# Infinitely Many Quasi-Coincidence Point Solutions of Multivariate Polynomial Problems

Authors: Yi-Chou Chen
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

LetF:ℝn×ℝ→ℝ be a real-valued polynomial function of the form F(x¯,y)=as(x¯)ys+as-1(x¯)ys-1+⋯+a0(x¯) where the degree s of y in F(x¯,y) is greater than 1. For arbitrary polynomial function f(x¯)∈ℝ[x¯], x¯∈ℝn, we will find a polynomial solution y(x¯)∈ℝ[x¯] to satisfy the following equation (⋆): F(x¯,y(x¯))=af(x¯) where a∈ℝ is a constant depending on the solution y(x¯), namely a quasi-coincidence (point) solution of (⋆), and a is called a quasi-coincidence value of (⋆). In this paper, we prove that (i) the number of all solutions in (⋆) does not exceed degyF(x¯,y)(2degf(x¯)+s+3)·2degf(x¯)+1 provided those solutions are of finitely many exist, (ii) if all solutions are of infinitely many exist, then any solution is represented as the form y(x¯)=-as-1(x¯)/sas(x¯)+λp(x¯) where λ is arbitrary and p(x¯)=(f(x¯)/as(x¯))1/s is also a factor of f(x¯), provided the equation (⋆) has infinitely many quasi-coincidence (point) solutions.


---

## Body

## 1. Introduction

In 1987, Lenstra [1] researched a polynomial function F(x,y)∈ℚ(α)  [x,y]  (α is an algebraic number) and attempted to search the factorization of F(x,y). Continuing his job, many scientists tried to find the roots of the polynomial equations (cf. [2–6]). Later, many authors also studied fixed point theory and fixed coincidence theory (cf. [7–11]). Recently, Lai and Chen ([12–15]) research the quasi-fixed (point) polynomial problem; they assumed F:ℝn×ℝ→ℝ a polynomial function and solved y(x)∈ℝ[x] to satisfy the polynomial equation as the form
(1)F(x,y(x))=apm(x),x∈ℝ,
where a∈ℝ, p(x) is an irreducible polynomial in ℝ[x], and the polynomial function F(x,y) is written by
(2)F(x,y)=∑i=0sai(x)yiwiths≥1,
where s=degy  F denotes the degree of y in F(x,y).Definition 1 (Lai and Chen [12]).
A polynomial functiony=y(x) satisfying (1) is called a quasi-fixed solution corresponding to some real number a. This number a is called a quasi-fixed value corresponding to the polynomial solutions y=y(x).Moreover, Chen and Lai [16] extended (1) to a more general coincidence (point) problem in which the f(x)∈ℝ[x] is replaced by the irreducible polynomial power pm(x)∈ℝ[x], where f(x) is an arbitrary polynomial. Then we restate (1) as the following equation:
(3)F(x,y)=af(x).
It is a new development coincidence point-like problem. We call the polynomial solution y=y(x) for (3) a quasi-coincidence (point) solution. Precisely, we give the following definition like Definition 1.Definition 2 (Chen and Lai [16]).
A polynomial functiony=y(x) satisfying (3) is called a quasi-coincidence (point) solution corresponding to some real number a. This number a is called a quasi-coincidence value corresponding to the polynomial solutions y=y(x).Furthermore, we consider a multivariate polynomial functionF:ℝn×ℝ→ℝ and extend (3) as a more general coincidence (point) problem in which the x¯=(x1,x2,…,xn) is replaced by x throughout this paper, where f(x¯) is a nonzero arbitrary polynomial in ℝ[x¯]. Then we restate (3) as the following equation:
(4)F(x¯,y)=af(x¯).
Thus, we can give some definitions like Definition 2 as follows.Definition 3.
A polynomial functiony=y(x¯) satisfying (4) is called a quasi-coincidence (point) solution corresponding to some real number a. This number a is called a quasi-coincidence value corresponding to the polynomial solutions y=y(x¯).The number of all solutions in (4) may be infinitely many, finitely many, or not solvable. In this paper, we solve all solutions of (4) if the number is infinitely many. Moreover, we provide an upper bound for the number of all solutions if the number is finitely many.In Section2, we derive some properties of quasi-coincidence solutions. If (4) has infinitely many quasi-coincidence solutions, the form of F(x¯,y) will be described in Section 3. In the last section, we solve all solutions if (4) has infinitely many solutions.


## 2. Preliminaries

For convenience, we denote the polynomial function by(5)F(x¯,y)=as(x¯)ys+as-1(x¯)ys-1+⋯+a1(x¯)y+a0(x¯)=∑i=0sai(x¯)yi
throughout this paper and since there may exist many solutions corresponding to the same number a, we use the similar notations like (Definition  2, [11]) to represent them.Notation 1.
(1)QcsF, the set of all solutions satisfying equation (4), the solution in QcsF is also called a quasi-coincidence solution in (4) (like Definition 2).
(2)QcvF, the set of all solutions a satisfying equation (4), the solution in QcvF is also called a quasi-coincidence value in (4) (like Definition 2).
(3)QcsF(a), the set of all quasi-coincidence solutions y(x¯) corresponding to a quasi-coincidence value a.
(4) For eacha∈ℝ, we denote |QcsF(a)| as the cardinal number of QcsF(a).Evidently, by Notation1, we have the following lemma.Lemma 4.
(i)QcsF=⋃a∈QcvF‍QcsF(a);
(ii)QcsF(a)⋂QcsF(b)=∅ for any a≠b in QcvF;
(iii)|QcsF(a)|≤degyF(x¯,y) for any a∈ℝ;
(iv)|QcsF|≤|QcvF||QcsF(a)| for any a∈QcvF.Proof.
(i)⋃a∈QcvFQcsF(a)⊆QcsF is obvious.
Conversely, for anyy(x¯)∈QcsF, by Notation 1(1), we have
(6)F(x¯,y(x¯))=af(x¯)
for some a∈QcvF. This means y(x¯)∈QcsF(a) and we obtain
(7)QcsF⊆⋃a∈QcvFQcsF(a).
(ii) Leta≠b in QcvF; if there exists y(x¯)∈ℝ[x¯] such that
(8)y(x¯)∈QcsF(a)⋂QcsF(b),
by Notation 1(3), we have
(9)af(x¯)=F(x¯,y(x¯))=bf(x¯).
This leads a contradiction to a≠b and we have QcsF(a)⋂QcsF(b)=∅.
(iii) For eacha∈ℝ, the number of all solutions y=y(x¯) to the polynomial equation F(x¯,y)-af(x¯)=0 is at most degy(F(x¯,y)-af(x¯))=degyF(x¯,y); then the result is obtained.
(iv) By (i), we have(10)QcsF=⋃a∈QcvFQcsF(a).
It follows that
(11)|QcsF|=|⋃a∈QcvFQcsF(a)|≤|QcvF||QcsF(a)|,foranya∈QcvF.In the following lemma, we explain some interesting properties of the relations of quasi-coincidence point solutions. Throughout this paper, we consider (4) for polynomial function (5) and nonzero arbitrary polynomial f(x¯) in ℝ[x¯].Lemma 5.
Let the cardinal number|QcvF|≥2 and a≠b in QcvF. Then for any y1(x¯)∈QcsF(a) and y2(x¯)∈QcsF(b), one has
(12)y1(x¯)-y2(x¯)=dp(x¯)forsomed∈ℝ,
and this p(x¯) is a factor of f(x¯), that is, p(x¯)∣f(x¯).Proof.
Sincey1(x¯), y2(x¯) correspond to a, b, respectively, we have
(13)F(x¯,y1(x¯))=af(x¯),(14)F(x¯,y2(x¯))=bf(x¯).
Subtracting (14) from (13) and using binomial formula, it yields that
(15)(a-b)f(x¯)=F(x¯,y1(x¯))-F(x¯,y2(x¯))=as(x¯)[y1s(x¯)-y2s(x¯)]+as-1(x¯)[y1s-1(x¯)-y2s-1(x¯)]+⋯+a1(x¯)[y1(x¯)-y2(x¯)]=[y1(x¯)-y2(x¯)]×[as(x¯)Gs(y1(x¯),y2(x¯))+as-1(x¯)Gs-1(y1(x¯),y2(x¯))+⋯+a1(x¯)[as(x¯)Gs(y1(x¯),y2(x¯))]=[y1(x¯)-y2(x¯)]Q(x,y1(x¯),y2(x¯)),
where
(16)Gj(y1(x¯),y2(x¯))=y1j-1(x¯)+y1j-2(x¯)y2(x¯)+⋯+y2j-1(x¯),
for j=s, s-1,…,2,1. Evidently, the factor y1(x¯)-y2(x¯) is divisible to the term (a-b)f(x¯) and since a≠b, we obtain
(17)y1(x¯)-y2(x¯)=dp(x¯)
for some real number d∈ℝ and factor p(x¯) of f(x¯).In Lemma5, the difference of any two distinct quasi-coincidence solutions corresponding to distinct values is a factor of f(x¯). Thus we may define a class of those factors in the following.Notation 2.
(i) DenoteΦ(p(x¯))={αp(x¯):α∈ℝ}.
(ii) Lety(x¯) be an arbitrary polynomial in ℝ[x¯], and we denote y(x¯)+Φ(p(x¯))={y(x¯)+αp(x¯):α∈ℝ}.
(iii)degf(x¯)=∑i=1n‍degxif(x¯).Ify1(x), y2(x)∈QcsF correspond to distinct quasi-coincidence values, by Lemma 5 and Notation 2, we have
(18)y1(x)-y2(x)∈⋃p(x¯)∣f(x¯)Φ(p(x¯)).Since the number of all factorsp(x¯) to f(x¯) is at most 2degf(x¯), by the definitions of “the pigeonhole principle” in [17], we have the following results.Lemma 6.
Suppose that(19)|QcvF|≥k·2degf(x¯)+2.
Then there exists y(x¯)∈QcsF and p(x¯) is a factor of f(x¯) such that
(20)|(y(x¯)+Φ(p(x¯)))∩QcsF|≥k.Proof.
Since|QcvF|≥k2degf(x¯)+2, by (18), there exists y1(x¯), y2(x¯),…,y{k2degf(x¯)+2}(x¯)∈QcsF such that
(21)yj(x¯)-y1(x¯)∈Φ(pj(x¯))
for some factor pj(x¯) of f(x¯), for j=2,3,…,k2degf(x¯)+2. Moreover, we have that the number of all factors to f(x¯) is at most 2degf(x¯). By “the pigeonhole principle,” there exists a subset {yji(x¯)}i=1k⊆{yj(x¯)}j=2k2degf(x¯)+2 such that
(22)yji(x¯)-y1(x¯)∈Φ(p(x¯)),
for i=1,2,…,k and some factor p(x¯) of F(x¯) (this p(x¯) is independent of the choice of i). Thus
(23){yji(x¯)}i=1k⊆(y1(x¯)+Φ(p(x¯)))∩QcsF,|(y1(x¯)+Φ(p(x¯)))∩QcsF|≥k.For convenience, we explain the relations ofQcsF and Φ(p(x¯)) in the following lemma.Lemma 7.
Lety(x¯)∈QcsF(a) for some a∈ℝ. Then
(24)QcsF=QcsF(a)⋃(⋃p(x¯)∣f(x¯)‍{(y(x¯)+Φ(p(x¯)))∩QcsF})
for some factor p(x¯) of f(x¯).Proof.
For anyy1(x¯)∈QcsF∖QcsF(a), then y1(x¯)∈Qcs(b) for some b∈QcvF. By Lemma 5, we have
(25)y1(x¯)-y(x¯)∈Φ(p(x¯))
for some factor p(x¯) of f(x¯). Then
(26)y1(x¯)∈⋃p(x¯)∣f(x¯){y(x¯)+Φ(p(x¯))}
and it follows that
(27)QcsF⊆QcsF(a)⋃(⋃p(x¯)∣f(x¯)‍{y(x¯)+Φ(p(x¯))}).
Moreover, by Lemma 4(i), QcsF(a)⊆QcsF; then we obtain
(28)QcsF=QcsF(a)⋃‍(⋃p(x¯)∣f(x¯)‍{(y(x¯)+Φ(p(x¯)))∩QcsF}).In order to let the number of all elements in the intersection of setsy1(x¯)+Φ(p1(x¯)) and y2(x¯)+Φ(p2(x¯)) be large enough, we find a lower bound for |QcvF| in the following theorem.Theorem 8.
Suppose that the cardinal number(29)|QcvF|≥(2degf(x¯)+s+3)(2degf(x¯))+2.
Then for any y1(x¯)≠y2(x¯)∈QcsF, there exist two factors p1(x¯) and p2(x¯) of f(x¯) such that
(30)|(y1(x¯)+Φ(p1(x¯)))∩(y2(x¯)+Φ(p2(x¯)))∩QcsF-{y1(x¯),y2(x¯)}|(y1(x¯)+Φ(p1(x¯)))∩(y2(x¯)+Φ(p2(x¯)))∩QcsF|≥2.Proof.
Lety1(x¯)∈QcsF and by assumption
(31)|QcvF|≥(2degf(x¯)+s+3)(2degf(x¯))+2,
and by Lemma 6, there exists a factor p1(x¯) of f(x¯) such that
(32)|(y1(x¯)+Φ(p1(x¯)))∩QcsF|≥2degf(x¯)+s+3.
This implies that
(33)|(y1(x¯)+Φ(p1(x¯)))∩QcsF-{y1(x¯)}|≥2degf(x¯)+s+2.
Moreover, for any y2(x¯)∈QcsF, we have
(34)(y1(x¯)+Φ(p1(x¯)))∩QcsF-{y1(x¯)}⊆(y1(x¯)+Φ(p1(x¯)))∩QcsF⊆QcsF(byLemma7)=QcsF(b)⋃‍(⋃p(x¯)∣f(x¯)‍(y2(x¯)+Φ(p(x¯)))∩QcsF),
for some constant b∈ℝ and it follows that
(35)2degf(x¯)+s+2=|(y1(x¯)+Φ(p1(x¯)))∩QcsF-{y1(x¯)}|≤|(y1(x¯)+Φ(p1(x¯)))∩QcsF|≤|QcsF(b)⋃‍(⋃p(x¯)∣f(x¯)‍(y2(x¯)+Φ(p(x¯)))∩QcsF)|≤|QcsF(b)|+∑p(x¯)∣f(x¯)‍|(y2(x¯)+Φ(p(x¯)))∩QcsF|(byLemma4(iii))≤s+∑p(x¯)∣f(x¯)|y2(x¯)+Φ(p(x¯))∩QcsF|.
Canceling both sides of the above inequality by “s”, it follows that
(36)2degf(x¯)+2≤∑p(x¯)∣f(x¯)|y2(x¯)+Φ(p(x¯))∩QcsF|,
this implies that
(37)2degf(x¯)+1≤∑p(x¯)∣f(x¯)|y2(x¯)+Φ(p(x¯))∩QcsF-{y2(x¯)}|;
By the pigeonhole’ principle and since the number of all factors p(x¯) to f(x¯) is at most 2degf(x¯), we have
(38)|(y2(x¯)+Φ(p2(x¯)))∩QcsF-{y2(x¯)}|≥2,|(y2(x¯)+Φ(p2(x¯))-{y2(x¯)})∩QcsF|(y1(x¯)+Φ(p1(x¯))-{y1(x¯)})∩(y2(x¯)+Φ(p2(x¯))-{y2(x¯)})∩QcsF|≥2
for some factor p2(x¯) of f(x¯). Thus we obtain
(39)|(y1(x¯)+Φ(p1(x¯)))∩(y2(x¯)+Φ(p2(x¯)))∩QcsF-{y1(x¯),y2(x¯)}|(y1(x¯)+Φ(p1(x¯)))∩(y2(x¯)+Φ(p2(x¯)))|≥2.Up to now, we have not shown that the factorp(x¯) uniquely existed eventually. In the following theorem, we would show the uniqueness property for the factor p(x¯) of f(x¯) if the number of all quasi-coincidence values is large enough.Theorem 9.
Assume that the cardinal number(40)|QcvF|≥(2degf(x¯)+s+3)(2degf(x¯))+2.
Then for any y1(x¯),y2(x¯)∈QcsF, one has
(41)y1(x¯)-y2(x¯)=λp(x¯),
where λ∈ℝ and p(x¯) is a factor of f(x¯) (this p(x¯) is independent of the choice of y1(x¯) and y2(x¯)).Proof.
Lety1(x¯)≠y2(x¯)∈QcsF; by Theorem 8, we have
(42)|(y1(x¯)+Φ(p1(x¯)))∩(y2(x¯)+Φ(p2(x¯)))∩QcsF-{y1(x¯),y2(x¯)}|(y1(x¯)+Φ(p1(x¯)))∩(y2(x¯)+Φ(p2(x¯)))|≥2
for some factors p1(x¯) and p2(x¯) of f(x¯). There exists g1(x¯)≠g2(x¯)∈QcsF, such that
(43)g1(x¯),g2(x¯)∈(y1(x¯)+Φ(p1(x¯)))∩(y2(x¯)+Φ(p2(x¯)))∩QcsF-{y1(x¯),y2(x¯)}.
This implies that
(44)g1(x¯)∈y1(x¯)+Φ(p1(x¯)),g2(x¯)∈y1(x¯)+Φ(p1(x¯)),g1(x¯)∈y2(x¯)+Φ(p2(x¯)),g2(x¯)∈y2(x¯)+Φ(p2(x¯)).
By Notation 2(ii), it yields that
(45)g1(x¯)-y1(x¯)=λ1p1(x¯),g2(x¯)-y1(x¯)=λ2p1(x¯),g1(x¯)-y2(x¯)=λ3p2(x¯),g2(x¯)-y2(x¯)=λ4p2(x¯)
for some constants λ1, λ2, λ3, and λ4∈ℝ and consequently
(46)g2(x¯)-g1(x¯)=(g2(x¯)-y1(x¯))-(g1(x¯)-y1(x¯))=(λ2-λ1)p1(x¯),g2(x¯)-g1(x¯)=(g2(x¯)-y2(x¯))-(g1(x¯)-y2(x¯))=(λ4-λ3)p2(x¯).
This implies that
(47)(λ2-λ1)p1(x¯)=(λ4-λ3)p2(x¯),
and p1(x¯)=p2(x¯). Therefore,
(48)y1(x¯)-y2(x¯)=(g1(x¯)-y2(x¯))-(g1(x¯)-y1(x¯))=(λ3-λ1)p1(x¯),
and this means that the factor p(x¯) of f(x¯) is uniquely determined independent of the choice of y1(x¯) and y2(x¯).Corollary 10.
Assume that the cardinal number(49)|QcsF|≥degyF(x¯,y)((2degf(x¯)+s+3)(2degf(x¯))+2).
Then for any y(x¯)∈QcsF, there exists h(x¯),p(x¯)∈ℝ[x¯] such that
(50)y(x¯)=h(x¯)+λp(x¯)
for some λ∈ℝ  (h(x¯),p(x¯) are independent of the choice of y(x¯)).Proof.
By Lemma4(iv), we have
(51)|QcsF|≤|QcvF||QcsF(a)|,foranya∈QcvF(byLemma4(iii))HHH≤degyF(x¯,y)|QcvF|.
By assumption, it follows that
(52)degyF(x¯,y)((2degf(x¯)+s+3)(2degf(x¯))+2)≤degyF(x¯,y)|QcvF|.
Dividing both sides of the above equation by degyF(x¯,y), we get
(53)|QcvF|≥((2degf(x¯)+s+3)(2degf(x¯))+2).
If h(x¯)∈QcsF, for any y(x¯)∈QcsF, by Theorem 9, we have
(54)y(x¯)=h(x¯)+dp(x¯)
for some factor p(x¯) of f(x¯).


## 3. The Type ofF(x¯,y) If the Number of All Quasi-Coincidence Solutions Is Infinitely Many

In this section, we consider (4) for polynomial function F(x¯,y) in (5); that is, let
(55)F(x¯,y)=∑i=0sai(x¯)yiwiths≥2.f(x¯)∈ℝ[x¯] and we assume that F(x¯,y) has at least s+1 distinct quasi-coincidence solutions satisfying some conditions, that is, y1(x¯),y2(x¯),y3(x¯),…,ys+1(x¯),… in the following theorem. According to the above assumptions, we could derive the following result.Theorem 11.
Suppose that the cardinal number|QcsF|≥s+1 and for each y(x¯)∈QcsF can be represented as the form
(56)y(x¯)=y1(x¯)+λip(x¯),λi∈ℝ,
for some y1(x¯),p(x¯)∈ℝ[x¯] and λi∈ℝ, for i=1,2,…,s+1,…. Then ps(x¯)∣f(x¯) and the polynomial F(x¯,y) can be represented as
(57)F(x¯,y)=∑i=0scif(x¯)pi(x¯)(y-y1(x¯))i
for constants ci∈ℝ, i=0,1,…,s.Proof.
Letyi(x¯) be distinct quasi-coincidence solutions of F(x¯,y) corresponding to quasi-coincidence values ai, 1≤i≤s+1, such that
(58)F(x¯,yi(x¯))=aif(x¯).
Choose i=1, F(x¯,y1(x¯))=a1f(x¯). When y-y1(x¯) divides the function F(x¯,y), we get
(59)F(x¯,y)=(y-y1(x¯))F1(x¯,y)+a1f(x¯),
where F1(x¯,y) is the quotient and a1f(x¯) is the remainder. From the above identity, take y=y2(x¯), and it becomes
(60)F(x¯,y2(x¯))=(y2(x¯)-y1(x¯))F1(x¯,y2(x¯))+a1f(x¯)=a2f(x¯).
Then
(61)(y2(x¯)-y1(x¯))F1(x¯,y2(x¯))=(a2-a1)f(x¯).
By (56), y2(x¯)-y1(x¯)=λ2p(x¯), it yields that
(62)F1(x¯,y2(x¯))=((a2-a1)λ2)f(x¯)p(x¯)=d2f(x¯)p(x¯)∈ℝ[x¯]ford2=(a2-a1)λ2.
Hence
(63)F1(x¯,y)=(y-y2(x¯))F2(x¯,y)+d2f(x¯)p(x¯).
Continuing this process from i=2 to s-1, we obtain
(64)Fi(x¯,y)=(y-yi+1(x¯))Fi+1(x,y)+di+1f(x¯)pi(x¯),
for some di+1∈ℝ, i=1,2,…,s-1. Finally, we could get
(65)Fs-1(x¯,y)=(y-ys(x¯))Fs(x¯)+dsf(x¯)ps-1(x¯).Fs(x¯) does not contain the variable y since degyF=s. By the assumption (58), F(x¯,ys+1(x¯))=as+1f(x¯). It follows that
(66)Fs(x¯)=λf(x¯)ps(x¯)∈ℝ[x¯],forsomeconstantλ∈ℝ.
Consequently,
(67)F(x¯,y)=(y-y1(x¯))F1(x¯,y)+a1f(x¯)=(y-y1(x¯))((y-y2(x¯))F2(x¯,y)+d2f(x¯)p(x¯))+a1f(x¯)=⋯=(y-y1(x¯))×(d2f(x¯)p(x¯))(y-y2(x¯))×(⋯((y-ys(x¯))Fs(x¯)+dsf(x¯)ps-1(x¯))⋯)+d2f(x¯)p(x¯))+a1f(x¯)=(y-y1(x¯))×(d2f(x¯)p(x¯))(y-y2(x¯))×(⋯(dsf(x¯)ps-1(x¯))(y-ys(x¯))λf(x¯)ps(x¯)+dsf(x¯)ps-1(x¯))⋯)+d2f(x¯)p(x¯))+a1f(x¯).
By (56), we have yi(x¯)=y1(x¯)+λip(x¯), i=2,3,…,s+1. Then F(x¯,y) can be expanded to a power series in the expression
(68)F(x¯,y)=(y-y1(x¯))×(dsf(x¯)ps-1(x¯))⋯)+d2f(x¯)p(x¯))(y-y1(x¯)-λ2p(x¯))×(⋯(dsf(x¯)ps-1(x¯))(y-y1(x¯)-λsp(x¯))λf(x¯)ps(x¯)+dsf(x¯)ps-1(x¯))⋯)+d2f(x¯)p(x¯))+a1f(x¯)=∑i=0scif(x¯)pi(x¯)(y-y1(x¯))i,
for some real numbers cj, j=0,1,…,s. Moreover, the leading coefficient of F(x¯,y), cs(f(x¯)/ps(x¯)) is contained to ℝ[x¯], and it follows ps(x¯)∣f(x¯).In the above theorem, if there exist at leasts+1 quasi-coincidence solutions with some relations, then F(x¯,y) has a fixed type. In the following theorem, if F(x¯,y) has a fixed type expressed as in Theorem 11, then the cardinal number |QcsF|=∞.Theorem 12.
The following three conditions are equivalent:(i)
F(x¯,y)=∑i=0s‍ci(f(x¯)/pi(x¯))(y-y1(x¯))i for some y1(x¯)∈ℝ[x¯], some factor p(x¯) of f(x¯) and ci∈ℝ, i=0,1,…,s;(ii)
|QcsF|=∞;(iii)
|QcvF|≥(2degf(x¯)+s+3)·2degf(x¯)+2.  (In fact, if |QcsF|=∞, then |QcsF|= the cardinal number of ℝ).Proof.
(
i
)
⇒
(
ii
) Suppose that (i) holds. Let y=y1(x¯)+λp(x¯) for any constant λ∈ℝ; we have
(69)F(x¯,y1(x¯)+λp(x¯))=∑i=0scif(x¯)pi(x¯)(λp(x¯))i=(∑i=0s‍ciλi)f(x¯)=cpm(x¯)forc=∑i=0sciλi∈ℝ.
This means that y1(x¯)+λp(x¯)∈QcsF  for all λ∈ℝ and we obtain
(70)∞=|{y1(x¯)+λp(x¯)}λ∈ℝ|≤|QcsF|.
It follows that the cardinal number |QcsF|=∞.
(
ii
)
⇒
(
iii
) can be obtained obvious from Lemma 4(iv).
(
iii
)
⇒
(
i
) For any y(x¯),y1(x¯)∈QcsF, by Theorem 9, we have
(71)y(x¯)-y1(x¯)=dp(x¯),
for some fixed factor p(x¯) of f(x¯) and by Theorem 11, we obtain
(72)F(x¯,y)=∑i=0scif(x¯)pi(x¯)(y-y1(x¯))i,
for some y1(x¯) and ci∈ℝ, i=0,1,…,s.Corollary 13.
If the number of all quasi-fixed solutions is finitely many, the number of all quasi-fixed values does not exceed an integerℓ. Actually,
(73)ℓ=(2degf(x¯)+s+3)·2degf(x¯)+1.Proof.
By the contrapositive of Theorem12(ii)⇒(iii), we have “if |QcsF|<∞, then |QcvF|<(2degf(x¯)+s+3)·2degf(x¯)+2.” Hence the number of all quasi-fixed values is at most ℓ; that is, ℓ≤(2degf(x¯)+s+3)·2degf(x¯)+1.Corollary 14.
If the number of all quasi-fixed solutions is finitely many, the number of all quasi-fixed solutions does not exceed(74)degyF(x¯,y)((2degf(x¯)+s+3)·2degf(x¯)+1).Proof.
By Lemma4(iv), we have for any a∈QcvF(75)|QcsF|≤|QcsF(a)||QcvF|(byLemma4(iii)andCorollary13)≤degyF(x¯,y)((2degf(x¯)+s+3)·2degf(x¯)+1).


## 4. Main Theorems and Some Corollaries

If theF(x¯,y) can be represented as the form (57), then any quasi-coincidence solution can be formed in this section.Lemma 15.
LetF(x¯,y) be represented as in (57). Then h(x¯)∈ℝ[x¯] is a quasi-coincidence solution of F(x¯,y) if and only if
(76)h(x¯)=y(x¯)+dp(x¯)
for some d∈ℝ and some factor p(x¯) of f(x¯).Proof.
Since(77)F(x¯,y)=∑i=0scif(x¯)pi(x¯)(y-y(x¯))i,
we let y=y(x¯), and then
(78)F(x¯,y(x¯))=∑i=0s‍cif(x¯)pi(x¯)(y(x¯)-y(x¯))i=c0f(x¯);
this means y(x¯)∈QcsF.
By Theorem12,
(79)F(x¯,y)=∑i=0scif(x¯)pi(x¯)(y-y(x¯))i,then|QcsF|=∞.
Assume that h(x¯) is a quasi-coincidence solution of F(x¯,y) and by Corollary 10, we obtain that for any quasi-coincidence solution h(x¯), we have
(80)h(x¯)=y(x¯)+dp(x¯),forsomed∈ℝ.
Conversely, suppose h(x¯)=y(x¯)+dp(x¯), for some factor p(x¯) of f(x¯) and d∈ℝ. Substituting this h(x¯) as y in (57), we have
(81)F(x¯,h(x¯))=F(x¯,y(x¯)+dp(x¯))=∑i=0scif(x¯)pi(x¯)(dp(x¯))i=(∑i=0scidi)f(x¯).
Therefore h(x¯)∈QcsF.Note that not any polynomial functionF(x¯,y) can be written as (57). Actually, almost all F(x¯,y) are expressed as the form of the next theorem. In this situation, any solution can be written as the next form (*) in this theorem under some condition.Theorem 16.
LetF(x¯,y) be a polynomial function with
(82)F(x¯,y)=as(x¯)ys+as-1(x¯)ys-1+⋯+a0(x¯),
and f(x¯) a polynomial. If the cardinal number |QcsF| is infinitely many, then each quasi-coincidence solution of (4) must be of the form
(*)-as-1(x¯)sas(x¯)+λp(x¯)
for arbitrary λ∈ℝ, where p(x¯)=(f(x¯)/as(x¯))1/s is a factor of f(x¯).Proof.
Assume|QcsF|=∞. By Theorem 12, we have
(83)F(x¯,y)=as(x¯)ys+as-1(x¯)ys-1+⋯+a0(x¯)=∑i=0scif(x¯)pi(x¯)(y-y(x¯))i,
for some ci∈ℝ, i=0,…,s, and y(x¯)∈QcsF. Comparing the coefficients of ys and ys-1 in both sides of the above equation, we get
(84)as(x¯)=csf(x¯)ps(x¯),as-1(x¯)=-sas(x¯)y(x¯)+cs-1f(x¯)ps-1(x¯).
Consequently, by (84), we get
(85)ps(x¯)=csf(x¯)as(x¯),y(x¯)=cs-1scsp(x¯)-as-1(x¯)sas(x¯)∈ℝ[x¯].
By Lemma 15, for any d∈ℝ, we have that any quasi-coincidence solution is represented by
(86)y(x¯)+dp(x¯)=cs-1scsp(x¯)-as-1(x¯)sas(x¯)+dp(x¯)=-as-1(x¯)sas(x¯)+(d-cs-1scs)p(x¯)=-as-1(x¯)sas(x¯)+λp(x¯),
where p(x¯)=(cs)1/s(f(x¯)/as(x¯))1/s (note that since d is arbitrary, then λ is arbitrary).
This completes the proof.Corollary 17.
LetF(x¯,y) be a polynomial function with
(87)F(x¯,y)=as(x¯)ys+as-1(x¯)ys-1+⋯+a0(x¯).
If the cardinal number |QcsF| of equation
(88)F(x¯,y)=a
is infinitely many, then the leading coefficient as(x¯) must be a real number and each quasi-coincidence point solution must be of the form
(★)λ-as-1(x¯)sas(x¯)
for arbitrary λ∈ℝ.Corollary 18.
LetF:ℝ×ℝ→ℝ be a polynomial function with
(89)F(x,y)=as(x)ys+as-1(x)ys-1+⋯+a0(x).
If the cardinal number |QcsF| of equation
(90)F(x,y)=ax
is infinitely many, then the leading coefficient as(x) must be a real number and each quasi-coincidence point solution must be of the form
(★)λ1+λ2as-1(x)x
for arbitrary λ1 and some λ2∈ℝ.Proof.
Assume that there exist infinitely many solutions; by Theorem16, any solution of (4) has the form
(91)-as-1(x)sas(x)+λp(x)
for arbitrary λ∈ℝ and p(x)=c(x/as(x))1/s∈ℝ[x], and then as(x) is a factor of x. This means that as(x¯)∈ℝ or as(x¯)=kx for some constant k∈ℝ, if as(x)∈ℝ; this implies p(x)=c(x/as(x))1/s∉ℝ[x], and this leads a contradiction. So we have as(x)=kx, for some k∈ℝ; then p(x)=c/k1/s∈ℝ and any solution (91) is represented as
(92)-as-1(x)sas(x)+λp(x)=λ1+λ2as-1(x)x
for arbitrary λ1=λc/k1/s and some λ2=-1/sk∈ℝ.Finally, we provide one example to explain Theorem16.Example 19.
Letx¯=(x1,x2)∈ℝ2, f(x¯)=(x1+x2)2, and
(93)F(x¯,y)=a2(x¯)y2+a1(x¯)y+a0(x¯)=y2-(2x1x2-x1-x2)y+(x12x22-x12x2-x1x22+x12+2x1x2+x22).
Can we solve all quasi-fixed solutions of F(x¯,y)=af(x¯)? This polynomial function has exactly 5(≥s+3, since s=2) quasi-fixed solutions as follows:
(94)F(x1,x2,x1x2-x1-x2)=1(x1+x2)2,F(x1,x2,x1x2)=1(x1+x2)2,F(x1,x2,x1x2+x1+x2)=3(x1+x2)2,F(x1,x2,x1x2+2x1+2x2)=3(x1+x2)2,F(x1,x2,x1x2+x12+x22)=34(x1+x2)2.
In fact, by Theorem 16, we can find any quasi-coincidence solution written as
(95)-a1(x¯)sa2(x¯)+λp(x¯)=2x1x2-x1-x22+λp(x¯)=x1x2+(λ-12)(x1-x2)=x1x2+μ(x1-x2),
where μ=λ-1/2∈ℝ is arbitrary and p(x¯)=(f(x)/as(x))1/s=x1+x2. This shows the quasi-coincidence (point) solutions have cardinal |QsF|=∞.
In practice, we have no idea to check the number of this equation is infinitely many or finitely many. But we provide an easy method to solve all solutions if the number of all solutions is infinitely many in this paper. Thus, we can solve those solutions directly and check whether those solutions are the solutions ofF(x¯,y)=af(x¯) and give an example in the following.Example 20.
Letx¯=(x1,x2)∈ℝ2, f(x¯)=x12(x2+x3)2, and
(96)F(x¯,y)=a2(x¯)y2+a1(x¯)y+a0(x¯)=y2-(2x1x2-x1(x2+x3))y+(x12x22+x12x2x3+x12x32).
We will solve all quasi-fixed solutions of F(x¯,y)=af(x¯) if the number of all solutions is infinitely many.
By Theorem16, we can find that any quasi-coincidence solution can be written as
(97)-a1(x¯)sa2(x¯)+λp(x¯)=2x1x2-x1(x2+x3)2+λp(x¯)=x1x2+μx1(x2+x3),
where μ=λ-1/2∈ℝ is arbitrary and p(x¯)=(f(x)/as(x))1/s=x1(x2+x3). We let y(x¯)=x1x2+μx1(x2+x3) and calculate F(x,y(x¯)) and obtain
(98)F(x,y(x¯))=(μ2+μ+1)x12(x2+x3)2.
This means that the quasi-coincidence (point) solutions have cardinal |QsF|=∞.We would like to provide one open problem as follows.Further Development. Let ℚ be a quotient field. Consider a quotient-valued polynomial function
(99)F:ℚn×ℚ⟶ℚ.
Can we find all quasi-coincidence solutions y=y(x¯)∈ℚ[x¯] to satisfy
(100)F(x¯,y)=af(x¯)
for some polynomials f(x¯)∈ℚ[x¯] by a co-NP hardness algorithm?



---

*Source: 307913-2013-12-31.xml*