On Minor Left Prime Factorization Problem for Multivariate Polynomial Matrices

A new necessary and sufficient condition for the existence of minor left prime factorizations of multivariate polynomial matrices without full row rank is presented.The key idea is to establish a relationship between a matrix and any of its full row rank submatrices.Based on the new result,the authors propose an algorithm for factorizing matrices and have implemented it on the computer algebra system Maple.Two examples are given to illustrate the effectiveness of the algorithm,and experimental data shows that the algorithm is efficient.

New Results on the Equivalence of Bivariate Polynomial Matrices

This paper investigates the equivalence problem of bivariate polynomial matrices.A necessary and sufficient condition for the equivalence of a square matrix with the determinant being some power of a univariate irreducible polynomial and its Smith form is proposed.Meanwhile,the authors present an algorithm that reduces this class of bivariate polynomial matrices to their Smith forms,and an example is given to illustrate the effectiveness of the algorithm.In addition,the authors generalize the main result to the non-square case.

An Improvement of the Rational Representation for High-Dimensional Systems

Based on the rational univariate representation of zero-dimensional polynomial systems,Tan and Zhang proposed the rational representation theory for solving a high-dimensional polynomial system,which uses so-called rational representation sets to describe all the zeros of a high-dimensional polynomial system.This paper is devoted to giving an improvement for the rational representation.The idea of this improvement comes from a minimal Dickson basis used for computing a comprehensive Grobner system of a parametric polynomial system to reduce the number of branches.The authors replace the normal Grobner basis G satisfying certain conditions in the original algorithm(Tan-Zhang’s algorithm)with a minimal Dickson basis G_(m) of a Grobner basis for the ideal,where G_(m) is smaller in size than G.Based on this,the authors give an improved algorithm.Moreover,the proposed algorithm has been implemented on the computer algebra system Maple.Experimental data and its performance comparison with the original algorithm show that it generates fewer branches and the improvement is rewarding.

Solving Multivariate Polynomial Matrix Diophantine Equations with Gr?bner Basis Method

Different from previous viewpoints,multivariate polynomial matrix Diophantine equations are studied from the perspective of modules in this paper,that is,regarding the columns of matrices as elements in modules.A necessary and sufficient condition of the existence for the solution of equations is derived.Using powerful features and theoretical foundation of Gr?bner bases for modules,the problem for determining and computing the solution of matrix Diophantine equations can be solved.Meanwhile,the authors make use of the extension on modules for the GVW algorithm that is a signature-based Gr?bner basis algorithm as a powerful tool for the computation of Gr?bner basis for module and the representation coefficients problem directly related to the particular solution of equations.As a consequence,a complete algorithm for solving multivariate polynomial matrix Diophantine equations by the Gr?bner basis method is presented and has been implemented on the computer algebra system Maple.