closed, there exists a subsequence ξnk⟶ξ0∈B2. Without loss of generality, assume ξn⟶ξ0. Since ∣xξ02−xξ01∣≥d0, the maximal path across Pξ01=ξ0,xξ01,lPξ01, exists for ∣ξ−ξ0∣≤δ0, where δ0 is a small constant. Let z1ξ,z2ξ,z3ξ be the three continuous bifurcations of roots of f. Then, lPξ01 must be on a continuous bifurcation. Without loss of generality, assume
(37)lPξ01⊂ξ,z1ξξ∈a,b.
Then,z1ξ0=xξ01. It follows that
(38)xξn1=z1ξn⟶z1ξ0=xξ01.
Noting thatz2ξ0=z3ξ0=xξ02 and
(39)xξn2=z2ξn=z3ξn⟶z2ξ0=z3ξ0=xξ02,we have xξn2−xξn1⟶xξ02−xξ01andthenxξ02−xξ01=d0.
Moreover, it follows easily that(40)fxξ,xξ1≥d02,∀ξ∈B2.
By Lemma14 and Remark 15, there exists δ>0, which is independent of ξ∈B2, such that for any η∈B2, a maximal path ℓη,xη1 is defined well for ∣ξ−η∣≤δ. More precisely, η−δ,η+δ is included in the definition interval of the maximal path.
Leta=ξ0<ξ1<⋯ξn=b such that ξj−ξj−1≤δ,j=1,2⋯n.
Ifξj−1,ξj∩B2≠∅, let
(41)aj=minξξ∈B2∩ξj−1,ξj,bj=maxξ∈B2∩ξj−1,ξj.
Obviously,ξj−1≤aj≤bj≤ξj and bj−aj≤δ. By Lemma 14 and using the same notation as in Lemma 16, the maximal path ℓPaj1 joins with Paj1 and Pbj1. Denote by ℓ¯j=ℓPaj1,Pbj1 the segment of ℓPaj1 joining with Paj1 and Pbj1. If aj=bj, we let lPaj1=Paj1. Obviously, there is no 2-multiple real root on the path ℓPaj1.
Ifξj−1,ξj∩B2=∅, we have no such path.
All these paths can be arranged aslpajk1k=1m with m≤n and
(42)a=aj1≤bj1≤⋯ajk≤bjk≤ajk+1≤bjk+1≤ajm≤bjm=b.
In particular, ifξ∈bjk,ajk+1,fξ,x has no 2-multiple real root. By Lemma 16, there exists a regular path
(43)ℓ~jk=ℓPbjk1,Pajk+11,such that ℓ~jk joins with Pbjk1 and Pajk+11.
Ifajk+1−bjk<δ, then ℓ~jk is a segment of the maximal path of the simple real root Pbjk, on which there is no 2-multiple real root. Only if ajk+1−bjk<δ can the 2-multiple real roots Pbjk2 and Pajk+12 be on ℓ~jk,k=1,2,⋯m−1.
It is easy to see that all the paths(44)ℓ¯j1,ℓ~j1,⋯ℓ¯jk,ℓ~jk⋯ℓ¯jm,ℓ~jm,can be continuously connected to form a regular path joining with Pa1 and Pb1. Moreover, it includes at most these 2-multiple real roots Pbjk2,Pajk+12k=1m.
By rescaling the parameter of the paths, we can define the path as a continuous mapping from0,1 to A, but we omit the details and refer to the proof of Lemma 26 below, where a more general case is considered.Lemma 18.
AssumeB3=a,b. Then, there exists a regular path ℓ in A joining with Pa and Pb; moreover, ℓ includes at most countable 2-multiple real roots.Proof.
Case 1.
Assume thatB2=∅. Let ξ0∈a,b and ξ0,x0∈A. Since fxξ0,x0≠0, by Lemma 14, the maximal path across ξ0,x0 must join with Pa and Pb.Case 2.
AssumeB2≠∅. Let
(45)a¯=infξ∈B2,b¯=supξ∈B2.
Ifa¯>a and b¯<b, then a¯,b¯∈B2. By Lemma 17, there exists a path joining with Pa¯1 and Pb¯1. Moreover, the maximal path across Pa¯1 reaches to ξ=a on the left and the maximal path across Pb¯1 reaches to ξ=b on the right. It is easy to see that three paths on a,a¯, a¯,b¯, and b¯,b form a regular path between ξ=a and ξ=b. Now we consider the case that a¯=a or b¯=b.
Leta¯=a. Take ξ0∈B2 and a sequence aj⊂B2 such that
(46)ξ0=a0>a1>a2>⋯>aj>aj+1⟶a.
Moreover,Paj1⟶Pa,Paj2⟶Pa. By Lemma 17, on aj−1,aj, we have a path ℓj joining with Paj−11 and Paj1. Then, all the paths ℓjj≥0 can define a path joining with Pa01 and Pa; for more details, we also refer to the proof of Lemma 26.
Ifb¯=b, in the same way, we can define a path connected with Pa01 and Pb. These two paths can be connected to form a regular path joining with Pa and Pb.Below, we consider the set of 3-multiple roots.Lemma 19.
For allξ,x∈A3, we have x=xξ=−c1ξ/3,ξ∈a,b. That means, all 3-multiple real roots of f are on the continuous curve
(47)l∗:ξ,xx=−c1ξ3,a≤ξ≤b.Proof.
Note that forξ,x∈A3, we have fxxξ,x=6x+2c1ξ=0; the result holds obviously.Now, we prove the following important lemma.Lemma 20.
There exists a regular pathℓ⊂A with two odd-multiple endpoints on Aa and Ab, respectively.Proof.
We first claim thatB3 is a closed set.
Letξn⟶ξ0,ξn∈B3. Then, there exists a subsequence xξnk of xξn, such that xξnk⟶x0. Since fξ,x,fxξ,x,fxxξ,x are continuous, by ξnk⟶ξ0 we have
(48)fξ0,x0=fxξ0,x0=fxxξ0,x0=0.
Thus,ξ0∈B3 and so B3 is closed.
Leta¯=minξ∣ξ∈B3,b¯=maxξ∣ξ∈B3. Then, we have a¯,b¯\B3=∪j≥1aj,bj where aj,bj∣j=1,2,⋯ are disjointed open intervals of each other.
By Lemma18, there exists a regular path ℓj between ξ=aj and ξ=bj such that Paj,Pbj∈ℓj.
Moreover, ifa<a¯, in the same way as the proof of Lemma 18, where we take ξ0=a and b=a¯, then there exists a regular path ℓl, which joins with Pa¯ and Pa∗, where Pa∗ is a simple root of fa,x. In the same way, if b¯<b, we have a regular path ℓr, which joins with Pb¯ and Pb∗, where Pb∗ is a simple root of fb,x. Now, we construct a path ℓ as follows:
(49)l=lj,ξ∈aj,bj,j≥1,l=l∗,ξ∈B3,l=ll,ξ∈a,a¯,l=lr,ξ∈b¯,b,where l∗ξ=ξ,−1/3c1ξ. Thus, ℓ is a path joining with Pa∗ and Pb∗.
Note that ifa=a¯ (or b=b¯), we have no path ℓl (or ℓr).
Below, we explain the continuity ofℓ. Let P0=ξ0,x0∈A3. By uniform continuity of continuous bifurcations of roots, it is easy to see that for any ϵ>0, there exists δ>0 such that if ∣ξ−ξ0∣≤δ and ξ,x is a real root of f, then x−x0≤ϵ. By this fact, the continuity of ℓ can be proven in the same way as in the extension of a continuous function on a closed set on ℝ to ℝ; here, we omit the details.
Moreover,ℓjj≥1,ℓl,ℓr include at most countable 2-multiple real roots; thus, the path ℓ includes at most countable 2-multiple real roots.
Furthermore, by the above construction, it is easy to see that the pathℓ has no basis point. Then, ℓ is a regular path in A, which connects from Aa to Ab.


## 3.5.2d+1-Degree Polynomials

In this subsection, we will prove our main results in the general case. To make an induction, we first give some definitions.Definition 21 (room of real roots and passage of odd-multiple real roots).
(1)
LetD=a,b×c,d be a rectangle in the ξ,x-plane and D°=a,b×c,d be the interior of D. Let I0=a0,b0⊂I=a,b and DI0=I0×c,d, where DI can be regarded as restriction of D for ξ to I0. Denote the section of D at ξ by Dξ=ξ×c,d⊂ℝ2. In particular, Da and Db are called the left side and the right side of D, respectively(2)
LetI=a,b and D be a rectangle as given above. A group f,m,D,I is called a room of real roots of f; it means that fξ,x is a real polynomial of degree m with respect to x and depends continuously in the parameter ξ on I; moreover, all real roots of fξ,x fall into D for all ξ∈I. If f has an m-multiple real root on D, f,m,D,I is called a minimal room of real roots of f(3)
If the left (right) side ofD contains an odd-multiple real root of f, the side is called a door of odd-multiple real roots. If a room of real roots of f has a door of odd-multiple real roots, it is called a passage of odd-multiple real roots. Otherwise, if the left (right) side of D contains no odd-multiple real root of f, the side is called a separating wall of odd-multiple real roots. If both the left side and the right side are separating walls of odd-multiple real roots, we call f,m,D,I an insulator of odd-multiple real roots. Let f,2d+1,D,I be a room of real roots. If fa,x and fb,x have 2d+1-multiple real roots, f,2d+1,D,I is called a one-way-passage of odd-multiple real roots(4)
IfD∘ contains no any real root of f, f,m,D,I is called a separating block of real roots(5)
Iff,m,D,I is a passage of odd-multiple real roots and f has an m-multiple real root in D, we call f,m,D,I a minimal passage of odd-multiple real roots. A minimal passage of odd-multiple real roots cannot be decomposed into smaller passagesA room of real roots is either a passage of odd-multiple real roots, or an insulator of odd-multiple real roots, or a separating block of real roots. Also, note that once a path enters into a one-way passage of odd-multiple real roots from one side, it can only go out of the one-way passage from the other side.Note that an odd-multiple real root is never isolated and an even-multiple real root can be isolated. Thus, there are a lot of odd-multiple real roots near to a door of odd-multiple real roots, so a passage is always neighbouring with some rooms of real roots. A separating block of real roots has no use in our problem, so it is usually ignored.Lemma 22 (decomposition of minimal rooms of real roots).
Letf,m,D,I be a room of real roots of f. Assume f has no any m-multiple root for all ξ∈I. Then, there exists δ>0, such that for any η∈I, if fη,x has a real root, there exist real polynomials fjξ,xj=1,2, with their degrees mj<m, which continuously depend on the parameter
(50)ξ∈I0=η−2δ,η+2δ∩I,and disjointed rectangles Dj⊂D,j=1,2, such that
(51)fξ,x=f1ξ,xf2ξ,x,ξ∈I0,and all real roots of fj fall into Dj,j=1,2. Moreover, f and fj have the same real roots on Dj,j=1,2.Remark 23.
In brief, the above statement can be written as(52)f,m,DI0,I0=f1,m1D1,I0×f2,m2,D2,I0.
Letf,m,D,I be a room of real roots. If fj,mj,Dj,I0 is a room of real roots, we call it a subroom of real roots of f,m,D,I. Similarly, we can define subpassage.
Also note thatfj,mj,Dj,I0 may not be a passage of odd-multiple real roots. In fact, fj,mj,Dj,I0 may be a separating block of real roots, which is usually ignored in our problem.Proof.
We first consider the case thatm=2d+1. Let xjξ,1≤j≤m be m continuous bifurcations of roots of f for ξ on I. Since f has no m-multiple real root, it follows that there exists α>0 such that
(53)maxi≠jxjξ−xiξ≥2d+1α,∀ξ∈I.
Letxj1η be a real root of fη,x. Then, there exist δ>0,c≥0,1≤r<m and two groups xj1,⋯,xjr and xjr+1,⋯,xjm such that
(54)xjkξ−xj1η<c,∀ξ∈I0,∀1≤k≤r<m,xjkξ−xj1η>c+α,∀ξ∈I0,∀r+1≤k≤m,where I0=η−2δ,η+2δ∩I. Since all xjξ are uniformly continuous in ξ on I, so δ can be independent of η∈I.
Note that forr+1≤k≤m,if∣Rexjkξ−xj1η∣<c+α, then xjkξ is a complex root.
Letx0=xj1η and M be a constant such that xjξ≤M,∀ξ∈I,j=1,2,⋯m.
Now we dividexjr+1,⋯xjm into two groups:
(55)xjkRexjk≤x0,r+1≤k≤m=x~j1≤j≤r~≤m−r,xjkRexjk>x0,r+1≤k≤m=x~jr~+1≤j≤m−r.
It is easy to see that for1≤j≤r~, if x~j is a real root, then x~j≤x0−c−α, and for r~+1≤j≤m−r, if x~j is a real root, then x~j>x0+c+α.
Let(56)g1ξ,x=x−xj1ξ⋯x−xjrξ,g2ξ,x=x−x~1ξ⋯x−x~r~ξ,g3ξ,x=x−x~r+1ξ⋯x−x~m−rξ.
Sincexj1η is a real number, it follows that xjkξ, x~jξ, and their conjugate complex roots belong to the same group; thus, all gjj=1,2,3 are real polynomials.
Ifr~≥1, let
(57)f1=g1g3,f2=g2,D1=I0×−M,x0+c,D2=I0×x0+c+α,M.
Ifr~=0, let
(58)f1=g1,f2=g3,D1=I0×x0−c,x0+c,D2=I0×−M,x0−c−α.The case thatm=2d can be considered in the same way by taking x0=xj1η to be a real root of fη,x, which is available by assumption.Iff has an m-multiple real root, the room of real roots f,m,D,I cannot be decomposed into smaller ones.The fact thatδ is independent of η guarantees a finite partition of a,b such that, on each small interval, we have the above decomposition. If some smaller room is not minimal, we can use the above lemma again until all small rooms are minimal. This result concerns with the following lemma.Lemma 24.
Considerf0,2d+1,D0,I0 and let f,m,D,I with I=a,b be a subroom of f0,2d+1,D0,I0 with m≤2d+1,D⊂D0,I⊂I0. Assume that f and f0 have same real roots on D; moreover, f has no any m-multiple root for all ξ∈a,b. Then, there exists finitely many minimal rooms of real roots of f,fj,mj,Dj,Ijj=1N with Ij=aj,bj, such that the following hold true:
(1)
f and fj have same real roots on Dj(2)
mj<m and fj has at least one mj-multiple real root on Dj(3)
Ifaj≠a and the left side Djaj contains a real root P of fj, then there is a room of real roots fi,mi,Di,Ii, which is neighboured with fj,mj,Dj,Ij such that P∈D°i and ai<aj<bi. For the right side, if bj≠b and Djbj contains a real root of fj, we have a similar result. Let δij=minai−bj,bi−bj, then we have δij>0. Since i,j≤N, the smallest distance of such δij is also positive(4)
Everyfj,mj,Dj,Ij is a minimal passage of odd-multiple real roots or an insulator of odd-multiple real roots; moreover, any real root of f on D is a real root of some fj on DjThe conclusion (3) is very important in extending of paths. By induction, a path can be always extended from an initial point in a passage of real rootsDj to arrive at an odd-multiple real root on a door of Dj. The new odd-multiple endpoint on the door of Dj is already in the interior of some Di. Then, we can do extension again in Di to arrive at another odd-multiple endpoint on a door of Di; in this process, the path must cross at least δij length of ξ. Since a path can only cross at most 2d+1b−a length about the parameter, the extension can repeat at most finitely many times before it cannot be extended again, which can only happen when the path has become a closed regular path or it arrives the boundary of the domain.Among the sequence of minimal roomsfj,mjDj,Ijj=1N in this lemma, there may be some minimal room which is completely included in a bigger one. In this case, we can drop away some smaller minimal rooms and the lemma still holds for the rest. So, without loss of generality, we can suppose Di\Dj≠∅ for all i≠j.Proof of Lemma 24.
By assumption,f has no any m-multiple real root. By Lemma 22, there exists δ>0 such that for η∈a,b, if fη,x has a real root, then there exist two real polynomials f1ξ,x and f2ξ,x with their degrees m1,m2<m, which continuously depend on the parameter ξ∈η−2δ,η+2δ∩a,b, and two disjointing rectangles D1 and D2 for ξ on η−2δ,η+2δ∩a,b, such that fξ,x=f1ξ,xf2ξ,x and all real roots of fi fall into Di, i=1,2.
Now, we choose a partition ofa,b:
(59)a=ξ0<ξ1<ξ2<⋯<ξn=b,such that ξj+1−ξj<δ. Then, let
(60)Ij1=ξj,ξj+2j=0,1,⋯n−2.
Iffη,x has a real root for some η∈ξj,ξj+2, Lemma 22 implies that on the small interval Ij1, we have the decomposition fj11,mj11,Dj11,Ij1 and fj21,mj21,Dj21,Ij1.
Iffη,x has no any real root for all η∈ξj,ξj+2, we drop away the room of real roots f,m,DIj1,Ij1. In this case, if there exists a real root P on Dξj or Dξj+2, P is an even-multiple real root and belongs to f,m,DIj−11,Ij−11 or f,m,DIj+11,Ij+11, so we do not lose any real root after dropping away f,mDIj1,Ij1.
Denote all these smaller rooms of real roots byfj1,mj1,Dj1,Ij1 for simplicity of notations. Note that some two intervals in Ij1 may be identical, but Dj1 are all different from each other. Let Ij1=aj1,bj1.
If somefj1 has no real root in the interior of Dj1, that is, (fj1,mj1,Dj1,Ij1) is a separating block of real roots, we drop it away. So without loss of generality, assume fj1 has at least one real root in the interior of Dj. Thus, fj,mj,Dj,Ij is a passage of odd-multiple real roots or an insulator of odd-multiple real roots. After dropping some separating blocks, for simplicity, we denote the subsequence by itself
(61)fj1,mj1,Dj1,Ij1j=1N1.
It is very important that for any side ofDj1, say Dj1aj1, if aj1≠a and P∈Dj1aj1 is a real root, then P must belong to the interior of some Di1i≠j. If bj1≠b and P∈Dj1bj1 is a real root, we have a similar result.
Now, if (f11,m11,D11,I11) is not minimal, in the same way as above, we decompose it into a sequence of smaller rooms of real roots
(62)f1k2,m1k2,D1k2,I1k2k≥1.
But we should note that when we make a partition ofI11=a11,b11, we require the points of the partition not to fall on M1, where
(63)M1=ξD11∩Dj1ξ≠ϕ,ξ∈aj1,bj1,1≤j≤N1.
SinceM1 is finite set, it is easy to have such partition of [a11,b11]. Such partition is very important, and it makes the sides of a neighbouring room of real roots to disjoint each other. Thus, any real root on a side must fall into the interior of a neighbouring room unless it falls into Da∪Db.
If (f11,m11,D11,I11) is minimal, we cannot decompose it and denote it by
(64)f112,m112,D112,I112=f11,m11,D11,I11.
Suppose we have decomposed(65)fjk1,mjk1,Djk1,Ijk1,j=1,2,⋯i−1,and obtained
(66)fjk2,mjk2,Djk2,Ijk2,j=1,2,⋯i−1.
Now, we consider (fi1,mi1,Di1,Ii1). In the same way, we make a partition of ai,bi as above; moreover, we require the points of the partition not to fall on Mi, where Mi=Mi1∪Mi2, where
(67)Mi1=ξDi1∩Dj1ξ≠ϕ,ξ∈aj1,bj1,j>i,Mi2=ξDi1∩Dj2ξ≠ϕ,ξ∈aj2,bj2,j=i,⋯i−1,k≥1.
If (fi1,mi1,Di1,Ii1) is minimal, we cannot decompose it and denote it by
(68)fi12,mi12,Di12,Ii12=fi1,mi1,Di1,Ii1.
In this way, we decompose all rooms (fj1,mj1,Dj1,Ij1) which are not minimal.
Then, we move away all separating blocks fromfjk2,mjk2,Djk2,Ijk2, and the rest is denoted by
(69)fj2,mj2,Dj2,Ij2j=1N2.
This step can iterate at mostm steps, and finally, we have a sequence of minimal rooms of real roots
(70)fjm,mjm,Djm,Ijmj=1Nm.
For simplicity, denote it by(71)fj,mj,Dj,Ijj=1N.According to the above decomposition, it is easy to see that if one side offj,mj,Dj,Ij does not fall on ξ=a or ξ=b, it may be divided into some parts such that each part is between the two sides of a neighbouring room fi,mi,Di,Ii, the overlapping region Dj∩Di can be regarded as a door way joining with fj,mj,Dj,Ij and fi,mi,Di,Ii. Any real root on the side of Dj is in the interior of some Di with i≠j.This is very important for our extension of paths. By induction assumption, a path of odd-multiple real roots can be extended inDj to a side of Dj. Since the new odd-multiple endpoint on the side of Dj falls into the interior of some Di, we can extend the path in Di again from the new endpoint to a side of Di. In this way, we can extend a path from the initial endpoint in some room to a neighbouring room and then go to the next neighbouring room. This process can go on unless it has become a closed regular path or arrived on ξ=a or ξ=b.We should note that once a path enters into an insulator of odd-multiple real roots, the whole path belongs to the insulator. Moreover, any extension of the path can only stay in the insulator forever and it can only be extended to be a closed path finally. So, only in some passages of odd-multiple real roots a path can be extended to be a desired path.A one-way passage of odd-multiple real roots can only allow one regular path to go through it. Once a segment of a path goes fromDa to Db, since there is a unique real root on Da and Db, respectively, according to the compatibility of paths, any segment of the path cannot enter into it again.We also note that a bigger room may be neighbouring with several smaller rooms. This case happens near to some multiple real roots, where some paths of lower-odd-multiple real roots bifurcate.The proof of Theorem2 is based on a special case that f has at least one 2d+1-multiple real root, which is considered in the following theorem:Theorem 25.
Consider a room of real rootsf,2d+1,D,I with I=a,b. If f has a 2d+1-multiple real root, then there exists a regular path of odd-multiple real roots in D, which joins with two odd-multiple endpoints on Da and Db, respectively.
The proof of Theorem25 is based on induction for degree of polynomials. So we first assume Theorem 25 already holds for polynomials of degree≤2d−1. Below, we prove that Theorem 25 also holds for polynomials of degree=2d+1. The proof is based on the following lemma.
Before giving the lemma, we first show some notations. Letm~≤2d+1 and f,m~,D,I be a room of real roots of f with I=a,b. Let Am~ be the set which consists of all m~-multiple real roots of f for all ξ∈I and Bm~=ξ∈I∣ξ,x∈Am~. For ξ∈Bm~, denote by Pξ the m~-multiple real root of fξ,x.Lemma 26.
Letf,m~,D,I be a subroom of real roots of f0,2d+1,D0,I0. Let γ0 be a regular path in f0,2d+1,D0,I0 with an odd-multiple endpoint P0∈D0\D and a simple odd-multiple endpoint P1∈D0. Let γ~j,⋯γ~k be paths in f0,2d+1,D0,I0, where k≤d.LetEj=γ~j0,γ~j1 be the endpoints of γ~j. Suppose that γ~j is a closed regular path or Ej⊂D0\D. Let γ~=γ~1∪⋯∪γ~ and Γ=γ0∪γ~. Suppose that {γ0,γ~1,⋯γ~k} are compatible on D.
(1)
Letm~=2d¯ and B2d¯=∅. Suppose that γ0 enters into D with P1∈D∘. Then γ0 can be extended in D to be a compatible path such that it has a simple odd-multiple endpoint P1+ on Da∪Db with outward direction. Here “outward direction” means, if P1+∈DaorDb, it is a left (or right) endpoint
Note that below the new endpoint in extension of paths always has outward direction in the above sense if without explanation.(2)
Letm~=2d¯ and P1∈D∘. Then, the following hold true:
(i)
LetB2d¯=b. Suppose that γ0 enters into D and arrives at P1=c,xc with a<c<b. Then, γ0 can be extended in D to be a compatible path such that it has a simple odd-multiple left endpoint on Da, or it reaches to Pb. Moreover, in the last situation, we can continue to extend γ0 with Pb as a reflecting point such that it has an odd-multiple left endpoint on Da; moreover, it is compatible with γ~(ii)
LetB2d¯≠∅. Then, γ0 can be extended to be a compatible path such that it has a simple odd-multiple endpoint on Da or Db with outward direction(3)
Letm~=2d¯+1 and B2d¯+1=∅. Suppose that γ0 enters into D with P1∈D∘. Then, γ0 can be extended to be a compatible path such that it has a simple odd-multiple endpoint on Da or Db with outward direction(4)
Letm~=2d¯+1 and B2d¯+1=b. Then, the following hold true:
(i)
Suppose thatγ0 enters into D with P1=Pc=c,xc∈D with a<c<b. Then, it can be extended to be a compatible path such that it has an odd-multiple left endpoint on Da, or it arrives at Pb. Moreover, in the last situation, if Pb is already a crossing point, γ0 can be extended to be a compatible path such that it has an odd-multiple left endpoint P1+ on Da(ii)
Cancel the existing pathsγ0 and γ~j in the assumption and consider f,m~,D,I to be an empty room of real roots. Then, there exists a regular path γ⊂D which joins with odd-multiple endpoints P0∈Da and P1=Pb with outward direction(5)
Letm~=2d¯+1 and B2d¯+1=a,b. Then, the following hold true:
(i)
Suppose that the pathγ0 reaches to D with P1∈D. Then, the path γ0 can be extended to be a compatible path such that it has a simple endpoint belonging to Pa,Pb with outward direction(ii)
In the same way as (4(ii)), there exists a regular pathγ⊂D, which joins with endpoints Pa and Pb with outward direction.(6)
Letm~=2d¯+1 and B2d¯+1≠∅. Then, the following hold true:
(i)
Letd¯≤d−1. Suppose that γ0 enters into D with P1∈D∘. Then, γ0 can be extended to be a compatible path such that it has a simple odd-multiple endpoint on Da∪Db with outward direction(ii)
Letd¯=d. Then, Theorem 25 holds trueRemark 27.
(i)
IfB2d¯=a, we have similar results to (2(i)) and (2(ii)). Moreover, if B2d¯+1=a, we have similar results to (4(i)) and (4(ii)). Here, we omit the details(ii)
We require endpoints ofγ~j to be outside of D to avoid γ0 being in a dilemma after it enters into D. Thus, when γ0 goes into D, it can always have some way to arrive at Da or DbProof.
We prove Lemma26 by induction. By the proof of Lemma 20, it is easy to see that Lemma 26 holds for d¯=1. Inductively suppose that Lemma 26 holds for all d¯≤m−1. Below, we prove it for d¯=m. Recall that we already make an induction assumption about d for Theorem 25. Here, we actually make an induction about both d and d¯ in Theorem 25 and Lemma 26.Proof of (1).
By assumption, we haveB2d¯=∅. Then, by Lemma 24, it follows that the room (f,2d¯,D,a,b) (d¯=m) can decompose into smaller minimal ones fj,mj,Dj,Ij with mj<2m.
Suppose that the pathγ0 has been extended to some room fj,mj,Dj,Ij such that P1∈D°j. By assumption it is easy to see that E=∪j=1kEj, the set of all the endpoints of {γ~j}, is outside of Dj. In addition, whatever mj=2m¯ or mj=2m¯+1, we have m¯≤m−1. By induction assumption of (2(ii)) and (6), we can extend γ0 in Dj to a side of Dj with a new odd-multiple left (or right) endpoint on Djaj (or Djbj).
Moreover, we can require the new odd-multiple endpoint to be simple and have outward direction about domainDj. In fact, we can always extend the path to reach to Djaj (or Djbj) such that the endpoint is neither a crossing point nor a reflecting point. If not so, the endpoint meets a crossing point or a reflecting point on Djaj (or Djbj), we can always continue the extension by reflecting at the crossing point or reflecting point such that the path enters into Dj again and then repeat the above process.
If the endpoint is already onDa∪Db, the result is proved. Otherwise, the odd-multiple endpoint is on a door of the passage fj,mj,Dj,Ij, which must be neighboured with another room of real roots, say, fi,mi,Di,Ii. Then, the path enters into Di and the odd-multiple endpoint on the door of fj,mj,Dj,Ij belongs to the interior of Di; thus, we can do extension in fi,mi,Di,Ii in the same way as in fj,mj,Dj,Ij.
If the extension does not go toDa∪Db, the process can repeat in the next neighbouring room. Each extension must go across the parameter of a positive length δ>0. It is easy to see that the total length of parameter of γ is no more than 2d¯b−a; thus, after at most finitely many extensions, the path must reach to Da∪Db with an odd-multiple endpoint with outward direction.
Thus, we haven segments of paths γ1,⋯,γn such that
(72)P0=γ10,P1=γ11,⋯,γn−11=Pn−1=γn0,Pn=γn1,where P0 is the initial endpoint of the path.
Note that eachγj is obtained when we do extension in some passage. Moreover, each γj is compatible with the existing paths since the extension is done according to the compatibility of paths by induction assumption. We should also note that once a path enters into an insulator, the path must totally fall into this insulator. However, since P0∉D, this situation cannot happen in our extension. Anyway, by induction assumption, the extension is well defined.
LetIj=i/n+1,j+1/n+1,j=0,1,⋯n and 0,1=∪j=0nIj. By rescaling the parameter t of paths, we suppose that the original path γ0t is defined for t on I0, and γjt is defined for t on Ij,j≥1. Then, we obtain an extension γ=∪j=0nγj with
(73)γt=γjt,t∈Ij,j=0,1⋯n.
By the above discussion, every extension follows the compatibility of paths, so all the paths are compatible. Thus, conclusion (1) is proven.Proof of (2).
Now, we turn to proving conclusion (2).Case of (2(i)).
(A) We first prove that onceγ0 enters into D, it can always be extended to arrive at Da or Pb; moreover, it is compatible with γ~=γ~∣j=1,2⋯k.
Take a partition(74)a=a0<c<a1<⋯<an<⋯bsuchthatan⟶b.
Sincef has no 2d¯-multiple real root for ξ∈a0,a1, by induction assumption and using the result (1) to the room of real roots f,2d¯,Da0,a1a0,a1, we can extend γ0 such that it goes to Da∪Da1. If it arrives at Da, the result is proved; otherwise, it goes to Da1. Denote the segment of extension by ℓ1. Note that here we can require the endpoint to be a simple one as discussed above; moreover, ℓ1 is compatible with Γ=γ~∪γ0.
Letγ1=γ0∪ℓ1, then γ1 is compatible with γ~. Now, we extend the path γ1 in Da0,a2 with a simple odd-multiple endpoint on Da1 as a new initial endpoint. In the same way as above, if it does not arrive at Da, it goes to Da2. Denote the segment of the extension by ℓ2. Moreover, ℓ2 is compatible with Γ1=γ~∪γ1. Let γ2=γ1∪ℓ2, then γ2 is compatible with γ~.
Then, we repeat the above process of extension. If the path is already extended toξ=an, we consider a0,an+1. In the same way, if it does not arrive at Da, it must arrive at ξ=an+1.
In this process, if the path does not arrive atDa, it must arrive at all ξ=an and have a sequence of segments of paths ln, which is obtained when extending on a0,an and given by a continuous mapping ℓn:t∈0,1⟶ℓnt with Pn=ℓn0∈Dan and Pn+1=ℓn1∈Dan+1, where Pn are all simple odd-multiple right endpoints of ℓn−1. Let
(75)0,1=∪n=1∞1−1n,1−1n+1∪1.
By rescaling the curve parametert, we can let γ0t,t∈0,1/2 and ℓn:t∈1−1/n+1,1−1/n+2⟶ℓnt with Pn=ℓn1−1/n+1andPn+1=ℓn1−1/n+2.
Define(76)γ^:γ^t=ℓnt,t∈1−1n+1,1−1n+2,∀n≥1,γ^1=Pb.
Then,γ^t,t∈1/2,1 is a continuous mapping with γ^0=P1 and γ^1=Pb. So γ=γ0∪γ^ is a continuous curve for t∈0,1 joining with P0 and Pb.
We should note that in the above extension,Pb may be already a reflecting point of γ~j or γ0. In this case, we can regard γ~j as two segments of paths with Pb as their endpoints. Obviously, all their endpoints are outside of Da0,an for all n≥1. Moreover, Γ is still compatible when γ~j is divided into two pieces of paths. Then, the assumption of (1) holds, so the above extension is reasonable.
Below, we show that the extensionγ is well defined and satisfies the definition of paths; thus, γ is a desired path.
By Lemma11, since the points on a path can recur to some parameter at most 2d¯ times, so ℓn can intersect with ℓ1,⋯,ℓn−1 at most 2d¯ times. So for any ℓm, there exists N>m such that for all n>N,ℓn∩ℓm=∅.
Letγn=γ0∪∪k=1n−1ℓk. By induction, all ℓn and γn follow the compatibility of paths.
It follows that allγn are compatible with γ~.
Letγ=γ0∪∪n=1∞ℓn∪Pb. Thus, for any P∈γ, if P≠Pb, there exists sufficiently large N>0 such that P∉∪n≥Nℓn and P∈γN=γ0∪∪n≤N−1ℓn. By induction γN is a well-defined compatible path, so P satisfies the definition of path as a point on γN. Since P∉∪n>Nℓn, then P also satisfies the definition of paths as a point on γ.
Below, we prove thatγt is continuous at t=1. Note that γ1=Pb. Let γt=ξt,xt. For any ϵ>0, there exists N>0 such that for all n>N, ℓn do not go across ξ=b−ϵ. Note that ℓn=ξnt,xnt,t∈1−1/n+1,1−1/n+2. Then, for all n>N,ξnt∈b−ϵ,b. Let δ=1/N+1. Then, if 0<1−t≤δ, there exists n>N such that t∈1−1/n+1,1−1/n+2,soξt=ξntandb−ξt=b−ξnt≤ϵ. Thus, ξt is continuous at t=1. Moreover, since Pb is a 2d¯-multiple real root, by the 2d¯ continuous bifurcations of roots of the polynomial f, one of which the real root ξt,xt must belong to; it follows easily that if t<1 and t⟶1, we have ξt⟶b and so xt⟶xb. Thus, γt⟶Pb,t⟶1, which implies that γt is continuous at t=1. Thus, γ is a compatible path with γ~, and Pb is a left even-multiple endpoint.
(B) Ifγ0 is already extended to arrive at Pb, we consider to extend it back from Pb to Da. We prove that there exists a path γ^, which joins also from an odd-multiple real root P1+∈Da to Pb, such that γ=γ0∪γ^ is a compatible path with γ~, which joins from P0 to Pb along γ0 and then goes back from Pb to P1+ along γ^.
Without loss of generality, suppose thatγ0 has arrived at Pb. Note that γ0 may meet Pb several times, that is, Pb is already a reflecting point of γ0. We first note that Lemma 13 implies that Γ=γ0∪γ~ has occupied odd number of crossing points on Da.
Sincef has even number of odd-multiple real roots on Da, we can take an odd multiple real root a,x1 on Da. By induction assumption of Theorem 25, there exists a small rectangle with a,x1 as the center and a small segment of path in the small rectangle, which goes across a,x1 from a left endpoint P0 on the left side to a right endpoint P1 on the right side. Then, P0∉D. Now, we extend the path from P1 to the right in the same way as in (A) of case (2(i)). Denote the path by γ1.
Ifγ1 is extended to Pb, let γ^=γ1 and then γ0∪γ^0 is a desired path. Otherwise, γ1 goes back to ξ=a without touching with Pb. We can require γ1 to be extended finally to an odd-multiple real root (a,x1′) with x1≠x1′. In fact, if the extension goes back to a,x1, it can also be extended to the right by reflecting at a,x1 and then repeat the above process. So, if it goes back to Da, we can always suppose it touches a different odd-multiple real root (a,x1′)on Da; moreover, we require it to cross Da a little with a simple endpoint of γ1 on the left of Da.
Denote the path joining witha,x1 and (a,x1′) still by γ1. Then, it has two endpoints of on the left of Da and occupies two odd-multiple real roots on Da. Note that Γ occupies odd number of odd-multiple real roots. Then Γ and γ1 have occupied odd number of odd-multiple real roots on Da.
In the same way as above, take another odd-multiple real roota,x2 on Da. By induction assumption of Theorem 25, there exists a small rectangle with a,x2 as the center and a small segment of path in the small rectangle, which goes across a,x2 from a left endpoint P0′ on the left side to a right endpoint P1′ on the right side. Then, P0′∉D. Now, we extend the path from P1′ to the right in the same way as in (A) of case (2(i)) to obtain a compatible path γ2.
We should note thatγ1 plays a role to restrict number of odd-multiple real roots on Da which can be taken as a crossing point of γ2. More precisely, since γ1 occupied at least two odd-multiple real roots on Da, a crossing point of γ2 on Da can be chosen in a smaller range.
Below, we analyze several cases of extension ofγ2.
(i)
γ2 does not recur to Da.
Ifγ2 does not go back to Da, it goes through all ξ=an, and then γ2 is extended from a,x2 to Pb in the same way as (A) of case (2(i)).
We should note thatγ2 is allowed to meet with γ0∪γ1 and γ~ according to the compatibility of paths.
Ifγ2 does not attach the path γ1, then γ2 is a desired path by taking off γ1.
Assume thatγ2 attaches γ1. Since γ1 is on the left of ξ=an1 with n1 being sufficiently big, so γ2 has no touching point with γ1 on the right of ξ=an1. Take the last crossing point of γ2 at ξ=an1 and denote it by an1,xn1.
Then, we drop awayγ1 and the segment of γ2 which joins from a,x2 to an1,xn1. The purpose of dropping γ1 away is to avoid influence of γ1 on γ2 such that γ2 can only have some touch with γ0 and γ~; thus, the next extension of γ2 can be a compatible path.
Then, we extend the segment ofγ2 between Pb and an1,xn1 to the left. Still denote by γ2 the segment of γ2 without confusion. If γ2 is extended to Da, then γ2 is a desired path. Otherwise, the extension goes through all ξ=an to arrive at Pb, then we have a closed path which reflects at Pb and denote it by γ¯1. Moreover, γ¯1 is compatible with γ0 and γ~.
(ii)
γ2 recurs to Da.
Ifγ2 goes back to ξ=a, we assume that it does not touch γ1 at ξ=a (if so, we can still extend it by reflecting at the touching point and finally γ2 can reach to an odd-multiple real root on Da, which does not belong to γ1). Moreover, in the same way as γ1, we require γ2 to cross Da a little with a simple endpoint on the left of Da so that its two endpoints are on the left of Da. Then, γ1∪γ2 occupy at least 4 odd-multiple real roots on Da and all their endpoints are on the left of Da. Moreover, γ0,γ1,γ2,γ~ are compatible.
At this stage, we can makeγ3. Using γ1∪γ2 in place of γ1 and γ3 in place of the above γ2, we can have a similar discussion as above.
Finally, if we have not obtained a desired path, either we obtain another closed pathγ¯2 withPb being a reflecting point, or γ1∪γ2∪γ3 occupies at least 6 odd-multiple real roots on Da; moreover, γ0,γ1,γ2,γ3,γ~ are compatible.
We1 should note thatγ¯2 is obtained by extension of a segment of the path after dropping γ1∪γ2. So γ0,γ¯1 and γ¯2 also follow the compatibility of paths with γ~, and so γ0,γ1,γ2,γ~ are compatible.
If we do not obtain a closed pathγ¯2, we can consider to find γ4 in the same way as above. Thus, if we do not find a desired path, we either find a closed path γ¯2 as above or a path γ4 as γj,1≤j≤3.
During this process, if we have not obtained a desired path, one of two situations happen: we find one more closed path withPb being a reflecting point or two more odd-multiple real roots on Da are costed by a new path. Moreover, all the paths are compatible.
Also note that we can have at mostγ1,⋯γd¯−1, which has occupied at least 2d¯−2 odd-multiple real roots on Da. At this stage, if we still have not obtained a desired path, we must obtain a closed path γ¯2 as above.
(iii)
Iteration of the above process
In this way, if the number of closed paths which reflect atPb stops increasing, then more and more odd-multiple real roots on Da are costed and after some n−1 steps, finally, only one odd-multiple real root on Da is left. Then, in n-th step, the new path γn cannot recur to Da, and it then must go to Pb. If we do not obtain a desired path, the number of closed paths which reflect at Pb finally increases by one.
In the same way, if, at some step, the number of closed paths reflecting atPb increases to d¯−1, in the next step, the number of closed paths which reflect at Pb cannot increase again and the new constructed path, denoted by γ^, has to go back to ξ=a. Finally, we obtain some closed paths γ¯1,⋯,γ¯mm≤d¯−1, which all reflect at Pb, and a path γ^, which joins with Pb and an odd-multiple real root on Da, such that all path γ0,γ¯1,⋯,γ¯m,γ^,γ~ are compatible. Hence, γ=γ0∪γ¯1∪⋯∪γ¯m∪γ^ is a desired path.
Note that these closed pathsγ¯1,⋯,γ¯m play as a role of bridge to join all the pathsγ0,γ¯1,⋯γ¯m,γ^ together to form a compatible path with γ~. In other words, these closed paths have no influence on the continuity of γ, but they may be important for γ to become a compatible path (see Figure 4).Case of (2(ii)).
Note thatB2d¯≠∅. Let a¯=infξ∣ξ∈B2d¯ and b¯=supξ∣ξ∈B2d¯. Obviously, a≤a¯<b¯<b and a¯,b¯∈B2d¯. Suppose P1=ξ1,x1 and a<ξ1<a¯ or b¯<ξ1<b. Then fξ,x has no 2d¯-multiple real root for ξ∈a,a¯ or ξ∈b¯,b. Then, this case is reduced to that of (2(i)).
Note thatP0∉D, then the case that ξ1∈a¯,b¯ cannot happen.
We should note that in a minimal room of real roots (f,2d¯,D,I), a path in D can meet at most two 2d¯-multiple real roots. This fact is very important for a path to include at most countable even-multiple real roots.Proof of (3).
The result (3) is about the case of odd-degree, the proof is in the same as that of the result (1) by using all induction assumptions and the result (2(ii)), which is just proven; here, we omit the details.Proof of (4).
Case of (4(i))
In the same way as the proof of (2(i)), the first part of the result can follow by induction assumption and the results (2) and (3); here, we omit the details. See Figures5–7.Figure 4
Simple extension at even-multiple real roots.Figure 5
Simple extension at odd-multiple real roots.Figure 6
Complicated extension at odd-multiple real roots.Figure 7
Complicated extension at odd-multiple real roots.Now, we consider the second part of the result and assume thatγ0 is already extended to Pb with P1=Pb. Below, we consider the two cases:
(i)
Pb is a crossing point of ΓBy assumption,P0∉D. Note that the path is extended in D to Pb. Since Pb is a crossing point of γ0 or γ~j, then γ0 cannot be extended to go through Pb and P1=Pb can only be a left endpoint according the compatibility of paths.IfPb is a crossing point of γ0, then all γ~j cannot go through D. If Pb is a crossing point of γ~j, then Pb is not a crossing point of γ0 or γ~j with i≠j. Anyway, Γ occupies even number of crossing points on Da. Since fa,x has odd number of odd-multiple real roots on Da, we can take an odd-multiple real root on Da and a small rectangle with this odd-multiple real root as the center. Then, take a small segment of path crossing this odd-multiple real root in the small rectangle and then extend it to the right in the same way as in the proof of the result (2(i)). If the path reaches to Pb, then γ0 can be extended from Pb to Da along the path; thus, the result is proven. Otherwise, the path does not reach to Pb and it goes back to Da. In the same way as in the proof of (2(i)), we take a new odd-multiple real root on Da and repeat the above process. If we do not obtain a desired path from Pb to Da, we have a closed path with Pb as a basis point, which is similar to the closed paths γ¯j in the proof of (2(i)), where Pb is a reflecting point of γ¯j. Finally, we obtain a desired path from Pb to Da.Here, we should note that the proof is essentially in the same way as that of the result (2(i)). In fact,Pb runs out of 1-multiplicity by a crossing point of γ~j and remains an even-multiplicity which can be used. So the crossing point Pb plays a role as an even-multiple real root, which allows any path arriving at Pb to be able to reflect at Pb.
(ii)
Pb is not a crossing point of ΓAlso note thatP0∉D and P0≠6=Pb. If the path is extended to the right to arrive at Pb, then Pb is a simple right endpoint and we need not do any extension in D. In fact, Pb will become a crossing point in the next extension of γ0 in a neighbouring room of real roots.Case of (4(ii)).
This case is a special situation of (4(i)) with absence of {γ~j}. By induction assumption, we take an odd-multiple real root on Da and extend a small segment of a path crossing this odd-multiple real root to the right. In the same way as the proof of (4(i)), it is easy to find a desired path; here, we omit the details.Proof of (5).
We first prove that, ifP1∉Pa,Pb, the path γ0 can be extended such that it arrives at Pa or Pb. Since P0∉D, then Pa or Pb is a crossing point of γ0 and P1∈D∘.
Note that oncePa is a crossing point of γ~j, then Pb is also a crossing point of γ~j; in this case, γ0 cannot enter into D. So D∩γ~=∅.
LetP1=ξ1,x1. Take a partition
(77)a<an<an−1<⋯<a0<ξ1<b0<⋯<bn−1<bn<b,such that an⟶a and bn⟶b. Since f has no 2d¯+1-multiple real root for ξ∈a0,b0, by induction assumption and the result (3), we can extend γ0 such that it arrives at ξ=a0 or ξ=b0. Denote the segment of extension by ℓ1.
Now, we considerγ1=γ0∪ℓ1 with an odd-multiple endpoint on ξ=a0 or ξ=b0 as a new initial endpoint and extend the path γ1 in a1,b1 such that it arrives to ξ=a1 or ξ=b1. Denote the segment of the extension by ℓ2.
Then, we can repeat the extension by the result (3) and in the same way as in the proof of (2(i)). If the path is already extended toξ=an or ξ=bn, we consider an+1,bn+1. Then, it can be extended to reach to ξ=an+1 or ξ=bn+1.
We should note that by Lemma24, there are at most finitely many 2d¯-degree minimal rooms of real roots in the decompositions of rooms on an,bn; thus, the extension of the path on an,bn includes at most finitely many 2d¯-multiple real roots.
Note that the path cannot oscillate toξ=a andξ=b more than 2d¯+1 times; thus, the path must go along one direction finally. If the path is extended to cross all ξ=an, then, in the same way as in the proof of the result (2(i)), it is extended to Pa finally. If the path is extended to cross all ξ=bn, it is extended to Pb.Case of (5(i)).
(i)
Pa and Pb are crossing points of Γ
Note thatP0∉D. If Pa is a crossing of γ~j, then Pb is also a crossing of γ~j, since its endpoints are outside of D. In this case, γ0 cannot enter into D. Since P1∈D, then P1=Pa or Pb. In this situation, P1 can only be a basis point in the extension. If Pa is a crossing of γ0, then Pa cannot be a crossing of γ~j, so Pb is also a crossing of γ0. Then, P1=Pa or Pb can also be a basis point in the extension. Whatever happens, the endpoint of γ0, P1∈Da∪Db always has outward direction; thus, the result holds.
(ii)
Only one ofPa and Pb is a crossing point of Γ
Without loss of generality, we assume thatPa is a crossing point and Pb is not. In this case, neither Pa nor Pb is a crossing point of γ~j,∀j. Actually, D∩γ~=∅, where γ~=∪jγ~j.
SinceP0∉D, then γ0 must enter into D from Pa. Since we can ignore the presence of γ~, this case is actually the case of (5(ii)); see the proof of the case of (5(ii)) below. Then, we can extend the path to Pb such that Pb is a right endpoint.
(iii)
NeitherPa nor Pb is a crossing point of Γ
In the same way as above, we haveD∩γ~=∅.Note thatP0∉D. Then, P1=Pa or Pb. Moreover, γ0∩Da,b=∅. Thus, this case is also reduced to that of the case (5(ii)).Case of (5(ii)).
Ifd¯<d−1, this result is from the induction assumption of Theorem 25.
Letd¯=d. First, take a section Dξ0, where a<ξ0<b. By the result (4), we have a regular path γ0 which joins with P0=Pa and an odd-multiple left endpoint P1∈Dξ0. Then, by the above discussion, it can be extended to Pb or Pa. In the first case, we have a desired path. Otherwise, it arrives at Pa, the path forms a closed path γ¯1 with Pa as a basis point. In the same way, by the result in the case (4(i)), we have a path γ0 which joins with P0=Paand an odd-multiple left endpoint P1∈Dξ0. Then, by the above discussion, it can be extended to Pb or Pa. By repeating the above process, either we obtain a desired path or we obtain a closed path γ¯2 with Pa as a basis point, which is compatible with γ¯1.In this way, if we do not obtain a desired path, then we obtain one more closed pathγ¯1 with Pa as a basis point, which is compatible with the previous ones γ¯1,⋯γ¯j−1. After at most d steps, when we have d number of such closed paths γ¯1,⋯γ¯d, which are compatible, we can have a path γ0, which joins with P0=Pa and an odd-multiple left endpoint P1∈Dξ0. Then, we extend γ0 from P1 to the left and it must go to Pb since there is no way for it to go back to Pa. Moreover, γ0 is compatible with γ¯1,⋯γ¯d. Let γ=γ0∪γ¯1∪⋯∪γ¯d, then γ is a desired path.Proof of (6).
Case of (6(i))
First note thatd¯≤d−1 and d¯=m, then it follows that m≤d−1. By induction assumption, Theorem 25 already holds.
By assumptionB2m+1 is a nonempty closed set. Let
(78)a¯=infξξ∈B2m+1,b¯=supξξ∈B2m+1.
Then,a¯,b¯∈B2m+1 and a≤a¯≤b¯≤b. Let
(79)a¯,b¯\B2m+1=∪j≥1aj,bj.
Then,fx,ξ has no 2m+1-multiple real root for ξ∈a,a¯∪b¯,b∪∪j≥1aj,bj.
NoteD=Da,a¯∪Db¯,b∪Da¯,b¯, where Da¯,b¯ is one-way passage of real roots. Below, we consider several cases according to distribution of the endpoints P0 and P1 about D.
By the induction assumption of Theorem25, there exists a regular path γ^ in Da¯,b¯ such that γ^ joins with Pa¯ and Pb¯. The path γ^ is useful in the proof of this part.
(i)
Pa¯ and Pb¯ are crossing points of Γ
By the discussion, we can suppose that there existsγ~j going through Pa¯ and Pb¯ or γ~j∩Da¯,b¯=∅,∀j. In the first case, γ0 cannot enter into D, so P1∈Da,a¯∪Db¯,b. Then, by the result of (4(i)), we can extend γ0 to Da or Db. In the second case, γ0 must enter into D and go through Da¯,b¯ with P1∈Da,a¯∪Db¯,b, then we can extend γ0 to Da or Db.
(ii)
Only one ofPa¯ and Pb¯ is a crossing point of Γ
Without loss of generality, supposePa¯ is a crossing point and Pb¯ is not. Since P0∉Da¯,b¯, then γ0 must enter into D from Pa¯. Then, we replace the segment of γ0 in Da¯,b¯ with γ~ to extend the path to Pb¯. Then, we extend the path to cross Pb¯ to reach to Db.
(iii)
NeitherPa¯ nor Pb¯ is a crossing point of Γ
SinceP0∉D, then P1∉D∘a¯,b¯ and γ0∩Da¯,b¯=∅. So P1 belongs to Da,a¯ or Db¯,b. If P1∈Da,a¯, the path γ0 can be extended to Pa¯ or Da. If it is extended to Pa¯, by joining the path γ^ with γ0, γ0 can be extended to Pb¯ and then to Db. If P1 belongs to Db¯,b, we have similar discussion.Case of (6(ii)) and Proof of Theorem 25.
Now, we are in a position to prove Theorem25. Recall
(80)A2d+1=ξ,xxisa2d+1‐multiplerealrootoffξ,x,B2d+1=ξξ,x∈A2d+1.
Then,B2d+1 is a closed set.
Let(81)a∗=minξξ∈B2d+1,b∗=maxξξ∈B2d+1.
Then,a∗,b∗∈B2d+1 and a≤a∗≤b∗≤b. Without loss of generality, suppose a<a∗ and b∗<b. By Lemma 26, there exist a path γl joining with (a∗,x∗) and Aa and another path γr joining with (b∗,y∗) and Ab. Let
(82)a∗,b∗\B2d+1=∪n≥1an,bn.
Letxn and yn be the 2d+1-multiple real root of fan,x=0 and fbn,x=0. By the result (5(ii)) of Lemma 26, there exists a path γn:γnt,t∈0,1, joining with an,xn and bn,yn.
By rescaling the parameter of the curveγn and suppose γn:γnt,t∈an,bn. Similarly, suppose γl:γlt,t∈a,a∗ and γr:γrt,t∈b∗,b.
Moreover, similar to Lemma19, it follows that A2d+1 is on the continuous curve x=−c1ξ/2d+1 determined by f2dξ,x=0,soxn=−c1an/2d+1,yn=−c1bn/2d+1.
Note thata,b=∪n≥1an,bn∪B2d+1∪a,a∗∪b∗,b. Then, define
(83)γt=γnt,an≤t≤bnn≥1,γt=t,−c1t2d+1,t∈B2d+1,γt=γlt,a≤t≤a∗,γt=γrt,b∗≤t≤b.
Then,γ:t∈a,b⟶A is a path joining with odd-multiple real roots on Da and Db, respectively. Of course, by rescaling of parameter we can write as γ:t∈0,1⟶A.
Below we show thatγ has at most countable reflecting points and basis points.
From the proof of the results (2(ii)) and (4(ii)), the pathγ includes at most countable number of 2d-multiple real roots as reflecting points on each interval an,bn. Moreover, only an,xn,bn,yn can be basis points of 2d+1-multiple real roots, so the path includes at most countable number of 2d+1-multiple real roots as basis points. Furthermore, the other reflecting points and basis points are lower-multiple real roots and they belong to some segments of the path in some lower-degree passages of odd-multiple real roots. By induction assumption, all reflecting points and basis points on these segments of the path are at most countable. Moreover, these lower-degree passages are also at most countable. Thus, all reflecting points and basis points on the path are also at most countable. For the idea of the proof, we refer to Figure 8.Proof of Theorem 2.
Now, we can prove our main result. By Theorem25, we only need to consider the case that f has no 2d+1-multiple real root in D. We reduce the case to the special one as in Theorem 25.
Leta0<a and b<b0, then a,b⊂a0,b0. It is easy to see that there exists a real polynomial of f0ξ,x of degree 2d+1, which depends continuously on ξ∈a0,b0, such that f0ξ,x=f0ξ,x,ξ∈a,b and both f0a0,x and f0b0,x has a 2d+1-multiple real root. Then, we use Theorem 25 or the result (5(ii)) of Lemma 26 to f0,2d+1,D0,I0, where I0=a0,b0. There exists a regular path γ0t,t∈0,1, such that γ00∈D0a0 and γ01∈D0b0.
Let(84)t0=maxt∈0,1∣γ0t∈Daisacrossingpointofγ0,t1=mint∈0,1∣γ0t∈Dbisacrossingpointofγ0.
Thus, it is easy to see thatγ0t0∈Da and γ0t1∈Db are odd-multiple real roots of f. Moreover, γt=γ0t∈D,t∈t0,t1 is a path on D with two odd-multiple real roots γ0t0 and γ0t1 on Da and Db, respectively.
RecallTP=t∈0,1γt=P. By construction of γ, the estimate (11) in Theorem 2 holds obviously. Moreover, for P∈γ, if TP#≥2, then P is a basis point or reflecting point; all these points are at most countable. Thus, Theorem 2 is proven.Remark 28.
From the above proof, we do not know if the pathγ in Theorem 2 is regular; however, it has no influence on our application.Figure 8
Passage of odd-multiple real roots.



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