Open access peer-reviewed chapter

An Interpretation of Rosenbrock's Theorem via Local Rings

By A. Amparan, S. Marcaida and I. Zaballa

Submitted: December 21st 2011Reviewed: April 16th 2012Published: July 11th 2012

DOI: 10.5772/46483

Downloaded: 1342

1. Introduction

Consider a linear time invariant system

x˙(t)=Ax(t)+Bu(t)uid1

to be identified with the pair of matrices (A,B)where A𝔽n×n, B𝔽n×mand 𝔽=or the fields of the real or complex numbers. If state-feedback u(t)=Fx(t)+v(t)is applied to system (), Rosenbrock's Theorem on pole assignment (see [1]) characterizes for the closed-loop system

x˙(t)=(A+BF)x(t)+Bv(t),uid2

the invariant factors of its state-space matrix A+BF. This result can be seen as the solution of an inverse problem; that of finding a non-singular polynomial matrix with prescribed invariant factors and left Wiener–Hopf factorization indices at infinity. To see this we recall that the invariant factors form a complete system of invariants for the finite equivalence of polynomial matrices (this equivalence relation will be revisited in Section ) and it will be seen in Section that any polynomial matrix is left Wiener–Hopf equivalent at infinity to a diagonal matrix Diag(sk1,...,skm), where the non-negative integers k1,...,km(that can be assumed in non-increasing order) form a complete system of invariants for the left Wiener–Hopf equivalence at infinity. Consider now the transfer function matrix G(s)=(sI-(A+BF))-1Bof (). This is a rational matrix that can be written as an irreducible matrix fraction description G(s)=N(s)P(s)-1, where N(s)and P(s)are right coprime polynomial matrices. In the terminology of [2], P(s)is a polynomial matrix representation of (), concept that is closely related to that of polynomial model introduced by Fuhrmann (see for example [3] and the references therein). It turns out that all polynomial matrix representations of a system are right equivalent (see [2], [3]), that is, if P1(s)and P2(s)are polynomial matrix representations of the same system there exists a unimodular matrix U(s)such that P2(s)=P1(s)U(s). Therefore all polynomial matrix representations of () have the same invariant factors, which are the invariant factors of sIn-(A+BF)except for some trivial ones. Furthermore, all polynomial matrix representations also have the same left Wiener– Hopf factorization indices at infinity, which are equal to the controllability indices of () and (), because the controllability indices are invariant under feedback. With all this in mind it is not hard to see that Rosenbrock's Theorem on pole assignment is equivalent to finding necessary and sufficient conditions for the existence of a non-singular polynomial matrix with prescribed invariant factors and left Wiener–Hopf factorization indices at infinity. This result will be precisely stated in Section once all the elements that appear are properly defined. In addition, there is a similar result to Rosenbrock's Theorem on pole assignment but involving the infinite structure (see [4]).

Our goal is to generalize both results (the finite and infinite versions of Rosenbrock's Theorem) for rational matrices defined on arbitrary fields via local rings. This will be done in Section and an extension to arbitrary fields of the concept of Wiener–Hopf equivalence will be needed. This concept is very well established for complex valued rational matrix functions (see for example [5], [6]). Originally it requires a closed contour, γ, that divides the extended complex plane ({}) into two parts: the inner domain (Ω+) and the region outside γ(Ω-), which contains the point at infinity. Then two non-singular m×mcomplex rational matrices T1(s)and T2(s), with no poles and no zeros in γ, are said to be left Wiener–Hopf equivalent with respect to γif there are m×mmatrices U-(s)and U+(s)with no poles and no zeros in Ω-γand Ω+γ, respectively, such that

T2(s)=U-(s)T1(s)U+(s).uid3

It can be seen, then, that any non-singular m×mcomplex rational matrix T(s)is left Wiener–Hopf equivalent with respect to γto a diagonal matrix

Diag(s-z0)k1,...,(s-z0)kmuid4

where z0is any complex number in Ω+and k1kmare integers uniquely determined by T(s). They are called the left Wiener–Hopf factorization indices of T(s)with respect to γ(see again [5], [6]). The generalization to arbitrary fields relies on the following idea: We can identify Ω+γand (Ω-γ){}with two sets Mand M', respectively, of maximal ideals of [s]. In fact, to each z0we associate the ideal generated by s-z0, which is a maximal ideal of [s]. Notice that s-z0is also a prime polynomial of [s]but Mand M', as defined, cannot contain the zero ideal, which is prime. Thus we are led to consider the set Specm([s])of maximal ideals of [s]. By using this identification we define the left Wiener–Hopf equivalence of rational matrices over an arbitrary field 𝔽with respect to a subset Mof Specm(𝔽[s]), the set of all maximal ideals of 𝔽[s]. In this study local rings play a fundamental role. They will be introduced in Section . Localization techniques have been used previously in the algebraic theory of linear systems (see, for example, [7]). In Section the algebraic structure of the rings of proper rational functions with prescribed finite poles is studied (i.e., for a fixed MSpecm(𝔽[s])the ring of proper rational functions p(s)q(s)with gcd(g(s),π(s))=1for all (π(s))M). It will be shown that if there is an ideal generated by a linear polynomial outside Mthen the set of proper rational functions with no poles in Mis an Euclidean domain and all rational matrices can be classified according to their Smith–McMillan invariants. In this case, two types of invariants live together for any non-singular rational matrix and any set MSpecm(𝔽[s]): its Smith–McMillan and left Wiener–Hopf invariants. In Section we show that a Rosenbrock-like Theorem holds true that completely characterizes the relationship between these two types of invariants.

2. Preliminaries

In the sequel 𝔽[s]will denote the ring of polynomials with coefficients in an arbitrary field 𝔽and Specm(𝔽[s])the set of all maximal ideals of 𝔽[s], that is,

Specm(𝔽[s])=(π(s)):π(s)𝔽[s],irreducible,monic,differentfrom1.uid5

Let π(s)𝔽[s]be a monic irreducible non-constant polynomial. Let S=𝔽[s](π(s))be the multiplicative subset of 𝔽[s]whose elements are coprime with π(s). We denote by 𝔽π(s)the quotient ring of 𝔽[s]by S; i.e., S-1𝔽[s]:

𝔽π(s)=p(s)q(s):p(s),q(s)𝔽[s],gcd(q(s),π(s))=1.uid6

This is the localization of 𝔽[s]at (π(s))(see [8]). The units of 𝔽π(s)are the rational functions u(s)=p(s)q(s)such that gcd(p(s),π(s))=1and gcd(q(s),π(s))=1. Consequentially,

𝔽π(s)=u(s)π(s)d:u(s)isaunitandd0{0}.uid7

For any MSpecm(𝔽[s]), let

𝔽M(s)=(π(s))M𝔽π(s)=p(s)q(s):p(s),q(s)𝔽[s],gcd(q(s),π(s))=1(π(s))M.uid8

This is a ring whose units are the rational functions u(s)=p(s)q(s)such that for all ideals (π(s))M, gcd(p(s),π(s))=1and gcd(q(s),π(s))=1. Notice that, in particular, if M=Specm(𝔽[s])then 𝔽M(s)=𝔽[s]and if M=then 𝔽M(s)=𝔽(s), the field of rational functions.

Moreover, if α(s)𝔽[s]is a non-constant polynomial whose prime factorization, α(s)=kα1(s)d1αm(s)dm, satisfies the condition that (αi(s))Mfor all i, we will say that α(s)factorizes in Mor α(s)has all its zeros in M. We will consider that the only polynomials that factorize in M=are the constants. We say that a non-zero rational function factorizes in Mif both its numerator and denominator factorize in M. In this case we will say that the rational function has all its zeros and poles in M. Similarly, we will say that p(s)q(s)has no poles in Mif p(s)0and gcd(q(s),π(s))=1for all ideals (π(s))M. And it has no zeros in Mif gcd(p(s),π(s))=1for all ideals (π(s))M. In other words, it is equivalent that p(s)q(s)has no poles and no zeros in Mand that p(s)q(s)is a unit of 𝔽M(s). So, a non-zero rational function factorizes in Mif and only if it is a unit in 𝔽Specm(𝔽[s])M(s).

Let 𝔽M(s)m×mdenote the set of m×mmatrices with elements in 𝔽M(s). A matrix is invertible in 𝔽M(s)m×mif all its elements are in 𝔽M(s)and its determinant is a unit in 𝔽M(s). We denote by Glm(𝔽M(s))the group of units of 𝔽M(s)m×m.

Remark 1

Let M1,M2Specm(𝔽[s]). Notice that

1. If M1M2then 𝔽M1(s)𝔽M2(s)and Glm(𝔽M1(s))Glm(𝔽M2(s)).

2. 𝔽M1M2(s)=𝔽M1(s)𝔽M2(s)and Glm(𝔽M1M2(s))=Glm(𝔽M1(s))Glm(𝔽M2(s)).

For any MSpecm(𝔽[s])the ring 𝔽M(s)is a principal ideal domain (see [9]) and its field of fractions is 𝔽(s). Two matrices T1(s),T2(s)𝔽(s)m×mare equivalent with respect to Mif there exist matrices U(s),V(s)Glm(𝔽M(s))such that T2(s)=U(s)T1(s)V(s). Since 𝔽M(s)is a principal ideal domain, for all non-singular G(s)𝔽M(s)m×m(see [10]) there exist matrices U(s),V(s)Glm(𝔽M(s))such that

G(s)=U(s)Diag(α1(s),...,αm(s))V(s)uid10

with α1(s)αm(s)(“” stands for divisibility) monic polynomials factorizing in M, unique up to multiplication by units of 𝔽M(s). The diagonal matrix is the Smith normal form of G(s)with respect to Mand α1(s),...,αm(s)are called the invariant factors of G(s)with respect to M. Now we introduce the Smith–McMillan form with respect to M. Assume that T(s)𝔽(s)m×mis a non-singular rational matrix. Then T(s)=G(s)d(s)with G(s)𝔽M(s)m×mand d(s)𝔽[s]monic, factorizing in M. Let G(s)=U(s)Diag(α1(s),...,αm(s))V(s)be the Smith normal form with respect to Mof G(s), i.e., U(s),V(s)invertible in 𝔽M(s)m×mand α1(s)αm(s)monic polynomials factorizing in M. Then

T(s)=U(s)Diagϵ1(s)ψ1(s),...,ϵm(s)ψm(s)V(s)uid11

where ϵi(s)ψi(s)are irreducible rational functions, which are the result of dividing αi(s)by d(s)and canceling the common factors. They satisfy that ϵ1(s)ϵm(s), ψm(s)ψ1(s)are monic polynomials factorizing in M. The diagonal matrix in () is the Smith–McMillan form with respect to M. The rational functions ϵi(s)ψi(s), i=1,...,m, are called the invariant rational functions of T(s)with respect to Mand constitute a complete system of invariants of the equivalence with respect to Mfor rational matrices.

In particular, if M=Specm(𝔽[s])then 𝔽Specm(𝔽[s])(s)=𝔽[s], the matrices U(s),V(s)Glm(𝔽[s])are unimodular matrices, () is the global Smith–McMillan form of a rational matrix (see [11] or [1] when 𝔽=or ) and ϵi(s)ψi(s)are the global invariant rational functions of T(s).

From now on rational matrices will be assumed to be non-singular unless the opposite is specified. Given any MSpecm(𝔽[s])we say that an m×mnon-singular rational matrix has no zeros and no poles in Mif its global invariant rational functions are units of 𝔽M(s). If its global invariant rational functions factorize in M, the matrix has its global finite structure localized in Mand we say that the matrix has all zeros and poles in M. The former means that T(s)Glm(𝔽M(s))and the latter that T(s)Glm(𝔽Specm(𝔽[s])M(s))because detT(s)=detU(s)detV(s)ϵ1(s)ϵm(s)ψ1(s)ψm(s)and detU(s),detV(s)are non-zero constants. The following result clarifies the relationship between the global finite structure of any rational matrix and its local structure with respect to any MSpecm(𝔽[s]).

Proposition 2 Let MSpecm(𝔽[s]). Let T(s)𝔽(s)m×mbe non-singular with α1(s)β1(s),...,αm(s)βm(s)its global invariant rational functions and let ϵ1(s)ψ1(s),...,ϵm(s)ψm(s)be irreducible rational functions such that ϵ1(s)ϵm(s), ψm(s)ψ1(s)are monic polynomials factorizing in M. The following properties are equivalent:

  1. There exist TL(s),TR(s)𝔽(s)m×msuch that the global invariant rational functions of TL(s)are ϵ1(s)ψ1(s),...,ϵm(s)ψm(s), TR(s)Glm(𝔽M(s))and T(s)=TL(s)TR(s).

  2. There exist matrices U1(s),U2(s)invertible in 𝔽M(s)m×msuch that

    T(s)=U1(s)Diagϵ1(s)ψ1(s),...,ϵm(s)ψm(s)U2(s),uid15

    i.e., ϵ1(s)ψ1(s),...,ϵm(s)ψm(s)are the invariant rational functions of T(s)with respect to M.

  3. αi(s)=ϵi(s)ϵi'(s)and βi(s)=ψi(s)ψi'(s)with ϵi'(s),ψi'(s)𝔽[s]units of 𝔽M(s), for i=1,...,m.

Proof.- 1 2. Since the global invariant rational functions of TL(s)are ϵ1(s)ψ1(s),...,ϵm(s)ψm(s), there exist W1(s),W2(s)Glm(𝔽[s])such that TL(s)=W1(s)Diagϵ1(s)ψ1(s),...,ϵm(s)ψm(s)W2(s).As 𝔽Specm(𝔽[s])(s)=𝔽[s], by Remark .1, W1(s),W2(s)Glm(𝔽M(s)). Therefore, putting U1(s)=W1(s)and U2(s)=W2(s)TR(s)it follows that U1(s)and U2(s)are invertible in 𝔽M(s)m×mand T(s)=U1(s)Diagϵ1(s)ψ1(s),...,ϵm(s)ψm(s)U2(s).

2 3. There exist unimodular matrices V1(s),V2(s)𝔽[s]m×msuch that

T(s)=V1(s)Diagα1(s)β1(s),...,αm(s)βm(s)V2(s)uid17

with αi(s)βi(s)irreducible rational functions such that α1(s)αm(s)and βm(s)β1(s)are monic polynomials. Write αi(s)βi(s)=pi(s)pi'(s)qi(s)qi'(s)such that pi(s),qi(s)factorize in Mand pi'(s),qi'(s)factorize in Specm(𝔽[s])M. Then

T(s)=V1(s)Diagp1(s)q1(s),...,pm(s)qm(s)Diagp1'(s)q1'(s),...,pm'(s)qm'(s)V2(s)uid18

with V1(s)and Diagp1'(s)q1'(s),...,pm'(s)qm'(s)V2(s)invertible in 𝔽M(s)m×m. Since the Smith–McMillan form with respect to Mis unique we get that pi(s)qi(s)=ϵi(s)ψi(s).

3 1. Write () as

T(s)=V1(s)Diagϵ1(s)ψ1(s),...,ϵm(s)ψm(s)Diagϵ1'(s)ψ1'(s),...,ϵm'(s)ψm'(s)V2(s).uid19

It follows that T(s)=TL(s)TR(s)with TL(s)=V1(s)Diagϵ1(s)ψ1(s),...,ϵm(s)ψm(s)and TR(s)=Diagϵ1'(s)ψ1'(s),...,ϵm'(s)ψm'(s)V2(s)Glm(𝔽M(s)).

Corollary 3 Let T(s)𝔽(s)m×mbe non-singular and M1,M2Specm(𝔽[s])such that M1M2=. If ϵ1i(s)ψ1i(s),...,ϵmi(s)ψmi(s)are the invariant rational functions of T(s)with respect to Mi, i=1,2, then ϵ11(s)ϵ12(s)ψ11(s)ψ12(s),...,ϵm1(s)ϵm2(s)ψm1(s)ψm2(s)are the invariant rational functions of T(s)with respect to M1M2.

1.08 Proof.- Let α1(s)β1(s),...,αm(s)βm(s)be the global invariant rational functions of T(s). By Proposition , αi(s)=ϵi1(s)ni1(s),βi(s)=ψi1(s)di1(s),with ni1(s),di1(s)𝔽[s]units of 𝔽M1(s). On the other hand αi(s)=ϵi2(s)ni2(s),βi(s)=ψi2(s)di2(s),with ni2(s),di2(s)𝔽[s]units of 𝔽M2(s). So, ϵi1(s)ni1(s)=ϵi2(s)ni2(s)or equivalently ni1(s)=ϵi2(s)ni2(s)ϵi1(s),ni2(s)=ϵi1(s)ni1(s)ϵi2(s).The polynomials ϵi1(s),ϵi2(s)are coprime because ϵi1(s)factorizes in M1, ϵi2(s)factorizes in M2and M1M2=. In consequence ϵi1(s)ni2(s)and ϵi2(s)ni1(s). Therefore, there exist polynomials a(s), unit of 𝔽M2(s), and a'(s), unit of 𝔽M1(s), such that ni2(s)=ϵi1(s)a(s),ni1(s)=ϵi2(s)a'(s).Since αi(s)=ϵi1(s)ni1(s)=ϵi1(s)ϵi2(s)a'(s)and αi(s)=ϵi2(s)ni2(s)=ϵi2(s)ϵi1(s)a(s). This implies that a(s)=a'(s)unit of 𝔽M1(s)𝔽M2(s)=𝔽M1M2(s). Following the same ideas we can prove that βi(s)=ψi1(s)ψi2(s)b(s)with b(s)a unit of 𝔽M1M2(s). By Proposition ϵ11(s)ϵ12(s)ψ11(s)ψ12(s),...,ϵm1(s)ϵm2(s)ψm1(s)ψm2(s)are the invariant rational functions of T(s)with respect to M1M2.

Corollary 4 Let M1,M2Specm(𝔽[s]). Two non-singular matrices are equivalent with respect to M1M2if and only if they are equivalent with respect to M1and with respect to M2.

Proof.- Notice that by Remark .2 two matrices T1(s),T2(s)𝔽(s)m×mare equivalent with respect to M1M2if and only if there exist U1(s),U2(s)invertible in 𝔽M1(s)m×m𝔽M2(s)m×msuch that T2(s)=U1(s)T1(s)U2(s). Since U1(s)and U2(s)are invertible in both 𝔽M1(s)m×mand 𝔽M2(s)m×mthen T1(s)and T2(s)are equivalent with respect to M1and with respect to M2.

Conversely, if T1(s)and T2(s)are equivalent with respect to M1and with respect to M2then, by the necessity of this result, they are equivalent with respect to M1(M1M2), with respect to M2(M1M2)and with respect to M1M2. Let ϵ11(s)ψ11(s),...,ϵm1(s)ψm1(s)be the invariant rational functions of T1(s)and T2(s)with respect to M1(M1M2), ϵ12(s)ψ12(s),...,ϵm2(s)ψm2(s)be the invariant rational functions of T1(s)and T2(s)with respect to M2(M1M2)and ϵ13(s)ψ13(s),...,ϵm3(s)ψm3(s)be the invariant rational functions of T1(s)and T2(s)with respect to M1M2. By Corollary ϵ11(s)ψ11(s)ϵ12(s)ψ12(s)ϵ13(s)ψ13(s),...,ϵm1(s)ψm1(s)ϵm2(s)ψm2(s)ϵm3(s)ψm3(s)must be the invariant rational functions of T1(s)and T2(s)with respect to M1M2. Therefore, T1(s)and T2(s)are equivalent with respect to M1M2.

Let 𝔽pr(s)be the ring of proper rational functions, that is, rational functions with the degree of the numerator at most the degree of the denominator. The units in this ring are the rational functions whose numerators and denominators have the same degree. They are called biproper rational functions. A matrix B(s)𝔽pr(s)m×mis said to be biproper if it is a unit in 𝔽pr(s)m×mor, what is the same, if its determinant is a biproper rational function.

Recall that a rational function t(s)has a pole (zero) at if t1shas a pole (zero) at 0. Following this idea, we can define the local ring at as the set of rational functions, t(s), such that t1sdoes not have 0 as a pole, that is, 𝔽(s)=t(s)𝔽(s):t1s𝔽s(s). If t(s)=p(s)q(s)with p(s)=atst+at+1st+1++apsp,ap0, q(s)=brsr+br+1sr+1++bqsq,bq0,p=d(p(s)),q=d(q(s)), where d(·)stands for “degree of”, then

t1s=atst+at+1st+1++apspbrsr+br+1sr+1++bqsq=atsp-t+at+1sp-t-1++apbrsq-r+br+1sq-r-1++bqsq-p=f(s)g(s)sq-p.uid22

As 𝔽s(s)=f(s)g(s)sd:f(0)0,g(0)0andd00, then

𝔽(s)=p(s)q(s)𝔽(s):d(q(s))d(p(s)).uid23

Thus, this set is the ring of proper rational functions, 𝔽pr(s).

Two rational matrices T1(s),T2(s)𝔽(s)m×mare equivalent at infinity if there exist biproper matrices B1(s),B2(s)Glm(𝔽pr(s))such that T2(s)=B1(s)T1(s)B2(s). Given a non-singular rational matrix T(s)𝔽(s)m×m(see [11]) there always exist B1(s),B2(s)Glm(𝔽pr(s))such that

T(s)=B1(s)Diag(sq1,...,sqm)B2(s)uid24

where q1qmare integers. They are called the invariant orders of T(s)at infinity and the rational functions sq1,...,sqmare called the invariant rational functions of T(s)at infinity.

1.05

3. Structure of the ring of proper rational functions with prescribed finite poles

Let M'Specm(𝔽[s]). Any non-zero rational function t(s)can be uniquely written as t(s)=n(s)d(s)n'(s)d'(s)where n(s)d(s)is an irreducible rational function factorizing in M'and n'(s)d'(s)is a unit of 𝔽M'(s). Define the following function over 𝔽(s){0}(see [11], [12]):

δ:𝔽(s){0}t(s)d(d'(s))-d(n'(s)).uid25

This mapping is not a discrete valuation of 𝔽(s)if M': Given two non-zero elements t1(s),t2(s)𝔽(s)it is clear that δ(t1(s)t2(s))=δ(t1(s))+δ(t2(s)); but it may not satisfy that δ(t1(s)+t2(s))min(δ(t1(s)),δ(t2(s))). For example, let M'={(s-a)Specm([s]):a[-2,-1]}. Put t1(s)=s+0.5s+1.5and t2(s)=s+2.5s+1.5. We have that δ(t1(s))=d(s+1.5)-d(1)=1, δ(t2(s))=d(s+1.5)-d(1)=1but δ(t1(s)+t2(s))=δ(2)=0.

However, if M'=and t(s)=n(s)d(s)𝔽(s)where n(s),d(s)𝔽[s], d(s)0, the map

δ:𝔽(s){+}uid26

defined via δ(t(s))=d(d(s))-d(n(s))if t(s)0and δ(t(s))=+if t(s)=0is a discrete valuation of 𝔽(s).

Consider the subset of 𝔽(s), 𝔽M'(s)𝔽pr(s), consisting of all proper rational functions with poles in Specm(𝔽[s])M', that is, the elements of 𝔽M'(s)𝔽pr(s)are proper rational functions whose denominators are coprime with all the polynomials π(s)such that (π(s))M'. Notice that g(s)𝔽M'(s)𝔽pr(s)if and only if g(s)=n(s)n'(s)d'(s)where:

  1. n(s)𝔽[s]is a polynomial factorizing in M',

  2. n'(s)d'(s)is an irreducible rational function and a unit of 𝔽M'(s),

  3. δ(g(s))-d(n(s))0or equivalently δ(g(s))0.

In particular (c)implies that n'(s)d'(s)𝔽pr(s). The units in 𝔽M'(s)𝔽pr(s)are biproper rational functions n'(s)d'(s), that is d(n'(s))=d(d'(s)), with n'(s),d'(s)factorizing in Specm(𝔽[s])M'. Furthermore, 𝔽M'(s)𝔽pr(s)is an integral domain whose field of fractions is 𝔽(s)provided that M'Specm(𝔽[s])(see, for example, Prop.5.22[11]). Notice that for M'=Specm(𝔽[s]), 𝔽M'(s)𝔽pr(s)=𝔽[s]𝔽pr(s)=𝔽.

Assume that there are ideals in Specm(𝔽[s])M'generated by linear polynomials and let (s-a)be any of them. The elements of 𝔽M'(s)𝔽pr(s)can be written as g(s)=n(s)u(s)1(s-a)dwhere n(s)𝔽[s]factorizes in M', u(s)is a unit in 𝔽M'(s)𝔽pr(s)and d=δ(g(s))d(n(s)). If 𝔽is algebraically closed, for example 𝔽=, and M'Specm(𝔽[s])the previous condition is always fulfilled.

The divisibility in 𝔽M'(s)𝔽pr(s)is characterized in the following lemma.

Lemma 5 Let M'Specm(𝔽[s]). Let g1(s),g2(s)𝔽M'(s)𝔽pr(s)be such that g1(s)=n1(s)n1'(s)d1'(s)and g2(s)=n2(s)n2'(s)d2'(s)with n1(s),n2(s)𝔽[s]factorizing in M'and n1'(s)d1'(s),n2'(s)d2'(s)irreducible rational functions, units of 𝔽M'(s). Then g1(s)divides g2(s)in 𝔽M'(s)𝔽pr(s)if and only if

n1(s)n2(s)in𝔽[s]uid31
δ(g1(s))-d(n1(s))δ(g2(s))-d(n2(s)).uid32

Proof.- If g1(s)g2(s)then there exists g(s)=n(s)n'(s)d'(s)𝔽M'(s)𝔽pr(s), with n(s)𝔽[s]factorizing in M'and n'(s),d'(s)𝔽[s]coprime, factorizing in Specm(𝔽[s])M', such that g2(s)=g(s)g1(s). Equivalently, n2(s)n2'(s)d2'(s)=n(s)n'(s)d'(s)n1(s)n1'(s)d1'(s)=n(s)n1(s)n'(s)n1'(s)d'(s)d1'(s). So n2(s)=n(s)n1(s)and δ(g2(s))-d(n2(s))=δ(g(s))-d(n(s))+δ(g1(s))-d(n1(s)). Moreover, as g(s)is a proper rational function, δ(g(s))-d(n(s))0and δ(g2(s))-d(n2(s))δ(g1(s))-d(n1(s)).

Conversely, if n1(s)n2(s)then there is n(s)𝔽[s], factorizing in M', such that n2(s)=n(s)n1(s). Write g(s)=n(s)n'(s)d'(s)where n'(s)d'(s)is an irreducible fraction representation of n2'(s)d1'(s)d2'(s)n1'(s), i.e., n'(s)d'(s)=n2'(s)d1'(s)d2'(s)n1'(s)after canceling possible common factors. Thus n2'(s)d2'(s)=n'(s)d'(s)n1'(s)d1'(s)and

δ(g(s))-d(n(s))=d(d'(s))-d(n'(s))-d(n(s))=d(d2'(s))+d(n1'(s))-d(n2'(s))-d(d1'(s))-d(n2(s))+d(n1(s))=δ(g2(s))-d(n2(s))-(δ(g1(s))-d(n1(s)))0.uid33

Then g(s)𝔽M'(s)𝔽pr(s)and g2(s)=g(s)g1(s).

Notice that condition () means that g1(s)g2(s)in 𝔽M'(s)and condition () means that g1(s)g2(s)in 𝔽pr(s). So, g1(s)g2(s)in 𝔽M'(s)𝔽pr(s)if and only if g1(s)g2(s)simultaneously in 𝔽M'(s)and 𝔽pr(s).

Lemma 6 Let M'Specm(𝔽[s]). Let g1(s),g2(s)𝔽M'(s)𝔽pr(s)be such that g1(s)=n1(s)n1'(s)d1'(s)and g2(s)=n2(s)n2'(s)d2'(s)as in Lemma . If n1(s)and n2(s)are coprime in 𝔽[s]and either δ(g1(s))=d(n1(s))or δ(g2(s))=d(n2(s))then g1(s)and g2(s)are coprime in 𝔽M'(s)𝔽pr(s).

Proof.- Suppose that g1(s)and g2(s)are not coprime. Then there exists a non-unit g(s)=n(s)n'(s)d'(s)𝔽M'(s)𝔽pr(s)such that g(s)g1(s)and g(s)g2(s). As g(s)is not a unit, n(s)is not a constant or δ(g(s))>0. If n(s)is not a constant then n(s)n1(s)and n(s)n2(s)which is impossible because n1(s)and n2(s)are coprime. Otherwise, if n(s)is a constant then δ(g(s))>0and we have that δ(g(s))δ(g1(s))-d(n1(s))and δ(g(s))δ(g2(s))-d(n2(s)). But this is again impossible.

It follows from this Lemma that if g1(s),g2(s)are coprime in both rings 𝔽M'(s)and 𝔽pr(s)then g1(s),g2(s)are coprime in 𝔽M'(s)𝔽pr(s). The following example shows that the converse is not true in general.

Example 7 Suppose that 𝔽=and M'=Specm([s]){(s2+1)}. It is not difficult to prove that g1(s)=s2s2+1and g2(s)=ss2+1are coprime elements in M'(s)pr(s). Assume that there exists a non-unit g(s)=n(s)n'(s)d'(s)M'(s)pr(s)such that g(s)g1(s)and g(s)g2(s). Then n(s)s2, n(s)sand δ(g(s))-d(n(s))=0. Since g(s)is not a unit, n(s)cannot be a constant. Hence, n(s)=cs, c0, and δ(g(s))=1, but this is impossible because d'(s)and n'(s)are powers of s2+1. Therefore g1(s)and g2(s)must be coprime. However n1(s)=s2and n2(s)=sare not coprime.

Now, we have the following property when there are ideals in Specm(𝔽[s])M', M'Specm(𝔽[s]), generated by linear polynomials.

Lemma 8 Let M'Specm(𝔽[s]). Assume that there are ideals in Specm(𝔽[s])M'generated by linear polynomials and let (s-a)be any of them. Let g1(s),g2(s)𝔽M'(s)𝔽pr(s)be such that g1(s)=n1(s)u1(s)1(s-a)d1and g2(s)=n2(s)u2(s)1(s-a)d2. If g1(s)and g2(s)are coprime in 𝔽M'(s)𝔽pr(s)then n1(s)and n2(s)are coprime in 𝔽[s]and either d1=d(n1(s))or d2=d(n2(s)).

Proof.- Suppose that n1(s)and n2(s)are not coprime in 𝔽[s]. Then there exists a non-constant n(s)𝔽[s]such that n(s)n1(s)and n(s)n2(s). Let d=d(n(s)). Then g(s)=n(s)1(s-a)dis not a unit in 𝔽M'(s)𝔽pr(s)and divides g1(s)and g2(s)because 0=d-d(n(s))d1-d(n1(s))and 0=d-d(n(s))d2-d(n2(s)). This is impossible, so n1(s)and n2(s)must be coprime.

Now suppose that d1>d(n1(s))and d2>d(n2(s)). Let d=min{d1-d(n1(s)),d2-d(n2(s))}. We have that d>0. Thus g(s)=1(s-a)dis not a unit in 𝔽M'(s)𝔽pr(s)and divides g1(s)and g2(s)because dd1-d(n1(s))and dd2-d(n2(s)). This is again impossible and either d1=d(n1(s))or d2=d(n2(s)).

The above lemmas yield a characterization of coprimeness of elements in 𝔽M'(s)𝔽pr(s)when M'excludes at least one ideal generated by a linear polynomial.

Following the same steps as in p. 11[12] and p. 271[11] we get the following result.

Lemma 9 Let M'Specm(𝔽[s])and assume that there is at least an ideal in Specm(𝔽[s])M'generated by a linear polynomial. Then 𝔽M'(s)𝔽pr(s)is a Euclidean domain.

The following examples show that if all ideals generated by polynomials of degree one are in M', the ring 𝔽M'(s)𝔽pr(s)may not be a Bezout domain. Thus, it may not be a Euclidean domain. Even more, it may not be a greatest common divisor domain.

Example 10 Let 𝔽=and M'=Specm([s]){(s2+1)}. Let g1(s)=s2s2+1,g2(s)=ss2+1M'(s)pr(s). We have seen, in the previous example, that g1(s),g2(s)are coprime. We show now that the Bezout identity is not fulfilled, that is, there are not a(s),b(s)M'(s)pr(s)such that a(s)g1(s)+b(s)g2(s)=u(s), with u(s)a unit in M'(s)pr(s). Elements in M'(s)pr(s)are of the form n(s)(s2+1)dwith n(s)relatively prime with s2+1and 2dd(n(s))and the units in M'(s)pr(s)are non-zero constants. We will see that there are not elements a(s)=n(s)(s2+1)d, b(s)=n'(s)(s2+1)d'with n(s)and n'(s)coprime with s2+1, 2dd(n(s))and 2d'd(n'(s))such that a(s)g1(s)+b(s)g2(s)=c, with cnon-zero constant. Assume that n(s)(s2+1)ds2s2+1+n'(s)(s2+1)d'ss2+1=c.We conclude that c(s2+1)d+1or c(s2+1)d'+1is a multiple of s, which is impossible.

Example 11 Let 𝔽=and M'=Specm([s]){(s2+1)}. A fraction g(s)=n(s)(s2+1)dM'(s)pr(s)if and only if 2d-d(n(s))0. Let g1(s)=s2(s2+1)3,g2(s)=s(s+1)(s2+1)4M'(s)pr(s). By Lemma :

  1. g(s)g1(s)n(s)s2and02d-d(n(s))6-2=4

  2. g(s)g2(s)n(s)s(s+1)and02d-d(n(s))8-2=6.

If n(s)s2and n(s)s(s+1)then n(s)=cor n(s)=cswith ca non-zero constant. Then g(s)g1(s)and g(s)g2(s)if and only if n(s)=cand d2or n(s)=csand 2d5. So, the list of common divisors of g1(s)and g2(s)is:

c,cs2+1,c(s2+1)2,css2+1,cs(s2+1)2:c𝔽,c0.uid42

If there would be a greatest common divisor, say n(s)(s2+1)d, then n(s)=csbecause n(s)must be a multiple of cand cs. Thus such a greatest common divisor should be either css2+1or cs(s2+1)2, but c(s2+1)2does not divide neither of them because

4=δc(s2+1)2-d(c)>maxδcss2+1-d(cs),δcs(s2+1)2-d(cs)=3.uid43

Thus, g1(s)and g2(s)do not have greatest common divisor.

3.1. Smith–McMillan form

A matrix U(s)is invertible in 𝔽M'(s)m×m𝔽pr(s)m×mif U(s)𝔽M'(s)m×m𝔽pr(s)m×mand its determinant is a unit in both rings, 𝔽M'(s)and 𝔽pr(s), i.e., U(s)Glm(𝔽M'(s)𝔽pr(s))if and only if U(s)Glm(𝔽M'(s))Glm(𝔽pr(s)).

Two matrices G1(s),G2(s)𝔽M'(s)m×m𝔽pr(s)m×mare equivalent in 𝔽M'(s)𝔽pr(s)if there exist U1(s),U2(s)invertible in 𝔽M'(s)m×m𝔽pr(s)m×msuch that

G2(s)=U1(s)G1(s)U2(s).uid45

If there are ideals in Specm(𝔽[s])M'generated by linear polynomials then 𝔽M'(s)𝔽pr(s)is an Euclidean ring and any matrix with elements in 𝔽M'(s)𝔽pr(s)admits a Smith normal form (see [10], [11] or [12]). Bearing in mind the characterization of divisibility in 𝔽M'(s)𝔽pr(s)given in Lemma we have

Theorem 12 (Smith normal form in 𝔽M'(s)𝔽pr(s)) Let M'Specm(𝔽[s]). Assume that there are ideals in Specm(𝔽[s])M'generated by linear polynomials and let (s-a)be one of them. Let G(s)𝔽M'(s)m×m𝔽pr(s)m×mbe non-singular. Then there exist U1(s),U2(s)invertible in 𝔽M'(s)m×m𝔽pr(s)m×msuch that

G(s)=U1(s)Diagn1(s)1(s-a)d1,...,nm(s)1(s-a)dmU2(s)uid47

with n1(s)||nm(s)monic polynomials factorizing in M'and d1,...,dmintegers such that 0d1-d(n1(s))dm-d(nm(s)).

Under the hypothesis of the last theorem n1(s)1(s-a)d1,...,nm(s)1(s-a)dmform a complete system of invariants for the equivalence in 𝔽M'(s)𝔽pr(s)and are called the invariant rational functions of G(s)in 𝔽M'(s)𝔽pr(s). Notice that 0d1dmbecause ni(s)divides ni+1(s).

Recall that the field of fractions of 𝔽M'(s)𝔽pr(s)is 𝔽(s)when M'Specm(𝔽[s]). Thus we can talk about equivalence of matrix rational functions. Two rational matrices T1(s),T2(s)𝔽(s)m×mare equivalent in 𝔽M'(s)𝔽pr(s)if there are U1(s),U2(s)invertible in 𝔽M'(s)m×m𝔽pr(s)m×msuch that

T2(s)=U1(s)T1(s)U2(s).uid48

When all ideals generated by linear polynomials are not in M', each rational matrix admits a reduction to Smith–McMillan form with respect to 𝔽M'(s)𝔽pr(s).

Theorem 13 (Smith–McMillan form in 𝔽M'(s)𝔽pr(s)) Let M'Specm(𝔽[s]). Assume that there are ideals in Specm(𝔽[s])M'generated by linear polynomials and let (s-a)be any of them. Let T(s)𝔽(s)m×mbe a non-singular matrix. Then there exist U1(s),U2(s)invertible in 𝔽M'(s)m×m𝔽pr(s)m×msuch that

T(s)=U1(s)Diagϵ1(s)(s-a)n1ψ1(s)(s-a)d1,...,ϵm(s)(s-a)nmψm(s)(s-a)dmU2(s)uid50

with ϵi(s)(s-a)ni,ψi(s)(s-a)di𝔽M'(s)𝔽pr(s)coprime for all isuch that ϵi(s), ψi(s)are monic polynomials factorizing in M', ϵi(s)(s-a)nidivides ϵi+1(s)(s-a)ni+1for i=1,...,m-1while ψi(s)(s-a)didivides ψi-1(s)(s-a)di-1for i=2,...,m.

The elements ϵi(s)(s-a)niψi(s)(s-a)diof the diagonal matrix, satisfying the conditions of the previous theorem, constitute a complete system of invariant for the equivalence in 𝔽M'(s)𝔽pr(s)of rational matrices. However, this system of invariants is not minimal. A smaller one can be obtained by substituting each pair of positive integers (ni,di)by its difference li=ni-di.

Theorem 14 Under the conditions of Theorem , ϵi(s)ψi(s)1(s-a)liwith ϵi(s), ψi(s)monic and coprime polynomials factorizing in M', ϵi(s)ϵi+1(s)while ψi(s)ψi-1(s)and l1,...,lmintegers such that l1+d(ψ1(s))-d(ϵ1(s))lm+d(ψm(s))-d(ϵm(s))also constitute a complete system of invariants for the equivalence in 𝔽M'(s)𝔽pr(s).

Proof.- We only have to show that from the system ϵi(s)ψi(s)1(s-a)li, i=1,...,m, satisfying the conditions of Theorem , the system ϵi(s)(s-a)niψi(s)(s-a)di, i=1,...,n, can be constructed satisfying the conditions of Theorem .

Suppose that ϵi(s), ψi(s)are monic and coprime polynomials factorizing in M'such that ϵi(s)ϵi+1(s)and ψi(s)ψi-1(s). And suppose also that l1,...,lmare integers such that l1+d(ψ1(s))-d(ϵ1(s))lm+d(ψm(s))-d(ϵm(s)). If li+d(ψi(s))-d(ϵi(s))0for all i, we define non-negative integers ni=d(ϵi(s))and di=d(ϵi(s))-lifor i=1,...,m. If li+d(ψi(s))-d(ϵi(s))>0for all i, we define ni=li+d(ψi(s))and di=d(ψi(s)). Otherwise there is an index k{2,...,m}such that

lk-1+d(ψk-1(s))-d(ϵk-1(s))0<lk+d(ψk(s))-d(ϵk(s)).uid52

Define now the non-negative integers ni,dias follows:

ni=d(ϵi(s))ifi<kli+d(ψi(s))ifikdi=d(ϵi(s))-liifi<kd(ψi(s))ifikuid53

Notice that li=ni-di. Moreover,

ni-d(ϵi(s))=0ifi<kli+d(ψi(s))-d(ϵi(s))ifikuid54
di-d(ψi(s))=-li-d(ψi(s))+d(ϵi(s))ifi<k0ifikuid55

and using (), ()

n1-d(ϵ1(s))==nk-1-d(ϵk-1(s))=0<nk-d(ϵk(s))nm-d(ϵm(s))uid56
d1-d(ψ1(s))dk-1-d(ψk-1(s))0=dk-d(ψk(s))==dm-d(ψm(s)).uid57

In any case ϵi(s)(s-a)niand ψi(s)(s-a)diare elements of 𝔽M'(s)𝔽pr(s). Now, on the one hand ϵi(s),ψi(s)are coprime and ni-d(ϵi(s))=0or di-d(ψi(s))=0. This means (Lemma ) that ϵi(s)(s-a)ni,ψi(s)(s-a)diare coprime for all i. On the other hand ϵi(s)ϵi+1(s)and 0ni-d(ϵi(s))ni+1-d(ϵi+1(s)). Then (Lemma ) ϵi(s)(s-a)nidivides ϵi+1(s)(s-a)ni+1. Similarly, since ψi(s)ψi-1(s)and 0di-d(ψi(s))di-1-d(ψi-1(s)), it follows that ψi(s)(s-a)didivides ψi-1(s)(s-a)di-1.

We call ϵi(s)ψi(s)1(s-a)li, i=1,...,m, the invariant rational functions of T(s)in 𝔽M'(s)𝔽pr(s).

There is a particular case worth considering: If M'=then 𝔽(s)𝔽pr(s)=𝔽pr(s)and (s)Specm(𝔽[s])M'=Specm(𝔽[s]). In this case, we obtain the invariant rational functions of T(s)at infinity (recall ()).

4. Wiener–Hopf equivalence

The left Wiener–Hopf equivalence of rational matrices with respect to a closed contour in the complex plane has been extensively studied ([5] or [6]). Now we present the generalization to arbitrary fields ([13]).

Definition 15 Let Mand M'be subsets of Specm(𝔽[s])such that MM'=Specm(𝔽[s]). Let T1(s),T2(s)𝔽(s)m×mbe two non-singular rational matrices with no zeros and no poles in MM'. The matrices T1(s),T2(s)are said to be left Wiener–Hopf equivalent with respect to (M,M')if there exist both U1(s)invertible in 𝔽M'(s)m×m𝔽pr(s)m×mand U2(s)invertible in 𝔽M(s)m×msuch that

T2(s)=U1(s)T1(s)U2(s).uid59

This is, in fact, an equivalence relation as it is easily seen. It would be an equivalence relation even if no condition about the union and intersection of Mand M'were imposed. It will be seen later on that these conditions are natural assumptions for the existence of unique diagonal representatives in each class.

The right Wiener–Hopf equivalence with respect to (M,M')is defined in a similar manner: There are invertible matrices U1(s)in 𝔽M'(s)m×m𝔽pr(s)m×mand U2(s)in 𝔽M(s)m×msuch that

T2(s)=U2(s)T1(s)U1(s).uid60

In the following only the left Wiener–Hopf equivalence will be considered, but, by transposition, all results hold for the right Wiener–Hopf equivalence as well.

The aim of this section is to obtain a complete system of invariants for the Wiener–Hopf equivalence with respect to (M,M')of rational matrices, and to obtain, if possible, a canonical form.

There is a particular case that is worth-considering: If M=Specm(𝔽[s])and M'=, the invertible matrices in 𝔽(s)m×m𝔽pr(s)m×mare the biproper matrices and the invertible matrices in 𝔽Specm(𝔽[s])(s)m×mare the unimodular matrices. In this case, the left Wiener–Hopf equivalence with respect to (M,M')=(Specm(𝔽[s]),)is the so-called left Wiener–Hopf equivalence at infinity (see [14]). It is known that any non-singular rational matrix is left Wiener–Hopf equivalent at infinity to a diagonal matrix Diag(sg1,...,sgm)where g1,...,gmare integers, that is, for any non-singular T(s)𝔽(s)m×mthere exist both a biproper matrix B(s)Glm(𝔽pr(s))and a unimodular matrix U(s)Glm(𝔽[s])such that

T(s)=B(s)Diag(sg1,...,sgm)U(s)uid61

where g1gmare integers uniquely determined by T(s). They are called the left Wiener–Hopf factorization indices at infinity and form a complete system of invariants for the left Wiener–Hopf equivalence at infinity. These are the basic objects that will produce the complete system of invariants for the left Wiener–Hopf equivalence with respect to (M,M').

For polynomial matrices, their left Wiener–Hopf factorization indices at infinity are the column degrees of any right equivalent (by a unimodular matrix) column proper matrix. Namely, a polynomial matrix is column proper if it can be written as PcDiag(sg1,...,sgm)+L(s)with Pc𝔽m×mnon-singular, g1,...,gmnon-negative integers and L(s)a polynomial matrix such that the degree of the ith column of L(s)smaller than gi, 1im. Let P(s)𝔽[s]m×mbe non-singular polynomial. There exists a unimodular matrix V(s)𝔽[s]m×msuch that P(s)V(s)is column proper. The column degrees of P(s)V(s)are uniquely determined by P(s), although V(s)is not (see [14], p. 388[15], [16]). Since P(s)V(s)is column proper, it can be written as P(s)V(s)=PcD(s)+L(s)with Pcnon-singular, D(s)=Diag(sg1,...,sgm)and the degree of the ith column of L(s)smaller than gi, 1im. Then P(s)V(s)=(Pc+L(s)D(s)-1)D(s). Put B(s)=Pc+L(s)D(s)-1. Since Pcis non-singular and L(s)D(s)-1is a strictly proper matrix, B(s)is biproper, and P(s)=B(s)D(s)U(s)where U(s)=V(s)-1.

The left Wiener–Hopf factorization indices at infinity can be used to associate a sequence of integers with every non-singular rational matrix and every MSpecm(𝔽[s]). This is done as follows: If T(s)𝔽(s)m×mthen it can always be written as T(s)=TL(s)TR(s)such that the global invariant rational functions of TL(s)factorize in Mand TR(s)Glm(𝔽M(s))or, equivalently, the global invariant rational functions of TR(s)factorize in Specm(𝔽[s])M(see Proposition ). There may be many factorizations of this type, but it turns out (see Proposition 3.2[4] for the polynomial case) that the left factors in all of them are right equivalent. This means that if T(s)=TL1(s)TR1(s)=TL2(s)TR2(s)with the global invariant rational functions of TL1(s)and TL2(s)factorizing in Mand the global invariant rational functions of TR1(s)and TR2(s)factorizing in Specm(𝔽[s])Mthen there is a unimodular matrix U(s)such that TL1(s)=TL2(s)U(s). In particular, TL1(s)and TL2(s)have the same left Wiener–Hopf factorization indices at infinity. Thus the following definition makes sense:

Definition 16 Let T(s)𝔽(s)m×mbe a non-singular rational matrix and MSpecm(𝔽[s]). Let TL(s),TR(s)𝔽(s)m×msuch that

  1. T(s)=TL(s)TR(s),

  2. the global invariant rational functions of TL(s)factorize in M, and

  3. the global invariant rational functions of TR(s)factorize in Specm(𝔽[s])M.

Then the left Wiener–Hopf factorization indices of T(s)with respect to Mare defined to be the left Wiener–Hopf factorization indices of TL(s)at infinity.

In the particular case that M=Specm(𝔽[s]), we can put TL(s)=T(s)and TR(s)=Im. Therefore, the left Wiener–Hopf factorization indices of T(s)with respect to Specm(𝔽[s])are the left Wiener–Hopf factorization indices of T(s)at infinity.

We prove now that the left Wiener–Hopf equivalence with respect to (M,M')can be characterized through the left Wiener–Hopf factorization indices with respect to M.

Theorem 17 Let M,M'Specm(𝔽[s])be such that MM'=Specm(𝔽[s]). Let T1(s), T2(s)𝔽(s)m×mbe two non-singular rational matrices with no zeros and no poles in MM'. The matrices T1(s)and T2(s)are left Wiener–Hopf equivalent with respect to (M,M')if and only if T1(s)and T2(s)have the same left Wiener–Hopf factorization indices with respect to M.

Proof.- By Proposition we can write T1(s)=TL1(s)TR1(s),T2(s)=TL2(s)TR2(s)with the global invariant rational functions of TL1(s)and of TL2(s)factorizing in MM'(recall that T1(s)and T2(s)have no zeros and no poles in MM') and the global invariant rational functions of TR1(s)and of TR2(s)factorizing in M'M.

Assume that T1(s),T2(s)have the same left Wiener–Hopf factorization indices with respect to M. By definition, T1(s)and T2(s)have the same left Wiener–Hopf factorization indices with respect to Mif TL1(s)and TL2(s)have the same left Wiener–Hopf factorization indices at infinity. This means that there exist matrices B(s)Glm(𝔽pr(s))and U(s)Glm(𝔽[s])such that TL2(s)=B(s)TL1(s)U(s). We have that T2(s)=TL2(s)TR2(s)=B(s)TL1(s)U(s)TR2(s)=B(s)T1(s)(TR1(s)-1U(s)TR2(s)).We aim to prove that B(s)=TL2(s)U(s)-1TL1(s)-1is invertible in 𝔽M'(s)m×mand TR1(s)-1U(s)TR2(s)Glm(𝔽M(s)). Since the global invariant rational functions of TL2(s)and TL1(s)factorize in MM', TL2(s),TL1(s)𝔽M'(s)m×mand B(s)𝔽M'(s)m×m. Moreover, detB(s)is a unit in 𝔽M'(s)m×mas desired. Now, TR1(s)-1U(s)TR2(s)Glm(𝔽M(s))because TR1(s),TR2(s)𝔽M(s)m×mand detTR1(s)and detTR2(s)factorize in M'M. Therefore T1(s)and T2(s)are left Wiener–Hopf equivalent with respect to (M,M').

Conversely, let U1(s)Glm(𝔽M'(s))Glm(𝔽pr(s))and U2(s)Glm(𝔽M(s))such that T1(s)=U1(s)T2(s)U2(s).Hence, T1(s)=TL1(s)TR1(s)=U1(s)TL2(s)TR2(s)U2(s). Put T¯L2(s)=U1(s)TL2(s)and T¯R2(s)=TR2(s)U2(s). Therefore,

  1. T1(s)=TL1(s)TR1(s)=T¯L2(s)T¯R2(s),

  2. the global invariant rational functions of TL1(s)and of T¯L2(s)factorize in M, and

  3. the global invariant rational functions of TR1(s)and of T¯R2(s)factorize in Specm(𝔽[s])M.

Then TL1(s)and T¯L2(s)are right equivalent (see the remark previous to Definition ). So, there exists U(s)Glm(𝔽[s])such that TL1(s)=T¯L2(s)U(s).Thus, TL1(s)=U1(s)TL2(s)U(s). Since U1(s)is biproper and U(s)is unimodular TL1(s), TL2(s)have the same left Wiener–Hopf factorization indices at infinity. Consequentially, T1(s)and T2(s)have the same left Wiener–Hopf factorization indices with respect to M.

In conclusion, for non-singular rational matrices with no zeros and no poles in MM'the left Wiener–Hopf factorization indices with respect to Mform a complete system of invariants for the left Wiener–Hopf equivalence with respect to (M,M')with MM'=Specm(𝔽[s]).

A straightforward consequence of the above theorem is the following Corollary

Corollary 18 Let M,M'Specm(𝔽[s])be such that MM'=Specm(𝔽[s]). Let T1(s), T2(s)𝔽(s)m×mbe non-singular with no zeros and no poles in MM'. Then T1(s)and T2(s)are left Wiener–Hopf equivalent with respect to (M,M')if and only if for any factorizations T1(s)=TL1(s)TR1(s)and T2(s)=TL2(s)TR2(s)satisfying the conditions (i)–(iii) of Definition , TL1(s)and TL2(s)are left Wiener–Hopf equivalent at infinity.

Next we deal with the problem of factorizing or reducing a rational matrix to diagonal form by Wiener–Hopf equivalence. It will be shown that if there exists in Man ideal generated by a monic irreducible polynomial of degree equal to 1 which is not in M', then any non-singular rational matrix, with no zeros and no poles in MM'admits a factorization with respect to (M,M'). Afterwards, some examples will be given in which these conditions on Mand M'are removed and factorization fails to exist.

Theorem 19 Let M,M'Specm(𝔽[s])be such that MM'=Specm(𝔽[s]). Assume that there are ideals in MM'generated by linear polynomials. Let (s-a)be any of them and T(s)𝔽(s)m×ma non-singular matrix with no zeros and no poles in MM'. There exist both U1(s)invertible in 𝔽M'(s)m×m𝔽pr(s)m×mand U2(s)invertible in 𝔽M(s)m×msuch that

T(s)=U1(s)Diag((s-a)k1,...,(s-a)km)U2(s),uid72

where k1kmare integers uniquely determined by T(s). Moreover, they are the left Wiener–Hopf factorization indices of T(s)with respect to M.

Proof.- The matrix T(s)can be written (see Proposition ) as T(s)=TL(s)TR(s)with the global invariant rational functions of TL(s)factorizing in MM'and the global invariant rational functions of TR(s)factorizing in Specm(𝔽[s])M=M'M. As k1,...,kmare the left Wiener–Hopf factorization indices of TL(s)at infinity, there exist matrices U(s)Glm(𝔽[s])and B(s)Glm(𝔽pr(s))such that TL(s)=B(s)D1(s)U(s)with D1(s)=Diag(sk1,...,skm). Put D(s)=Diag((s-a)k1,...,(s-a)km)and U1(s)=B(s)Diagsk1(s-a)k1,...,skm(s-a)km.Then TL(s)=U1(s)D(s)U(s).If U2(s)=U(s)TR(s)then this matrix is invertible in 𝔽M(s)m×mand T(s)=U1(s)Diag((s-a)k1,...,(s-a)km)U2(s). We only have to prove that U1(s)is invertible in 𝔽M'(s)m×m𝔽pr(s)m×m. It is clear that U1(s)is in 𝔽pr(s)m×mand biproper. Moreover, the global invariant rational functions of TL(s)U1(s)=TL(s)(D(s)U(s))-1factorize in MM'. Therefore, U1(s)is invertible in 𝔽M'(s)m×m.

We prove now the uniqueness of the factorization. Assume that T(s)also factorizes as

T(s)=U˜1(s)Diag((s-a)k˜1,...,(s-a)k˜m)U˜2(s),uid73

with k˜1k˜mintegers. Then,

Diag((s-a)k˜1,...,(s-a)k˜m)=U˜1(s)-1U1(s)Diag((s-a)k1,...,(s-a)km)U2(s)U˜2(s)-1.uid74

The diagonal matrices have no zeros and no poles in MM'(because (s-a)MM') and they are left Wiener–Hopf equivalent with respect to (M,M'). By Theorem , they have the same left Wiener–Hopf factorization indices with respect to M. Thus, k˜i=kifor all i=1,...,m.

Following [5] we could call left Wiener–Hopf factorization indices with respect to (M,M')the exponents k1kmappearing in the diagonal matrix of Theorem . They are, actually, the left Wiener–Hopf factorization indices with respect to M.

Several examples follow that exhibit some remarkable features about the results that have been proved so far. The first two examples show that if no assumption is made on the intersection and/or union of Mand M'then existence and/or uniqueness of diagonal factorization may fail to exist.

Example 20

If P(s)is a polynomial matrix with zeros in MM'then the existence of invertible matrices U1(s)Glm(𝔽M'(s))Glm(𝔽pr(s))and U2(s)Glm(𝔽M(s))such that P(s)=U1(s)Diag((s-a)k1,...,(s-a)km)U2(s)with (s-a)MM'may fail. In fact, suppose that M={(s),(s+1)}, M'=Specm𝔽[s]{(s)}. Therefore, MM'={(s+1)}and (s)MM'. Consider p1(s)=s+1. Assume that s+1=u1(s)sku2(s)with u1(s)a unit in 𝔽M'(s)𝔽pr(s)and u2(s)a unit in 𝔽M(s). Thus, u1(s)=ca nonzero constant and u2(s)=1cs+1skwhich is not a unit in 𝔽M(s).

Example 21

If MM'Specm𝔽[s]then the factorization indices with respect to (M,M')may be not unique. Suppose that (β(s))MM', (π(s))MM'with d(π(s))=1and p(s)=u1(s)π(s)ku2(s), with u1(s)a unit in 𝔽M'(s)𝔽pr(s)and u2(s)a unit in 𝔽M(s). Then p(s)can also be factorized as p(s)=u˜1(s)π(s)k-d(β(s))u˜2(s)with u˜1(s)=u1(s)π(s)d(β(s))β(s)a unit in 𝔽M'(s)𝔽pr(s)and u˜2(s)=β(s)u2(s)a unit in 𝔽M(s).

The following example shows that if all ideals generated by polynomials of degree equal to one are in M'Mthen a factorization as in Theorem may not exist.

Example 22 Suppose that 𝔽=. Consider M={(s2+1)}Specm([s])and M'=Specm([s]){(s2+1)}. Let

P(s)=s0-s2(s2+1)2.uid78

Notice that P(s)has no zeros and no poles in MM'=. We will see that it is not possible to find invertible matrices U1(s)M'(s)2×2pr(s)2×2and U2(s)M(s)2×2such that

U1(s)P(s)U2(s)=Diag((p(s)/q(s))c1,(p(s)/q(s))c2).uid79

We can write p(s)q(s)=u(s)(s2+1)awith u(s)a unit in M(s)and a. Therefore,

Diag((p(s)/q(s))c1,(p(s)/q(s))c2)=Diag((s2+1)ac1,(s2+1)ac2)Diag(u(s)c1,u(s)c2).uid80

Diag(u(s)c1,u(s)c2)is invertible in M(s)2×2and P(s)is also left Wiener–Hopf equivalent with respect to (M,M')to the diagonal matrix Diag((s2+1)ac1,(s2+1)ac2).

Assume that there exist invertible matrices U1(s)M'(s)2×2pr(s)2×2and U2(s)M(s)2×2such that U1(s)P(s)U2(s)=Diag((s2+1)d1,(s2+1)d2), with d1d2integers. Notice first that detU1(s)is a nonzero constant and since detP(s)=s(s2+1)2and detU2(s)is a rational function with numerator and denominator relatively prime with s2+1, it follows that cs(s2+1)2detU2(s)=(s2+1)d1+d2. Thus, d1+d2=2. Let

U1(s)-1=b11(s)b12(s)b21(s)b22(s),U2(s)=u11(s)u12(s)u21(s)u22(s).uid81

From P(s)U2(s)=U1(s)-1Diag((s2+1)d1,(s2+1)d2)we get

su11(s)=b11(s)(s2+1)d1,uid82
-s2u11(s)+(s2+1)2u21(s)=b21(s)(s2+1)d1,uid83
su12(s)=b12(s)(s2+1)d2,uid84
-s2u12(s)+(s2+1)2u22(s)=b22(s)(s2+1)d2.uid85

As u11(s)M(s)and b11(s)M'(s)pr(s), we can write u11(s)=f1(s)g1(s)and b11(s)=h1(s)(s2+1)q1with f1(s),g1(s),h1(s)[s], gcd(g1(s),s2+1)=1and d(h1(s))2q1. Therefore, by (), sf1(s)g1(s)=h1(s)(s2+1)q1(s2+1)d1. Hence, u11(s)=f1(s)or u11(s)=f1(s)s. In the same way and using (), u12(s)=f2(s)or u12(s)=f2(s)swith f2(s)a polynomial. Moreover, by (), d2must be non-negative. Hence, d1d20. Using now () and () and bearing in mind again that u21(s),u22(s)M(s)and b21(s),b22(s)M'(s)pr(s), we conclude that u21(s)and u22(s)are polynomials.

We can distinguish two cases: d1=2, d2=0and d1=d2=1. If d1=2and d2=0, by (), b12(s)is a polynomial and since b12(s)is proper, it is constant: b12(s)=c1. Thus u12(s)=c1s. By (), b22(s)=-c1s+(s2+1)2u22(s). Since u22(s)is polynomial and b22(s)is proper, b22(s)is also constant and then u22(s)=0and c1=0. Consequentially, b22(s)=0, and b12(s)=0. This is impossible because U1(s)is invertible.

If d1=d2=1then , using (),

b21(s)=-s2u11(s)+(s2+1)2u21(s)s2+1=-s2b11(s)s(s2+1)+(s2+1)2u21(s)s2+1=-sb11(s)+(s2+1)u21(s)=-sh1(s)(s2+1)q1+(s2+1)u21(s)=-sh1(s)+(s2+1)q1+1u21(s)(s2+1)q1.uid86

Notice that d(-sh1(s))1+2q1and d((s2+1)q1+1u21(s))=2(q1+1)+d(u21(s))2q1+2unless u21(s)=0. Hence, if u21(s)0, d(-sh1(s)+(s2+1)q1+1u21(s))2q1+2which is greater than d((s2+1)q1)=2q1. This cannot happen because b21(s)is proper. Thus, u21(s)=0. In the same way and reasoning with () we get that u22(s)is also zero. This is again impossible because U2(s)is invertible. Therefore no left Wiener–Hopf factorization of P(s)with respect to (M,M')exits.

We end this section with an example where the left Wiener–Hopf factorization indices of the matrix polynomial in the previous example are computed. Then an ideal generated by a polynomial of degree 1 is added to Mand the Wiener–Hopf factorization indices of the same matrix are obtained in two different cases.

Example 23 Let 𝔽=and M={(s2+1)}. Consider the matrix

P(s)=s0-s2(s2+1)2,uid88

which has a zero at 0. It can be written as P(s)=P1(s)P2(s)with

P1(s)=10-s(s2+1)2,P2(s)=s001,uid89

where the global invariant factors of P1(s)are powers of s2+1and the global invariant factors of P2(s)are relatively prime with s2+1. Moreover, the left Wiener–Hopf factorization indices of P1(s)at infinity are 3, 1 (add the first column multiplied by s3+2sto the second column; the result is a column proper matrix with column degrees 1 and 3). Therefore, the left Wiener–Hopf factorization indices of P(s)with respect to Mare 3, 1.

Consider now M˜={(s2+1),(s)}and M˜'=Specm([s])M˜. There is a unimodular matrix U(s)=1s2+201, invertible in M˜(s)2×2, such that P(s)U(s)=ss3+2s-s21is column proper with column degrees 3 and 2. We can write

P(s)U(s)=01-10s200s3+s2s01=B(s)s200s3,uid90

where B(s)is the following biproper matrix

B(s)=01-10+s2s01s-200s-3=1ss2+2s2-11s3.uid91

Moreover, the denominators of its entries are powers of sand detB(s)=(s2+1)2s4. Therefore, B(s)is invertible in M˜'(s)2×2pr(s)2×2. Since B(s)-1P(s)U(s)=Diag(s2,s3), the left Wiener–Hopf factorization indices of P(s)with respect to M˜are 3, 2.

If M˜={(s2+1),(s-1)}, for example, a similar procedure shows that P(s)has 3,1as left Wiener–Hopf factorization indices with respect to M˜; the same indices as with respect to M. The reason is that s-1is not a divisor of detP(s)and so P(s)=P1(s)P2(s)with P1(s)and P2(s)as in () and P1(s)factorizing in M˜.

Remark 24 It must be noticed that a procedure has been given to compute, at least theoretically, the left Wiener–Hopf factorization indices of any rational matrix with respect to any subset Mof Specm(𝔽[s]). In fact, given a rational matrix T(s)and M, write T(s)=TL(s)TR(s)with the global invariant rational functions of TL(s)factorizing in M, and the global invariant rational functions of TR(s)factorizing in Specm(𝔽[s])M(for example, using the global Smith–McMillan form of T(s)). We need to compute the left Wiener–Hopf factorization indices at infinity of the rational matrix TL(s). The idea is as follows: Let d(s)be the monic least common denominator of all the elements of TL(s). The matrix TL(s)can be written as TL(s)=P(s)d(s), with P(s)polynomial. The left Wiener–Hopf factorization indices of P(s)at infinity are the column degrees of any column proper matrix right equivalent to P(s). If k1,...,kmare the left Wiener–Hopf factorization indices at infinity of P(s)then k1+d,...,km+dare the left Wiener–Hopf factorization indices of TL(s), where d=d(d(s))(see [4]). Free and commercial software exists that compute such column degrees.

5. Rosenbrock's Theorem via local rings

As said in the Introduction, Rosenbrock's Theorem ([1]) on pole assignment by state feedback provides, in its polynomial formulation, a complete characterization of the relationship between the invariant factors and the left Wiener–Hopf factorization indices at infinity of any non-singular matrix polynomial. The precise statement of this result is the following theorem:

Theorem 25 Let g1gmand α1(s)αm(s)be non-negative integers and monic polynomials, respectively. Then there exists a non-singular matrix P(s)𝔽[s]m×mwith α1(s),...,αm(s)as invariant factors and g1,...,gmas left Wiener–Hopf factorization indices at infinity if and only if the following relation holds:

(g1,...,gm)(d(αm(s)),...,d(α1(s))).uid94

Symbol appearing in () is the majorization symbol (see [17]) and it is defined as follows: If (a1,...,am)and (b1,...,bm)are two finite sequences of real numbers and a[1]a[m]and b[1]b[m]are the given sequences arranged in non-increasing order then (a1,...,am)(b1,...,bm)if

i=1ja[i]i=1jb[i],1jm-1uid95

with equality for j=m.

The above Theorem can be extended to cover rational matrix functions. Any rational matrix T(s)can be written as N(s)d(s)where d(s)is the monic least common denominator of all the elements of T(s)and N(s)is polynomial. It turns out that the invariant rational functions of T(s)are the invariant factors of N(s)divided by d(s)after canceling common factors. We also have the following characterization of the left Wiener– Hopf factorization indices at infinity of T(s): these are those of N(s)plus the degree of d(s)(see [4]). Bearing all this in mind one can easily prove (see [4])

Theorem 26 Let g1gmbe integers and α1(s)β1(s),...,αm(s)βm(s)irreducible rational functions, where αi(s),βi(s)𝔽[s]are monic such that α1(s)αm(s)while βm(s)β1(s). Then there exists a non-singular rational matrix T(s)𝔽(s)m×mwith g1,...,gmas left Wiener–Hopf factorization indices at infinity and α1(s)β1(s),...,αm(s)βm(s)as global invariant rational functions if and only if

(g1,...,gm)(d(αm(s))-d(βm(s)),...,d(α1(s))-d(β1(s))).uid97

Recall that for MSpecm(𝔽[s])any rational matrix T(s)can be factorized into two matrices (see Proposition ) such that the global invariant rational functions and the left Wiener–Hopf factorization indices at infinity of the left factor of T(s)give the invariant rational functions and the left Wiener–Hopf factorization indices of T(s)with respect to M. Using Theorem on the left factor of T(s)we get:

Theorem 27 Let MSpecm(𝔽[s]). Let k1kmbe integers and ϵ1(s)ψ1(s),...,ϵm(s)ψm(s)be irreducible rational functions such that ϵ1(s)ϵm(s), ψm(s)ψ1(s)are monic polynomials factorizing in M. Then there exists a non-singular matrix T(s)𝔽(s)m×mwith ϵ1(s)ψ1(s),...,ϵm(s)ψm(s)as invariant rational functions with respect to Mand k1,...,kmas left Wiener–Hopf factorization indices with respect to Mif and only if

(k1,...,km)(d(ϵm(s))-d(ψm(s)),...,d(ϵ1(s))-d(ψ1(s))).uid99

Theorem relates the left Wiener–Hopf factorization indices with respect to Mand the finite structure inside M. Our last result will relate the left Wiener–Hopf factorization indices with respect to Mand the structure outside M, including that at infinity. The next Theorem is an extension of Rosenbrock's Theorem to the point at infinity, which was proved in [4]:

Theorem 28 Let g1gmand q1qmbe integers. Then there exists a non-singular matrix T(s)𝔽(s)m×mwith g1,...,gmas left Wiener–Hopf factorization indices at infinity and sq1,...,sqmas invariant rational functions at infinity if and only if

(g1,...,gm)(q1,...,qm).uid101

Notice that Theorem can be obtained from Theorem when M=Specm(𝔽[s]). In the same way, taking into account that the equivalence at infinity is a particular case of the equivalence in 𝔽M'(s)𝔽pr(s)when M'=, we can give a more general result than that of Theorem . Specifically, necessary and sufficient conditions can be provided for the existence of a non-singular rational matrix with prescribed left Wiener–Hopf factorization indices with respect to Mand invariant rational functions in 𝔽M'(s)𝔽pr(s).

Theorem 29 Let M,M'Specm(𝔽[s])be such that MM'=Specm(𝔽[s]). Assume that there are ideals in MM'generated by linear polynomials and let (s-a)be any of them. Let k1kmbe integers, ϵ1(s)ψ1(s),...,ϵm(s)ψm(s)irreducible rational functions such that ϵ1(s)||ϵm(s), ψm(s)||ψ1(s)are monic polynomials factorizing in M'Mand l1,...,lmintegers such that l1+d(ψ1(s))-d(ϵ1(s))lm+d(ψm(s))-d(ϵm(s)). Then there exists a non-singular matrix T(s)𝔽(s)m×mwith no zeros and no poles in MM'with k1,...,kmas left Wiener–Hopf factorization indices with respect to Mand ϵ1(s)ψ1(s)1(s-a)l1,...,ϵm(s)ψm(s)1(s-a)lmas invariant rational functions in 𝔽M'(s)𝔽pr(s)if and only if the following condition holds:

(k1,...,km)(-l1,...,-lm).uid103

The proof of this theorem will be given along the following two subsections. We will use several auxiliary results that will be stated and proved when needed.

5.1. Necessity

We can give the following result for rational matrices using a similar result given in Lemma 4.2 in [18] for matrix polynomials.

Lemma 30 Let M,M'Specm(𝔽[s])be such that MM'=Specm(𝔽[s]). Let T(s)𝔽(s)m×mbe a non-singular matrix with no zeros and no poles in MM'with g1gmas left Wiener–Hopf factorization indices at infinity and k1kmas left Wiener–Hopf factorization indices with respect to M. If ϵ1(s)ψ1(s),...,ϵm(s)ψm(s)are the invariant rational functions of T(s)with respect to M'then

(g1-k1,...,gm-km)(d(ϵm(s))-d(ψm(s)),...,d(ϵ1(s))-d(ψ1(s))).uid106

It must be pointed out that (g1-k1,...,gm-km)may be an unordered m-tuple.

Proof.- By Proposition there exist unimodular matrices U(s),V(s)𝔽[s]m×msuch that

T(s)=U(s)Diagα1(s)β1(s),...,αm(s)βm(s)Diagϵ1(s)ψ1(s),...,ϵm(s)ψm(s)V(s)uid107

with αi(s)αi+1(s), βi(s)βi-1(s), ϵi(s)ϵi+1(s), ψi(s)ψi-1(s), αi(s),βi(s)units in 𝔽M'M(s)and ϵi(s),ψi(s)factorizing in M'Mbecause T(s)has no poles and no zeros in MM'. Therefore T(s)=TL(s)TR(s), where TL(s)=U(s)Diagα1(s)β1(s),...,αm(s)βm(s)has k1,...,kmas left Wiener–Hopf factorization indices at infinity and TR(s)=Diagϵ1(s)ψ1(s),...,ϵm(s)ψm(s)V(s)has ϵ1(s)ψ1(s),...,ϵm(s)ψm(s)as global invariant rational functions. Let d(s)=β1(s)ψ1(s). Hence,

d(s)T(s)=U(s)Diag(α¯1(s),...,α¯m(s))Diag(ϵ¯1(s),...,ϵ¯m(s))V(s)uid108

with α¯i(s)=αi(s)βi(s)β1(s)units in 𝔽M'M(s)and ϵ¯i(s)=ϵi(s)ψi(s)ψ1(s)factorizing in M'M. Put P(s)=d(s)T(s). Its left Wiener–Hopf factorization indices at infinity are g1+d(d(s)),...,gm+d(d(s))Lemma 2.3[4]. The matrix P1(s)=U(s)Diag(α¯1(s),...,α¯m(s))=β1(s)TL(s)has k1+d(β1(s)),...,km+d(β1(s))as left Wiener–Hopf factorization indices at infinity. Now if P2(s)=Diag(ϵ¯1(s),...,ϵ¯m(s))V(s)=ψ1(s)TR(s)then its invariant factors are ϵ¯1(s),...,ϵ¯m(s), P(s)=P1(s)P2(s)and, by Lemma 4.2[18],

(g1+d(d(s))-k1-d(β1(s)),...,gm+d(d(s))-km-d(β1(s)))(d(ϵ¯m(s)),...,d(ϵ¯1(s))).uid109

Therefore, () follows.

5.1.1. Proof of Theorem : Necessity

If ϵ1(s)ψ1(s)1(s-a)l1,...,ϵm(s)ψm(s)1(s-a)lmare the invariant rational functions of T(s)in 𝔽M'(s)𝔽pr(s)then there exist matrices U1(s),U2(s)invertible in 𝔽M'(s)m×m𝔽pr(s)m×msuch that

T(s)=U1(s)Diagϵ1(s)ψ1(s)1(s-a)l1,...,ϵm(s)ψm(s)1(s-a)lmU2(s).uid111

We analyze first the finite structure of T(s)with respect to M'. If D1(s)=Diag((s-a)-l1,...,(s-a)-lm)𝔽M'(s)m×m, we can write T(s)as follows:

T(s)=U1(s)Diagϵ1(s)ψ1(s),...,ϵm(s)ψm(s)D1(s)U2(s),uid112

with U1(s)and D1(s)U2(s)invertible matrices in 𝔽M'(s)m×m. Thus ϵ1(s)ψ1(s),...,ϵm(s)ψm(s)are the invariant rational functions of T(s)with respect to M'. Let g1gmbe the left Wiener–Hopf factorization indices of T(s)at infinity. By Lemma we have

(g1-k1,...,gm-km)(d(ϵm(s))-d(ψm(s)),...,d(ϵ1(s))-d(ψ1(s))).uid113

As far as the structure of T(s)at infinity is concerned, let

D2(s)=Diagϵ1(s)ψ1(s)sl1+d(ψ1(s))-d(ϵ1(s))(s-a)l1,...,ϵm(s)ψm(s)slm+d(ψm(s))-d(ϵm(s))(s-a)lm.uid114

Then D2(s)Gl(𝔽pr(s))and

T(s)=U1(s)Diags-l1-d(ψ1(s))+d(ϵ1(s)),...,s-lm-d(ψm(s))+d(ϵm(s))D2(s)U2(s)uid115

where U1(s)𝔽pr(s)m×mand D2(s)U2(s)𝔽pr(s)m×mare biproper matrices. Therefore s-l1-d(ψ1(s))+d(ϵ1(s)),...,s-lm-d(ψm(s))+d(ϵm(s))are the invariant rational functions of T(s)at infinity. By Theorem

(g1,...,gm)(-l1-d(ψ1(s))+d(ϵ1(s)),...,-lm-d(ψm(s))+d(ϵm(s))).uid116

Let σΣm(the symmetric group of order m) be a permutation such that gσ(1)-kσ(1)gσ(m)-kσ(m)and define ci=gσ(i)-kσ(i), i=1,...,m. Using () and () we obtain

j=1rkj+j=1r(d(ϵj(s))-d(ψj(s)))j=1rkj+j=m-r+1mcjj=1rkj+j=1r(gj-kj)=j=1rgjj=1r-lj+j=1r(d(ϵj(s))-d(ψj(s)))uid117

for r=1,...,m-1. When r=mthe previous inequalities are all equalities and condition () is satisfied.

Remark 31 It has been seen in the above proof that if a matrix has ϵ1(s)ψ1(s)1(s-a)l1,...,ϵm(s)ψm(s)1(s-a)lmas invariant rational functions in 𝔽M'(s)𝔽pr(s)then ϵ1(s)ψ1(s),...,ϵm(s)ψm(s)are its invariant rational functions with respect to M'and s-l1-d(ψ1(s))+d(ϵ1(s)),...,s-lm-d(ψm(s))+d(ϵm(s))are its invariant rational functions at infinity.

5.2. Sufficiency

Let a,b𝔽be arbitrary elements such that ab1. Consider the changes of indeterminate

f(s)=a+1s-b,f˜(s)=b+1s-auid120

and notice that f(f˜(s))=f˜(f(s))=s. For α(s)𝔽[s], let 𝔽[s](α(s))denote the multiplicative subset of 𝔽[s]whose elements are coprime with α(s). For a,b𝔽as above define

ta,b:𝔽[s]𝔽[s]s-bπ(s)s-bd(π(s))πa+1s-b=s-bd(π(s))π(f(s)).uid121

In words, if π(s)=pd(s-a)d+pd-1(s-a)d-1++p1(s-a)+p0(pd0) then

ta,b(π(s))=p0(s-b)d+p1(s-b)d-1++pd-1(s-b)+pd.uid122

In general d(ta,b(π(s)))d(π(s))with equality if and only if π(s)𝔽[s]s-a. This shows that the restriction ha,b:𝔽[s](s-a)𝔽[s](s-b)of ta,bto 𝔽[s](s-a)is a bijection. In addition ha,b-1is the restriction of tb,ato 𝔽[s](s-b); i.e.,

ha,b-1:𝔽[s](s-b)𝔽[s]s-aα(s)s-ad(α(s))αb+1s-a=s-ad(α(s))α(f˜(s))uid123

or ha,b-1=hb,a.

In what follows we will think of a,bas given elements of 𝔽and the subindices of ta,b, ha,band ha,b-1will be removed. The following are properties of h(and h-1) that can be easily proved.

Lemma 32 Let π1(s),π2(s)𝔽[s](s-a). The following properties hold:

  1. h(π1(s)π2(s))=h(π1(s))h(π2(s)).

  2. If π1(s)π2(s)then h(π1(s))h(π2(s)).

  3. If π1(s)is an irreducible polynomial then h(π1(s))is an irreducible polynomial.

  4. If π1(s),π2(s)are coprime polynomials then h(π1(s)), h(π2(s))are coprime polynomials.

As a consequence the map

H:Specm𝔽[s]){s-aSpecm𝔽[s]){s-b(π(s))(1p0h(π(s)))uid129

with p0=π(a), is a bijection whose inverse is

H-1:Specm𝔽[s]){s-bSpecm𝔽[s]){s-a(α(s))(1a0h-1(α(s)))uid130

where a0=α(b). In particular, if M'Specm(𝔽[s]){(s-a)}and M˜=Specm(𝔽[s])(M'{(s-a)})(i.e. the complementary subset of M'in Specm𝔽[s]){s-a) then

H(M˜)=Specm𝔽[s])(H(M'){s-b).uid131

In what follows and for notational simplicity we will assume b=0.

Lemma 33 Let M'Specm𝔽[s]){s-awhere a𝔽is an arbitrary element of 𝔽.

  1. If π(s)𝔽[s]factorizes in M'then h(π(s))factorizes in H(M').

  2. If π(s)𝔽[s]is a unit of 𝔽M'(s)then t(π(s))is a unit of 𝔽H(M')(s).

Proof.- 1. Let π(s)=cπ1(s)g1πm(s)gmwith c0constant, (πi(s))M'and gi1. Then h(π(s))=c(h(π1(s)))g1(h(πm(s)))gm. By Lemma h(πi(s))is an irreducible polynomial (that may not be monic). If ciis the leading coefficient of h(πi(s))then 1cih(πi(s))is monic, irreducible and (1cih(πi(s)))H(M'). Hence h(π(s))factorizes in H(M').

2. If π(s)𝔽[s]is a unit of 𝔽M'(s)then it can be written as π(s)=(s-a)gπ1(s)where g0and π1(s)is a unit of 𝔽M'{(s-a)}(s). Therefore π1(s)factorizes in Specm(𝔽[s])(M'{(s-a)}). Since t(π(s))=h(π1(s)), it factorizes in (recall that we are assuming b=0) H(Specm(𝔽[s])(M'{(s-a)})=Specm(𝔽[s])(H(M'){(s)}). So, t(π(s))is a unit of 𝔽H(M')(s).

Lemma 34 Let a𝔽be an arbitrary element. Then

  1. If M'Specm𝔽[s]){s-aand U(s)Glm(𝔽M'(s))then U(f(s))Glm(𝔽H(M')(s)).

  2. If U(s)Glm(𝔽s-a(s))then U(f(s))Glm(𝔽pr(s)).

  3. If U(s)Glm(𝔽pr(s))then U(f(s))Glm(𝔽s(s)).

  4. If (s-a)M'Specm𝔽[s]and U(s)Glm(𝔽M'(s))then the matrix U(f(s))Glm(𝔽H(M'{(s-a)})(s))Glm(𝔽pr(s))

Proof.- Let p(s)q(s)with p(s),q(s)𝔽[s].

p(f(s))q(f(s))=sd(p(s))p(f(s))sd(q(s))q(f(s))sd(q(s))-d(p(s))=t(p(s))t(q(s))sd(q(s))-d(p(s)).uid140

1. Assume that U(s)Glm(𝔽M'(s))and let p(s)q(s)be any element of U(s). Therefore q(s)is a unit of 𝔽M'(s)and, by Lemma .2, t(q(s))is a unit of 𝔽H(M')(s). Moreover, sis also a unit of 𝔽H(M')(s). Hence, p(f(s))q(f(s))𝔽H(M')(s). Furthermore, if detU(s)=p˜(s)q˜(s), it is a unit of 𝔽M'(s)and detU(f(s))=p˜(f(s))q˜(f(s))is a unit of 𝔽H(M')(s).

2. If p(s)q(s)is any element of U(s)Glm(𝔽s-a(s))then q(s)𝔽[s](s-a)and so d(h(q(s)))=d(q(s)). Since s-amay divide p(s)we have that d(t(p(s)))d(p(s)). Hence, d(h(q(s)))-d(q(s))d(t(p(s))-d(p(s))and p(f(s))q(f(s))=t(p(s))h(q(s))sd(q(s))-d(p(s))𝔽pr(s). Moreover if detU(s)=p˜(s)q˜(s)then p˜(s),q˜(s)𝔽[s](s-a), d(h(p˜(s)))=d(p˜(s))and d(h(q˜(s)))=d(q˜(s)). Thus, detU(f(s))=h(p˜(s))h(q˜(s))sd(q˜(s))-d(p˜(s))is a biproper rational function, i.e., a unit of 𝔽pr(s).

3. If U(s)Glm(𝔽pr(s))and p(s)q(s)is any element of U(s)then d(q(s))d(p(s)). Since p(f(s))q(f(s))=t(p(s))t(q(s))sd(q(s))-d(p(s))and t(p(s)),t(q(s))𝔽[s](s)we obtain that U(f(s))𝔽s(s)m×m. In addition, if detU(s)=p˜(s)q˜(s), which is a unit of 𝔽pr(s), then d(q˜(s))=d(p˜(s))and since t(p˜(s)),t(q˜(s))𝔽[s](s)we conclude that detU(f(s))=t(p˜(s))t(q˜(s))is a unit of 𝔽s(s).

4. It is a consequence of 1., 2. and Remark .2.

Proposition 35 Let MSpecm(𝔽[s])and (s-a)M. If T(s)𝔽(s)m×mis non-singular with ni(s)di(s)=(s-a)giϵi(s)ψi(s)ϵi(s),ψi(s)𝔽[s](s-a)as invariant rational functions with respect to Mthen T(f(s))T𝔽(s)m×mis a non-singular matrix with 1cih(ϵi(s))h(ψi(s))s-gi+d(ψi(s))-d(ϵi(s))as invariant rational functions in 𝔽H(M{(s-a)})(s)m×m𝔽pr(s)m×mwhere ci=ϵi(a)ψi(a).

Proof.- Since (s-a)giϵi(s)ψ(s)are the invariant rational functions of T(s)with respect to M, there are U1(s),U2(s)Glm(𝔽M(s))such that

T(s)=U1(s)Diag(s-a)g1ϵ1(s)ψ1(s),...,(s-a)gmϵm(s)ψm(s)U2(s).uid142

Notice that f(s)-agiϵi(f(s))ψi(f(s))=h(ϵi(s))h(ψi(s))s-gi+d(ψi(s))-d(ϵi(s)). Let ci=ϵi(a)ψi(a), which is a non-zero constant, and put D=Diagc1,...,cm. Hence,

T(f(s))T=U2(f(s))TDL(s)U1(f(s))Tuid143

with

L(s)=Diag1c1h(ϵ1(s))h(ψ1(s))s-g1+d(ψ1(s))-d(ϵ1(s)),...,1cmh(ϵm(s))h(ψm(s))s-gm+d(ψm(s))-d(ϵm(s)).uid144

By 4 of Lemma matrices U1(f(s))T, U2(f(s))TGlm(𝔽H(M{(s-a)})(s))Glm(𝔽pr(s))and the Proposition follows.

Proposition 36 Let M,M'Specm(𝔽[s])such that MM'=Specm(𝔽[s]). Assume that there are ideals in MM'generated by linear polynomials and let (s-a)be any of them. If T(s)𝔽(s)m×mis a non-singular rational matrix with no poles and no zeros in MM'and k1,...,kmas left Wiener–Hopf factorization indices with respect to Mthen T(f(s))T𝔽(s)m×mis a non-singular rational matrix with no poles and no zeros in H(MM')and -km,...,-k1as left Wiener–Hopf factorization indices with respect to H(M'){(s)}.

Proof.- By Theorem there are matrices U1(s)invertible in 𝔽M'(s)m×m𝔽pr(s)m×mand U2(s)invertible in 𝔽M(s)m×msuch that T(s)=U1(s)Diags-ak1,...,s-akmU2(s).By Lemma U2(f(s))Tis invertible in 𝔽H(M{(s-a)})(s)m×m𝔽pr(s)m×mand U1(f(s))Tis invertible in 𝔽H(M')(s)m×m𝔽s(s)m×m=𝔽H(M'){(s)}(s)m×m. Moreover, H(M{(s-a)})H(M'){(s)}=Specm(𝔽[s])and H(M{(s-a)})(H(M'){(s)})=H(MM'). Thus, T(f(s))T=U2(f(s))TDiags-k1,...,s-kmU1(f(s))Thas no poles and no zeros in H(MM')and -km,...,-k1are its left Wiener–Hopf factorization indices with respect to H(M'){(s)}.

5.2.1. Proof of Theorem : Sufficiency

Let k1kmbe integers, ϵ1(s)ψ1(s),...,ϵm(s)ψm(s)irreducible rational functions such that ϵ1(s)ϵm(s), ψm(s)ψ1(s)are monic polynomials factorizing in M'Mand l1,...,lmintegers such that l1+d(ψ1(s))-d(ϵ1(s))lm+d(ψm(s))-d(ϵm(s))and satisfying ().

Since ϵi(s)and ψi(s)are coprime polynomials that factorize in M'Mand (s-a)MM', by Lemmas and , h(ϵ1(s))h(ψ1(s))sl1+d(ψ1(s))-d(ϵ1(s)),...,h(ϵm(s))h(ψm(s))slm+d(ψm(s))-d(ϵm(s))are irreducible rational functions with numerators and denominators polynomials factorizing in H(M'){(s)}(actually, in H(M'M){(s)}) and such that each numerator divides the next one and each denominator divides the previous one.

By () and Theorem there is a matrix G(s)𝔽(s)m×mwith -km,...,-k1as left Wiener–Hopf factorization indices with respect to H(M'){(s)}and 1c1h(ϵ1(s))h(ψ1(s))sl1+d(ψ1(s))-d(ϵ1(s)),...,1cmh(ϵm(s))h(ψm(s))slm+d(ψm(s))-d(ϵm(s))as invariant rational functions with respect to H(M'){(s)}where ci=ϵi(a)ψi(a), i=1,...,m. Notice that G(s)has no zeros and poles in H(MM')because the numerator and denominator of each rational function h(ϵi(s))h(ψi(s))sli+d(ψi(s))-d(ϵi(s))factorizes in H(M'M){(s)}and so it is a unit of 𝔽H(MM')(s).

Put M^=H(M'){(s)}and M^'=H(M{(s-a)}). As remarked in the proof of Proposition , M^M^'=Specm(𝔽[s])and M^M^'=H(MM'). Now (s)M^so that we can apply Proposition to G(s)with the change of indeterminate f˜(s)=1s-a. Thus the invariant rational functions of G(f˜(s))Tin 𝔽M'(s)𝔽pr(s)are ϵ1(s)ψ1(s)1(s-a)l1,...,ϵm(s)ψm(s)1(s-a)lm.

On the other hand M^'=H(M{(s-a)})Specm(𝔽[s]){(s)}and so (s)M^M^'. Then we can apply Proposition to G(s)with f˜(s)=1s-aso that G(f˜(s))Tis a non-singular matrix with no poles and no zeros in H-1(M^M^')=H-1(H(MM'))=MM'and k1,...,kmas left Wiener–Hopf factorization indices with respect to H-1(M^'){(s-a)}=(M{(s-a)}){(s-a)}=M. The theorem follows by letting T(s)=G(f˜(s))T.

Remark 37 Notice that when M'=and M=Specm(𝔽[s])in Theorem we obtain Theorem (qi=-li).

© 2012 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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A. Amparan, S. Marcaida and I. Zaballa (July 11th 2012). An Interpretation of Rosenbrock's Theorem via Local Rings, Linear Algebra - Theorems and Applications, Hassan Abid Yasser, IntechOpen, DOI: 10.5772/46483. Available from:

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