Multivariate Polynomials, Matrices, and Matrix-Fraction Descriptions

Abstract

In this chapter the polynomial/polynomial matrix approach to multidimensional systems theory is presented, based on ideals, varieties, and modules. The factorization algorithms for bivariate polynomial matrices are now well developed but not presented elsewhere in sufficient detail. The generalizations or lack of adequate and full generalizations of the bivariate results to the multivariate case are explained and, to the extent possible, justified, based on current knowledge. The previous chapter should also prepare the reader to implement the algorithmic theory presented here. The n-variate polynomial ring K[z], where K is an arbitrary but fixed field and z = (z ₁ , z ₂ , ..., z _n ), is the base ring of interest. A result of crucial importance in polynomial algebra is Hilbert’s basis theorem, according to which any ideal I ⊂ K[z] is finitely generated in the sense that there exist polynomials, g _i (z) ∈ K[z], i = 1, 2, . . . , k such that I = (g ₁ , g ₁ , ..., g _k ). Furthermore, the ascending chain condition is satisfied on K[z] i. e. if(Math)is an ascending chain of ideals of K[z], then there exists a positive integer N such that I _N = I _N+1 = I _N +1....A commutadve ring that satisfies the ascending chain condition is called a (Noetherian ring). The ring K[z] is a Noetherian ring. A ring that satisfies a condition dual to the Noetherian condition : the descending chain condition of ideals, is called an Artinian ring i. e. a ring is Artinian if every descending chain of ideals is finite. An Artinian ring is also Noetherian.