# Polynomials in Control Theory Parametrized by Their Roots

Authors: Baltazar Aguirre-Hernández; José Luis Cisneros-Molina; Martín-Eduardo Frías-Armenta
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

---

## Abstract

The aim of this paper is to introduce the space of roots to study the topological properties of the spaces of polynomials. Instead of identifying a monic complex polynomial with the vector of its coefficients, we identify it with the set of its roots. Viète's map gives a homeomorphism between the space of roots and the space of coefficients and it gives an explicit formula to relate both spaces. Using this viewpoint we establish that the space of monic (Schur or Hurwitz) aperiodic polynomials is contractible. Additionally we obtain a Boundary Theorem.


---

## Body

## 1. Introduction

It is well known that for the stability of a linear systemx˙=Ax it is required that all the roots of the corresponding characteristic polynomial p(t) have negative real part in other words, p(t) is a Hurwitz (stable) polynomial. There are various approaches to decide if a given polynomial is Hurwitz. Maybe the most popular of such methods is the Routh-Hurwitz criterion. Other important approaches are Lienard-Chipart conditions and the Hermite-Biehler Theorem (see Gantmacher [1], Lancaster and Tismenetsky, [2] and Bhattacharyya et al. [3]). On the other hand, to have the stability of a discrete time linear system xn+1=Axn it is necessary that all of the roots of the characteristic polynomial are strictly within the unit disc. A polynomial with this property is named Schur polynomial. Maybe Jury’s test is the most studied criterion for checking if a given polynomial is a Schur polynomial [4], but also there exists the corresponding Hermite-Biehler Theorem for Schur polynomials [5] or we can mention as well the Schur stability test [3]. In addition to the stability of polynomials another important problem is the aperiodicity condition, which consists in obtaining from a (continuous or discrete) stable system a response that has no oscillations or has only a finite number of oscillations. Mathematically this requires that all the roots of the characteristic polynomial p(t) are distinct and on the negative real axis, for the case of continuous systems; and distinct and in the real interval (0,1) for the discrete case. Criteria to decide if a system is aperiodic are given for instance in [6–12]. An important reference where Hurwitz and Schur stability and aperiodicity are studied is the book of Jury [13].However, if a continuous or discrete system is modeling a physical phenomenon then it is affected by disturbances. Consequently it is convenient to think that there are uncertainties in the elements of the matrixA and then there are uncertainties in the coefficients of the polynomial p(t); that is, we have a family of polynomials and we like to know if all of the polynomials are Schur or Hurwitz polynomials (if the family is stable). The study of the stability of various families of polynomials has attracted the attention of a lot of researchers. The most famous result about families of polynomials is without doubt the Kharitonov Theorem which gives conditions for Hurwitz stability of interval polynomials [14]. There exists an analogous result to Kharitonov’s Theorem for Schur polynomials [15] and also results about segments of Schur polynomials (see [16] or [3]). Results on balls of Schur and Hurwitz polynomials can be found in [17]. In the case of Hurwitz polynomials, the stability of segments of polynomials has been studied in [18–21]. The stability of rays and cones of polynomials has been studied in [22, 23]. Other studied families are polytopes of polynomials and here the most important result is the Edge Theorem [24] which says that the stability of a polytope is determined by the stability of its edges. Good references about families of Hurwitz and Schur polynomials are the books of Ackermann [25], Barmish [26], and Bhattacharyya [3]. The problem of robustness for the aperiodicity condition has been worked in [27–30]. About the specific case of intervals of aperiodic polynomials we can mentioned the works of Foo and Soh [31] and Mori et al. [32].To define and study these families, one uses the fact that a real polynomialp(t)=a0+a1t+⋯+antn can be identified with the vector of its coefficients (a0,a1,…,an). Then the set of real polynomials up to degree n can be identified with ℝn+1 and the sets of Schur and Hurwitz polynomials can be seen as sets contained in ℝn+1. Using this approach, several topological and geometric properties of the spaces of Schur and Hurwitz polynomials have been studied. For instance, it is known that the space of Schur polynomials is an open set [3, Theorem 1.3]; it is not a convex set [16] and it is a contractible set [33]. On the other hand, it is known that the space ℋn of Hurwitz polynomials of degree n is an open set (see [3]) and it is not connected, since the coefficients of a Hurwitz polynomial have the same sign ([1]). However, the set of Hurwitz polynomials with positive coefficients, ℋn+, is connected ([3, 34]) and it is a contractible space [35]. Furthermore it is known that ℋn is not convex ([18, 19, 21, 23]).The aim of this paper is to present a The aim of this paper is to present a different viewpoint in the study of topological properties of the spaces of Schur and Hurwitz (aperiodic) polynomials which is more natural since the definition of such polynomials is in terms of their roots. Instead of identifying a monic complex polynomialp(t)=a0+a1t+⋯+an-1tn-1+tn with the vector of its coefficients, we identify it with the set of its roots {z1,…,zn}, where zi∈ℂ and p(zi)=0 for i=1,…,n. It is well known that the space of roots is homeomorphic to the space of the coefficients and Viète’s map gives an explicit formula to relate both spaces.Using this viewpoint we give simple proofs of some known results about the topology of the spaces of Schur and Hurwitz polynomials; for instance, we give a direct proof (see Section6) that the space of Hurwitz polynomials of degree n with positive (resp. negative) coefficients is contractible. Additionally we prove that the space of monic (Schur or Hurwitz) aperiodic polynomials is contractible and we establish a Boundary Theorem; these results have not been reported in control literature. We would like to emphasize that despite Viète’s map and the space of roots are well known in Mathematics, they have not been used explicitly (see Remark 7.2) to prove the aforementioned results and our contribution is to give simpler proofs using them. Compare Theorem 4.3 with [33, Lemma 1] and Theorem 4.4 with [35, Corrolary 4.1.28] and see Section 6 of the present paper.One could also study a broad variety of families of polynomials directly in the space of roots and via Viète’s map  to get the corresponding results in the space of coefficients. In this way we use a topological approach to study the spaces of polynomials. Other works where topological and geometric ideas have been applied in control theory are the papers [36–44]. The rest of the paper is organized as follow: in Section 2 we introduce the concept of Hurwitz or Schur (aperiodic) vector and we give some notation which will be useful in the proofs; in Section 3 we write some known results; in Section 4 we define the space of roots, which is the approach that we suggest in this paper; in Section 5 we establish the main results; in Section 6 some observations about the relation between Schur and Hurwitz polynomials are given; in Section 7 two remarks are included to explain the relation of our paper with other previous works; finally, some final conclusions are established  in Section 8.


## 2. The Spaces of Schur, Hurwitz, and Aperiodic Polynomials

Let𝔽 be either the real or the complex numbers. Consider the set
(2.1)𝒫≤n𝔽={a0+a1t+⋯+antn∣ai∈𝔽}
of polynomials in one variable, of degree less than or equal to n, with coefficients in 𝔽. The set 𝒫≤n𝔽 is a vector space and choosing the monomials {1,t,…,tn-1,tn} as a basis, we can give explicitly an isomorphism between 𝒫≤n𝔽 and 𝔽n+1 identifying the polynomial a0+a1t+⋯+antn with the vector (a0,a1,…,an) in 𝔽n+1. Clearly 𝒫≤n-1𝔽⊂𝒫≤n𝔽 and under the isomorphism 𝒫≤n-1𝔽 corresponds to the hyperplane defined by the equation an=0. Thus the set of polynomials of degree n with coefficients in 𝔽, denoted by 𝒫n𝔽, corresponds to the set {(a0,a1,…,an)∈𝔽n+1∣an≠0}. If we denote by ℳ𝒫n𝔽 the set of monic polynomials of degree n, then it corresponds to the hyperplane in 𝔽n+1 defined by the equation an=1, that is, vectors in 𝔽n+1 of the form (a0,a1,…,an-1,1). Usually we will identify ℳ𝒫n𝔽 directly with 𝔽n, identifying the monic polynomial a0+a1t+⋯+an-1tn-1+tn with the vector [a0,a1,…,an-1]∈𝔽n, we shall use square brackets to avoid confusion with its corresponding vector
(2.2)(a0,a1,…,an-1,1)∈𝔽n+1.Using this isomorphism we can endow𝒫≤nℂ (respectively 𝒫≤nℝ) with the Hermitian (respectively Euclidean) inner product and its induced topology. Also we can think of the inclusion of the set of real polynomials 𝒫≤nℝ in the set of complex polynomials 𝒫≤nℂ as the inclusion of ℝn+1 in ℂn+1.A (real or complex) polynomialp(t)=a0+a1t+⋯+antn is called a Schur polynomial if all its roots are in the open unit disk 𝔻={z∈ℂ∣∥z∥<1}. The polynomial p(t) is called a Schur aperiodic polynomial if all its roots are distinct, real, and in the interval (0,1).A (real or complex) polynomialp(t)=a0+a1t+⋯+antn is called a Hurwitz polynomial if all its roots have negative real part. The polynomial p(t) is called a Hurwitz aperiodic polynomial if all its roots are distinct, real, and negative.Following [34], we call a vector (a0,a1,…,an)∈𝔽n+1   Schur, Schur aperiodic, Hurwitz, or Hurwitz aperiodic if it corresponds, respectively, to a Schur, Schur aperiodic, Hurwitz, or Hurwitz aperiodic polynomial under the aforementioned isomorphism.Let𝒮≤n𝔽 be the sets of Schur vectors in 𝔽n+1 and let  𝒮n𝔽 denote the set of Schur vectors which correspond to Schur polynomials of degree n. Then we have that
(2.3)𝒮≤n𝔽=𝒮n𝔽∪𝒮≤n-1𝔽.Analogously, letℋ≤n𝔽 be the sets of Hurwitz vectors in 𝔽n+1 and let ℋn𝔽 denote the set of Hurwitz vectors which correspond to Hurwitz polynomials of degree n. Then we have that
(2.4)ℋ≤n𝔽=ℋn𝔽∪ℋ≤n-1𝔽.In the same way, let𝒮𝒜≤n𝔽 and ℋ𝒜≤n𝔽 be, respectively, the sets of Schur aperiodic and Hurwitz aperiodic polynomials. Then the previous decompositions restrict to the subsets of aperiodic polynomials
(2.5)𝒮𝒜≤n𝔽=𝒮𝒜n𝔽∪𝒮𝒜≤n-1𝔽,ℋ𝒜≤n𝔽=ℋ𝒜n𝔽∪ℋ𝒜≤n-1𝔽.From the fact that ifa∈𝔽n+1 the polynomials corresponding to a and λa have the same roots for any λ∈𝔽 with λ≠0, we have that if a∈𝒮n𝔽 (resp., a∈ℋn𝔽, a∈𝒮𝒜n𝔽, a∈ℋ𝒜n𝔽), then λa∈𝒮n𝔽 (resp. λa∈ℋn𝔽, λa∈𝒮𝒜n𝔽, λa∈ℋ𝒜n𝔽) for any λ≠0. Therefore, to study the space 𝒮n𝔽 (resp. ℋn𝔽, 𝒮𝒜n𝔽, ℋ𝒜n𝔽) it is enough to study the space of monic Schur (resp. Hurwitz, Schur aperiodic, Hurwitz aperiodic) polynomials of degree n with coefficients in 𝔽, which we will denote by ℳ𝒮n𝔽 (resp. ℳℋn𝔽, ℳ𝒮𝒜n, ℳℋ𝒜n). In the case of ℳ𝒮𝒜n and ℳℋ𝒜n we drop the superscript 𝔽 since the coefficients of a monic (Schur or Hurwitz) aperiodic polynomial are real, because all of its roots are real. If 𝒬n𝔽 is any of 𝒮n𝔽, ℋn𝔽, 𝒮𝒜n𝔽,  or  ℋ𝒜n𝔽, and ℳ𝒬n𝔽 is the corresponding set of monic polynomials, we have that
(2.6)𝒬n𝔽ishomeomorphictoℳ𝒬n𝔽×𝔽*,
where 𝔽*=𝔽-{0}.In the case of real polynomials we can say more. Since we are mainly interested in real polynomials we shall denoteℋnℝ simply by ℋn, 𝒮nℝ simply by 𝒮n, and so forth. As before, if 𝒬n=𝒬nℝ is any of 𝒮n, ℋn, 𝒮𝒜n,  or  ℋ𝒜n, and ℳ𝒬n is the corresponding set of monic polynomials, we have that the space 𝒬n is homeomorphic to the disjoint union of two cylinders over the space of corresponding real monic polynomials ℳ𝒬n, that is
(2.7)𝒬n≅ℳ𝒬n×(-∞,0)∪ℳ𝒬n×(0,∞).For the case of real Hurwitz polynomialsℋn we can give an explicit description of each of such cylinders. The coefficients of a real Hurwitz polynomial have the same sign [1], therefore we can express it as the union of two sets
(2.8)ℋn=ℋn+∪ℋn-,
where ℋn+ and ℋn- are, respectively, the set of real Hurwitz vectors with positive and negative coefficients. If a=(a0,a1,…,an)∈ℋn+, then -a∈ℋn-, that is, ℋn-=-ℋn+. Hence, to study the space ℋn it is enough to study ℋn+ (compare with [34, Proposition 2.1]). In fact, topologically the space ℋn+ is homeomorphic to one of the aforementioned cylinders over the space ℳℋn, that is,
(2.9)ℋn+≅ℳℋn×(0,∞),
geometrically it corresponds to a cone in ℝn+1 over ℳℋn with vertex at the origin (not including the vertex). This is expressed by the following map which maps ℳℋn×(0,∞) diffeomorphically onto ℋn+ in ℝn+1(2.10)ℳℋn×(0,∞)→ℝn+1([a0,a1,…,an-1],λ)↦(λa0,λa1,…,λan-1,λ),
where we identify a monic polynomial with a vector in ℝn as before.Hence we have that(2.11)ℋ≤n=ℋn+∪ℋn-∪ℋ≤n-1.
Analogously,
(2.12)ℋ𝒜≤n=ℋ𝒜n+∪ℋ𝒜n-∪ℋ𝒜≤n-1,
and ℋ𝒜n+≅ℳℋ𝒜n×(0,∞).


## 3. Symmetric Products, Configuration Spaces, and Viète’s Theorems

LetΣn be the symmetric or permutation group of the set {1,2,…,n}. Let X be a topological space and let Xn=X×⋯×X be its nth Cartesian product for n≥1. Consider the action of Σn on Xn given by permutation of coordinates
(3.1)Σn×Xn→Xn,τ·(x1,…,xn)=(xτ(1),…,xτ(n)).
The orbit space of this action
(3.2)Symn(X)=XnΣn,
endowed with the quotient topology, is called the nthsymmetric product of X. The equivalence class of the n-tuple (x1,…,xn) will be denoted by {x1,…,xn}. Notice that an element {x1,…,xn}∈Symn(X) is a set of n elements of X without order.Denote byF(X,n) the set of n-tuples of distinct points in X, that is,
(3.3)F(X,n)={(x1,…,xn)∈Xn∣xi≠xjifi≠j}.
By definition F(X,n)⊂Xn and clearly F(X,n) are invariant under the action of Σn, hence; the orbit space
(3.4)B(X,n)=F(X,n)Σn,
called the nthconfiguration space of X, consists of the sets of n distinct elements of X without order. Also by definition we have that B(X,n)⊂Symn(X).Consider the2-sphere S2 as the Riemann sphere consisting of the complex numbers together with the point at infinity, denoted by ∞. Also consider the complex projective space of dimension n as the quotient space
(3.5)ℂℙn=ℂn+1-{0}~,
identifying (a0,…,an) with (λa0,…,λan) for nonzero λ∈ℂ. We denote the equivalence class of (a0,…,an) using homogeneous coordinates (a0:⋯:an). If as before we identify (a0,…,an)∈ℂn+1 with a polynomial of degree less than or equal to n, then the class (a0:⋯:an) consists of all the polynomials which have the same roots as (a0,…,an). Since in every class (a0:⋯:an) there is a vector which represents a monic polynomial, we can think of ℂℙn as the space of complex monic polynomials of degree less than or equal to n and we shall denote it by ℳ𝒫≤nℂ.A point inSymn(S2) is an unordered n-tuple {z1,…,zn} of complex numbers or ∞. There exists a nonzero polynomial, unique up to a nonzero complex factor, of degree less than or equal to n whose roots are precisely {z1,…,zn}, where we consider ∞ to be a root of the polynomial if its degree is less than n. Considering the coefficients of this polynomial as homogeneous coordinates on ℂℙn we get a map from Symn(S2) to ℂℙn called Viète’s projective map. This map can be written explicitly as follows.Letσkn, k=1,…,n be the elementary symmetric polynomials in n variables σkn:ℂn→ℂ which are defined as follows:
(3.6)σ0n(z1,…,zn)=1,σkn(z1,…,zn)=∑1≤j1<j2<⋯<jk≤n‍zj1zj2⋯zjk,1≤k≤n.
Let τ∈Σn, then we have that
(3.7)σkn(τ·(z1,…,zn))=σkn(zτ(1),…,zτ(n))=σkn(z1,…,zn),
so the polynomials σkn descend to the nth symmetric product Sn(ℂ) giving continuous functions
(3.8)σkn:Symn(ℂ)→ℂ.Viète’s projective map assigns to {z1,…,zn}∈Symn(S2) the class in ℂℙn corresponding to all the complex polynomials with roots {z1,…,zn}. To simplify notation we omit the argument of the polynomials σkl since the superindex l means to evaluate in the l finite roots:
(3.9)ν:Symn(S2)→ℂℙn,{z1,…,zn}↦((-1)nσnn:⋯:-σ1n:σ0n),{z1,…,zn-1,∞}↦((-1)n-1σn-1n-1:⋯:σ0n-1:0),{z1,…,zn-2,∞,∞}↦((-1)n-2σn-2n-2:⋯:σ0n-2:0:0),⋮{∞,…,∞}↦(1:0:0:⋯:0).
We have the following known theorem; see, for instance, [45, Bei. 3.2], [46, Exa. 5.2.4] or [47, Section 10.2  and App. A].Theorem 3.1.
Viète’s projective map is a homeomorphism betweenSymn(S2) and ℂℙn.Proof (Sketch).
Viète’s projective mapν is a continuous bijection from a compact space to a Hausdorff space and therefore a homeomorphism.ClearlySymn(ℂ) is contained in Symn(S2) as the n-tuples {z1,…,zn} which consist only of complex numbers, without ∞’s. We have that under the Viète’s projective map the image of Symn(ℂ) consists of the classes in ℂℙn which corresponds to polynomials of degree exactly n, that is, classes of the form (a0:⋯:an-1:1). Since each class has a monic polynomial as a representative, we can identify the image of Symn(ℂ) with the space ℳ𝒫nℂ of monic polynomials of degree n, which we know is homeomorphic to ℂn.The restrictionν:Symn(ℂ)→ℂn of Viète’s projective map to Symn(ℂ) is given by
(3.10)ν:Symn(ℂ)→ℂn,{z1,…,zn}↦[(-1)nσnn,…,σ2n,-σ1n],
where the vector in ℂn corresponds to the monic complex polynomial with roots {z1,…,zn}. We denote it also by ν and we call it Viète’s map.Hence, as a corollary of Theorem3.1 we get the classical Viète’s TheoremTheorem 3.2.
Viète’s map is a homeomorphism betweenSymn(ℂ) and ℂn.There are results related to Viète’s Theorems which are consequence of a classical theorem by Maxwell [48]; see [49] or [47, Section 10.2 and App. A].


## 4. The Space of Roots

Letℛn denote the nth symmetric product Symn(ℂ). We call ℛn the space of roots of complex polynomials of degree n. Notice as before, that an element of ℛn is a set of n complex numbers {z1,…,zn} without order.What Theorem3.2 says is that to study the topological properties of a subspace of monic complex polynomials, it is equivalent to parametrize them in terms of  their coefficients or in terms of  their  roots.We define the following subspaces ofℛn. (i)
The space ofroots of real polynomials:
(4.1)ℛnℝ={{z1,…,zn}∈ℛn∣z2j=z¯2j-1,j=1,…,kwith2k≤n,zl∈ℝ,2k+1≤l≤n.}.Clearly its image under Viète’s map isℝn which is homeomorphic to the space ℳ𝒫nℝ of monic real polynomials.(ii)
The space ofroots of complex Schur polynomials:
(4.2)𝒮ℛnℂ={{z1,…,zn}∈ℛn∣zi∈𝔻,i=1,…,n}.Its image under Viète’s map is the set of monic complex Schur polynomialsℳ𝒮nℂ.(iii)
The space ofroots of real Schur polynomials:
(4.3)𝒮ℛn=𝒮ℛnℝ=ℛnℝ∩𝒮ℛnℂ.Its image under Viète’s map is the set of monic real Schur polynomialsℳ𝒮n.(iv)
The space ofroots of Schur aperiodic polynomials: let J=(0,1),
(4.4)𝒮𝒜ℛn=B(J,n).Its image under Viète’s map is the set of monic Schur aperiodic polynomialsℳ𝒮𝒜n.(v)
The space ofroots of complex Hurwitz polynomials:
(4.5)ℋℛnℂ={{z1,…,zn}∈ℛn∣ℜ(zi)<0,i=1,…,n}.Its image under Viète’s map is the set of monic complex Hurwitz polynomialsℳℋnℂ.(vi)
The space ofroots of real Hurwitz polynomials:
(4.6)ℋℛn=ℋℛnℝ=ℛnℝ∩ℋℛnℂ.Its image under Viète’s map is the set of monic real Hurwitz polynomialsℳℋn.(vii)
The space ofroots of Hurwitz aperiodic polynomials: denote by ℝ- the negative real axis,
(4.7)ℋ𝒜ℛn=B(ℝ-,n).Its image under Viète’s map is the set of monic Hurwitz aperiodic polynomialsℳℋ𝒜n.As an example of the use of the space of rootsℛn we get simple proofs of some known results about the topology of the spaces ℳ𝒮n𝔽 and ℳℋn𝔽.Proposition 4.1.
The spacesℳ𝒮n𝔽 and ℳℋn𝔽 are open in 𝔽n.Proof.
It is clear from the definition of the spaces𝒮ℛnℂ and ℋℛnℂ that they are open subset of ℛn; therefore under Viète’s homeomorphism ℳ𝒮nℂ and ℳℋnℂ are open in ℂn. On the other hand, 𝒮ℛn and ℋℛn are open in ℛnℝ and therefore ℳ𝒮n and ℳℋn are open in ℝn.Proposition 4.2.
The boundary∂ℳ𝒮n𝔽 consists of all coefficient vectors in 𝔽n which correspond to polynomials with all their roots in 𝔻- and which have at least one root on ∂𝔻.
The boundary∂ℳℋn𝔽 consists of all coefficient vectors in 𝔽n which correspond to polynomials with all their roots in ℂ-- and which have at least one root on ∂ℂ-=iℝ.Proof.
By the definition of the spaces𝒮ℛnℂ, 𝒮ℛn, ℋℛnℂ, and ℋℛn, their boundaries are given by
(4.8)∂𝒮ℛnℂ={{z1,…,zn}∈ℛn∣zi∈𝔻-,i=1,…,n,andatleastoneisin∂𝔻},∂𝒮ℛn=∂𝒮ℛnℂ∩ℛnℝ,∂ℋℛnℂ={{z1,…,zn}∈ℛn∣zi∈ℂ--,i=1,…,n,andatleastoneisiniℝ},∂ℋℛn=∂ℋℛnℂ∩ℛnℝ.
Using the homeomorphism given by Viète’s map we get the proposition.LetI=[0,1], a topological space X is contractible if there exists a homotopy F:X×I→X that starts with the identity and ends with the constant map c(x)=x0, for some x0∈X. Such a homotopy is called a contraction.The following theorem is proved in [33, Lemma 1] (see Remark 7.2).Theorem 4.3.
The spaces𝒮ℛnℂ and 𝒮ℛn are contractible. Therefore the spaces ℳ𝒮nℂ and ℳ𝒮n are contractible.Proof.
The following homotopy gives a contraction ofℛn to the point {0,…,0}(4.9)G:ℛn×I→ℛn,G({z1,…,zn},r)={(1-r)z1,…,(1-r)zn}.
If {z1,…,zn}∈𝒮ℛnℂ, by definition we have that ∥zi∥<1, i=1,…,n and since (1-r)<1, we have that ∥(1-r)zi∥<1 for all r∈I and therefore the contraction G restricts to a contraction G:𝒮ℛnℂ×I→𝒮ℛnℂ proving that 𝒮ℛnℂ is contractible.
For the case{z1,…,zn}∈𝒮ℛn we just need to check that if we have a pair of conjugate roots, say z2j=z¯2j-1, they stay a conjugate pair through all the homotopy, but clearly (1-r)z2j=(1-r)z2j-1¯ for all r∈I. Therefore the contraction F restricts to a contraction F:𝒮ℛn×I→𝒮ℛn proving that 𝒮ℛn is contractible.The following Theorem is indicated in [35, Corrolary 1.4.28, Ex. 13] (see Section 6).Theorem 4.4.
The spacesℋℛnℂ and ℋℛn are contractible. Therefore the spaces ℳℋnℂ and ℳℋn are contractible.Proof.
The following homotopy is a contraction ofℛn to the point {-1,…,-1}:
(4.10)F:ℛn×I→ℛn,F({z1,…,zn},r)={(z1+1)(1-r)-1,…,(zn+1)(1-r)-1}.
If {z1,…,zn}∈ℋℛnℂ, by definition we have that
(4.11)zk=-ak+ibk,ak,bk∈ℝ,ak>0,k=1,…,n.
Hence
(4.12)(zk+1)(1-r)-1=-[ak(1-r)+r]+ibk(1-r),k=1,…,n,
which has negative real part for all r∈I. Therefore the contraction F restricts to a contraction F:ℋℛnℂ×I→ℋℛnℂ proving that ℋℛnℂ is contractible.
For the case when{z1,…,zn}∈ℋℛn we just need to check that if we have a pair of conjugate roots, say z2j-1=-a+ib and z2j=-a-ib, they stay a conjugate pair through all the homotopy. From (4.12) we have that
(4.13)(z2j-1+1)(1-r)-1=-[a(1-r)+r]+ib(1-r),(z2j+1)(1-r)-1=-[a(1-r)+r]-ib(1-r),
which are conjugate for all r∈I. Therefore the contraction F restricts to a contraction F:ℋℛn×I→ℋℛn proving that ℋℛn is contractible.Recall that a topological spaceX is said simply connected if it is path connected and for some base point x0∈X the fundamental group π1(X,x0) is trivial (see [46, Section 2.5] for the definition of fundamental group).Corollary 4.5.
The spaces𝒮ℛn𝔽, ℳ𝒮n𝔽, ℋℛn𝔽, and ℳℋn𝔽 are connected and simply connected.Proof.
The space𝒮ℛn𝔽 is connected because the contraction G in the proof of Theorem 4.3 gives a path contained in 𝒮ℛn𝔽 from any set of Schur roots {z1,…,zn} in 𝒮ℛn𝔽 to the set {0,…,0}.
All the homotopy groups of a contractible space are trivial [46, Theorem 3.5.8 (g)], in particular the fundamental group, therefore 𝒮ℛn𝔽 is simply connected. Since ℳ𝒮n𝔽 is homeomorphic to 𝒮ℛn𝔽, it is also connected and simply connected.
The proof for the spacesℋℛn𝔽 and ℳℋn𝔽 is analogous using the contraction F in the proof of Theorem 4.4 and the set {-1,…,-1}.Corollary4.5 for the space ℳℋn is proved in the article [34, Lemma A1, Theorem 2.1] but there is a mistake in the part of the simply connectedness (see Remark 7.1).Corollary 4.6.
The spaces𝒮nℂ and ℋnℂ are homotopically equivalent to a circle S1.Proof.
Let𝒬nℂ be 𝒮nℂ or ℋnℂ. By (2.6) we have that 𝒬nℂ is homeomorphic to ℳ𝒬nℂ×ℂ* which is homotopically equivalent to a circle S1, since by Theorem 4.3, ℳ𝒬nℂ is contractible and ℂ*=ℂ-{0} is homotopically equivalent to a circle S1.Corollary 4.7.
The spaces𝒮n and ℋn consist of two contractible connected components. For ℋn these contractible connected components are ℋn+ and ℋn-.Proof.
Let𝒬n be 𝒮n or ℋn. By (2.7) we have that 𝒬n≅ℳ𝒬n×(-∞,0)∪ℳ𝒬n×(0,∞); by Corollary 4.5, ℳ𝒬n is connected and therefore ℳ𝒬n×(-∞,0) and ℳ𝒬n×(0,∞) are the connected components of 𝒬n,  with  each of them  being  contractible, since by Theorems 4.3 and 4.4, ℳ𝒬n is contractible, and the product of two contractible spaces is contractible. By (2.8) and (2.9), ℋn± are the connected components of ℋn.


## 5. The Topology of the Spaces of Aperiodic Polynomials

In this section we shall study the topology of the spaces of Schur and Hurwitz aperiodic polynomials𝒮𝒜n𝔽 and ℋ𝒜n𝔽. As we saw at the end of Section 3, it is enough to study the spaces of monic polynomials ℳ𝒮𝒜n and ℳℋ𝒜n; these in turn, by Theorem 3.2 and Section 4 are, respectively, homeomorphic to the spaces of roots of Schur aperiodic polynomials 𝒮𝒜ℛn and roots of Hurwitz aperiodic polynomials ℋ𝒜ℛn. Recall that
(5.1)𝒮𝒜ℛn=B(J,n),ℋ𝒜ℛn=B(ℝ-,n),
where B(X,n) is the nthconfiguration space ofX, J=(0,1), and ℝ- is the negative real axis.Theorem 5.1.
The spaces𝒮𝒜ℛn and ℋ𝒜ℛn are contractible. Therefore the spaces ℳ𝒮𝒜n and ℳℋ𝒜n are contractible.Proof.
Since the spaces(0,1) and ℝ- are homeomorphic, and these in turn are homeomorphic to the real line ℝ, we have that
(5.2)𝒮𝒜ℛn≅ℋ𝒜ℛn≅B(ℝ,n),
therefore, it is enough to prove that the space B(ℝ,n) is contractible.
The spaceF(ℝ,n) is precisely ℝn minus all the hyperplanes of the form xi=xj with i≠j, which consists of n! connected components all of them contractible, since they are convex subspaces of ℝn. When we take the quotient of F(ℝ,n) by the action of the symmetric groupΣn to get B(ℝ,n), all the n! connected components are identified in single contractible connected component.The fact that the spaceB(ℝ,n) is contractible is a very well-known result in topology but its identification with the space of monic aperiodic polynomials, we believe, is new in control theory.The proofs of the following corollaries are analogous to the proofs of Corollaries4.5, 4.6, and 4.7, respectively.Corollary 5.2.
The spaces𝒮𝒜ℛn, ℋ𝒜ℛn, ℳ𝒮𝒜n, and ℳℋ𝒜n are simply connected.Corollary 5.3.
The spaces𝒮𝒜nℂ and ℋ𝒜nℂ are homotopically equivalent to a circle S1.Corollary 5.4.
The spaces𝒮𝒜n and ℋ𝒜n consist of two contractible connected components. For ℋ𝒜n these contractible connected components are ℋ𝒜n+ and ℋ𝒜n-.If𝒬n is any of 𝒮n, ℋn±, 𝒮𝒜n, or ℋ𝒜n±, as a consequence of Corollaries 4.7 and 5.4, all the homotopy groups of 𝒬n are trivial (see [46, Theorem 3.5.8 (g)]). In other words, we have the following theorem (see [46, Lemma 3.1.5]).Theorem 5.5 (Boundary Theorem).
IfS⊂𝒬n is the image of an m-sphere under a continuous map f:Sm→𝒬n, for any m∈ℕ, then f can be extended to a continuous map F:Dm+1→𝒬n, where Dm+1 is a closed ball of dimension m+1 and Sm=∂Dm+1.


## 6. The Relation between Schur and Hurwitz Polynomials

Theorem4.3 was proved by Fam and Meditch in [33, Lemma 1] giving an explicit contraction in the space of monic Schur vectors. Their result is stated only for the space of real monic Schur polynomialsℳ𝒮n although their contraction also works for the space of complex monic Schur polynomialsℳ𝒮nℂ. If we compose the contraction G of Theorem 4.3 with Viète’s map we obtain the contraction G′ given by Fam and Meditch (actually the reversed contraction, interchanging r by 1-r; also compare with [35, Proposition 4.1.25]).Theorem4.4 is set as an exercise in [35, Corrolary 1.4.28, Ex. 13]; it is only stated for real monic Hurwitz polynomials but it can also be proved for complex monic Hurwitz polynomials. The proof indicated is an indirect one since it is based on the contractibility of the space ℳ𝒮n𝔽 and the following transformation to relate Schur and Hurwitz polynomials. Consider the Möbius transformation
(6.1)m:S2→S2,m(z)=z+1z-1.
It is a biholomorphism from the Riemann sphere onto itself which transforms the open left half plane ℂ- onto the open unit disk 𝔻 and vice versa, since m-1=m. In particular we have that m(0)=-1, m(1)=∞. It is also important to notice that if z∈ℝ⊂S2, with z≠1, then m(z)∈ℝ and if z2=z-1 then m(z2)=m(z1)¯.Letp(t)=a0+a1t+⋯+antn∈𝒫≤nℂ and suppose it has roots {z1,…,zn}. Define the Möbius transform(6.2)~:𝒫≤nℂ→𝒫≤nℂ,p↦p~.
By
(6.3)p~(s)=(s-1)np(s+1s-1)=∑i=0n‍ai(z+1)i(z-1)n-i=∑k=0n‍∑j=0n-k‍∑i=jk+j‍ai(ij)(n-in-k-j)sk.
This is a linear isomorphism; it is involutive modulo a non-zero constant, that is, p~~=2np, and one of its main properties is that if p∈𝒫nℂ of degree n with p(1)≠0, p is a Schur polynomial if and only if p~ is a Hurwitz polynomial of degree n; moreover p~ has roots {m(z1),…,m(zn)} (see [35, Lemma 3.4.81] for further properties).From the fact that ifz∈ℝ then m(z)∈ℝ and if z2=z-1 then m(z2)=m(z1)¯ we have that if p is a real polynomial then its Möbius transform p~ is again a real polynomial. However from (6.3) we can see that the Möbius transform of a monic polynomial p in general is not a monic polynomial since the leading coefficient of p~ in this case (an=1) is given by a~n=1+∑i=0n-1‍ai. Therefore we cannot use directly the Möbius transform to relate ℳ𝒮n𝔽 with ℳℋn𝔽. To avoid this problem one defines the normalized Möbius transformp˘ of p by
(6.4)p˘(s)=(1+∑l=0n-1‍al)-1p~(s).
In this way we get a continuous map
(6.5)˘:ℳ𝒫≤nℂ→ℳ𝒫≤nℂ.
By the aforementioned property of p~ we have that restricting to ℳℋn𝔽 we get the homeomorphism
(6.6)˘:ℳℋn𝔽→ℳ𝒮n𝔽,p↦p˘,
which is its own inverse, that is, p˘˘=p.Now we can use thenormalized Möbius transform to prove that ℳℋn𝔽 is contractible. Let G′:ℳ𝒮n𝔽×I→ℳ𝒮n𝔽 be the contraction of ℳ𝒮n𝔽 given by the image of the contraction G in the proof of Theorem 4.3 under Viète’s map.Define the contractionH:ℳℋn𝔽×I→ℳℋn𝔽 by H(p,r)=G′˘(p˘,r), that is, following the diagram.
(6.7)
The contraction G′ contracts the space ℳ𝒮n𝔽 to the Schur vector [0,…,0]∈ℝn, which corresponds to the polynomial tn, then the contraction H contracts the space ℳℋn𝔽 to the Hurwitz vector [(nn),(nn-1),…,(n1)] which corresponds to the polynomial (s+1)n, since the normalized Möbius transform of tn is (s+1)n.Using the space of roots we can give a simpler proof of the homeomorphism between the space of degreen monic Hurwitz polynomials ℳℋn𝔽 and the space of degree n monic Schur polynomials ℳ𝒮n𝔽. The proof is simpler in the sense that one does not need any of the properties of the Möbius transform given in [35, Lemma 3.4.81] but only the fact that the Möbius transformation (6.1) is a homeomorphism which transforms ℂ- into 𝔻. The Möbius transformation m given in (6.1) induces a homeomorphism from the nth Cartesian product S2×⋯S2 of the Riemann sphere onto itself. This homeomorphism is equivariant with respect to the action of Σn and therefore it induces a homeomorphism on the nth symmetric product of S2(6.8)m~:Symn(S2)→Symn(S2),{z1,…,zn}↦{m(z1),…,m(zn)},
such that m~-1=m~. Since m maps homeomorphically the open left half plane ℂ- onto the open unit disk 𝔻, it is clear that m~ maps the space of Hurwitz roots ℋℛn𝔽 homeomorphically onto the space of Schur roots 𝒮ℛn𝔽. The space ℋℛn𝔽 is homeomorphic to the space ℳℋn𝔽 and the space 𝒮ℛn𝔽 is homeomorphic to the space ℳ𝒮n𝔽 via Viète’s map. Therefore ℳℋn𝔽 and ℳ𝒮n𝔽 are homeomorphic.Remember thatℂℙn can be thought as the space of complex monic polynomials of degree less than or equal to n, denoted by ℳ𝒫≤nℂ, and that Viète’s projective map is a homeomorphism ν:Symn(S2)→ℳ𝒫≤nℂ. Combining Viète’s projective map with the homeomorphism m~, we get a natural homeomorphism h from ℳ𝒫≤nℂ to itself given by the following diagram:
(6.9)
given explicitly as follows. Let p∈ℳ𝒫≤nℂ and suppose it has roots {z1,…,zn}, that is, p(t)=∏i=1n‍(t-zi), then
(6.10)h(p)(s)=∏i=1n‍(s-m(zi)),
the monic polynomial with roots {m(z1),…,m(zn)}. Remember that we consider ∞ to be a root of the polynomial if its degree is less than n. Therefore the homeomorphism h is precisely the normalized Möbius transform·˘; to see this, compare (6.10) with (70) in the proof of Lemma  3.4.81 in [35] when p is monic of degree n.


## 7. Two Remarks

We have the following two remarks about previous works.Remark 7.1.
The proof of Theorem  2.1 in [34] that the space ℳℋn (H1+n in notation of [34]) is simply connected is not correct. In Lemma A3 the space
(7.1)ℝ1+n+1={(a0,…,an-1,1)∈ℝn+1∣ai>0,i=0,…,n-1},
which is homeomorphic to ℝn, is expressed as the union of two open connected subsets H1+n and U1+n with common boundary B1+n, that is,
(7.2)ℝ1+n+1=H1+n∪B1+n∪U1+n.
Then the argument is that if H1+n is not simply connected, then U1+n is not connected and this contradicts the connectivity of U1+n. This argument is valid only in dimension 2 (i.e., in ℝ1+3≅ℝ2) but it is not true in higher dimensions. One counterexample is to consider the subset of ℝ3 given by a closed solid torus
(7.3)T={(x,y,z)∈ℝ3∣((x2+y2)-2)2+z2≤1},
which is the solid obtained rotating the 2-disk D2 in the plane xz with center in (2,0,0), that is, D2={(x,0,z)∈ℝ3∣(x-2)2+z2≤1}, around the z-axis, giving the shape of a “doughnut.” The space T is homeomorphic to D2×S1. Denote by H the interior of the torus, by B its boundary, and by U the complement of T in ℝ3. Then ℝ3=H∪B∪U, with H and U being open and connected with common boundary B. The space H is not simply connected because it is homotopically equivalent to a circle S1. Also in ℝ3, even if B was homeomorphic to a 2-sphere S2, the unbounded component of ℝ3-B is not necessarily simply connected; an example of this is the famous Alexander Horned Sphere (see [50, Example 2B.2]).
A counterexample in dimensionn>2 is similar taking T=Dn-1×S1. The same mistake is also made in [51, Theorem 3.2].
Now with the proof thatℋn+ is contractible (Corollary 4.7), in particular simply connected, the Edge and Boundary Theorems of [34], which use as a main ingredient the simple connectivity of ℋn+, remain valid.Remark 7.2.
The approach of using the roots spaceℛn has been used implicitly in previous studies of spaces of polynomials. For instance, in [33, Lemma 1] to prove the contractibility of the space of real monic Schur polynomials ℳ𝒮n, implicitly they proved the contractibility of the space of roots which are in the open disk in the complex plane. Also in [34, Lemma A1] to prove that H1+n is connected, they implicitly constructed a path in the space of roots.


## 8. Conclusions

Viète’s map gives an explicit homeomorphism between the space of rootsℛn and the space of monic complex polynomials ℳ𝒫nℂ. Restricting this homeomorphism to the spaces of Schur and Hurwitz (aperiodic) roots one can study in a more natural way some topological properties of the spaces of Schur and Hurwitz (aperiodic) polynomials. Using this viewpoint we give simple proofs of some topological properties of the spaces of Schur and Hurwitz polynomials; in particular we prove that the spaces ℋn+ and ℋn- of Hurwitz polynomials of degree n with positive and negative coefficients, respectively, are contractible and therefore simply connected. As a new result we prove that the spaces of monic Schur aperiodic polynomialsℳ𝒮𝒜n and of monic Hurwitz aperiodic polynomialsℳℋ𝒜n are contractible. As a consequence of the contractibility of the spaces 𝒮n, ℋn±, 𝒮𝒜n, or ℋ𝒜n±, we get the Boundary Theorem given in Theorem 5.5.Also using the space of roots and Viète’s projective map we see that thenormalized Möbius transform is a natural transformation from the space ℳ𝒫≤nℂ of complex monic polynomials of degree less than or equal to n to itself, instead of just being seen as a “correction” to the Möbius transform to get monic polynomials. It gives a homeomorphism between the space of monic (complex or real) Schur polynomials and the space of (complex or real) Hurwitz polynomials.



---

*Source: 595076-2012-12-24.xml*