This paper is concerned with the factorization and equivalence problems of multivariate polynomial matrices. We present some new criteria for the existence of matrix factorizations for a class of multivariate polynomial matrices, and obtain a necessary and sufficient condition for the equivalence of a square polynomial matrix and a diagonal matrix. Based on the constructive proof of the new criteria, we give a factorization algorithm and prove the uniqueness of the factorization. We implement the algorithm on Maple, and two illustrative examples are given to show the effectiveness of the algorithm. 

　　Publication: 

　　Journal of Symbolic Computation, 2023, Volume 115, pp: 266-284. 

　　Author: 

　　Dong Lu 

　　School of Mathematics, Southwest Jiaotong University, Chengdu 610031, China 

　　School of Mathematical Sciences, Beihang University, Beijing 100191, China 

　　Dingkang Wang 

　　KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China 

　　School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China 

　　Email: dwang@mmrc.iss.ac.cn 

　　Fanghui Xiao 

　　MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, China