Factorizations Of Multivariate Polynomial Matrices

Many problems in multidimensional systems theory come down to problems for multivariate polynomial matrices.Among these problems,the factorization problems for multivariate polynomial matrices are very important.This is a doctoral dissertation on factorizations for multivariate polynomial matrices.It consists of the following three parts.1.In the first chapter,we introduce the required notions and main results.In the second chapter,we introduce some preliminaries and fundamental results of polynomial matrix factorizations.2.For polynomial matrices of full row rank,Wang and Kwong gave a characterization of existence of minor left prime factorizations.In the third chapter,we generalize this result to the general case,and for any polynomial matrix,we give a characterization of existence of minor left prime factorizations.Based on this result,we give an algorithm to decide if a polynomial matrix has minor left prime factorizations and compute them(if they exist).Finally,we give two examples to illustrate this algorithm.3.In the fourth chapter,we prove a polynomial matrix has rank factoriza-tions if and only if it has factor left prime factorizations,and every polynomial matrix of full row rank has factor left prime factorizations.Next,we obtain a result on factor factorizations of polynomial matrices.Based on this result,for a class of polynomial matrices,we give an algorithm to decide if a given one of them has factor left prime factorizations and rank factorizations and compute them(if they exist).We give an example to illustrate this algorithm.Finally,we study irreducible matrices over polynomial rings in one variable over an ar-bitrary field and irreducible factorizations of square matrices over polynomial rings in one variable over an arbitrary field.