Study Of Zero Left Prime Over Polynomial Ring

Multivariate polynomial matrix of multivariate polynomial ring is an important content in algebra,and most engineering problem can be solved by multivariate polynomial matrix,but solving the multivariate polynomial matrix is very difficulty in the polynomial matrix theory.At present,many scholars have carried on the research to it,and have obtained many significant results.Especially for the unary and binary polynomial matrices,the problems have been a better solution.But for the decomposition and equivalence of multivariate polynomial matrix,there are still controversial.In this paper,we focus on the properties of zero-prime matrix on polynomial rings,and the related conditions of the equivalence of matrices,mainly including:? Looking for the new ways for zero-left matrix elements embedded unimodular matrix;? Exploring the effective ways of decomposing the unitary matrix into elementary matrices and giving the corresponding basic algorithm;?Studying the effective sufficient and necessary conditions of the polynomial matrix on the univariate and multivariate polynomial rings equivalent to the matrix respectively.We conclude that greatly improved the references of the relevant conclusions,and provided an effective basis for better solve engineering problems.This article is divided into six chapters.The first chapter is the introduction,including historical background and research status.The second chapter is the basic knowledge,mainly used in the algebra of the relevant knowledge.The third chapter discusses the property of the zero-left matrix on polynomial ring,and concluded the following: For any unimodular matrix A can be decomposed into the product of elementary matrices;for any zero-left prime matrix A,we can find a unimodular matrix F,so that A is the sub-matrix formed by the forward l line of F,and these conclusions are given corresponding algorithm.In chapter four,we discuss the equivalence conditions of matrices over unary polynomial rings,and come to the following conclusion: The necessary and sufficient condition for any unary polynomial matrix A to be equivalent to matrice are that all the q-level minor of A generating units.In Chapter five,we extend the equivalence of matrices over a unary polynomial ring.We also discuss the equivalence of multivariate polynomial matrices.The main results are as follows: If for arbitrary multivariate polynomial matrix A is equivalent to the matrix,the necessary and sufficient condition is that the matrix M formed by a q-line of the row vector generation sub-module in A is zero-left.Chapter VI conclusion and prospection.