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 Gröbner 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 Gröbner basis algorithm as a powerful tool for the computation of Gröbner 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 Gröbner basis method is presented and has been implemented on the computer algebra system Maple.

与以前的观点不同，本文从模块的角度研究多元多项式矩阵Diophantine方程，即将矩阵的列视为模块中的元素。推导了方程解存在的充要条件。利用模块的Gröbner基的强大功能和理论基础，可以解决确定和计算矩阵Diophantine方程解的问题。同时，作者利用基于签名的Gröbner基算法GVW算法的模块扩展功能，作为计算模块Gröbner基和表示系数问题的有效工具，该问题直接与方程的特定解相关。作为结果，