This paper is concerned with Smith forms of bivariate polynomial matrices. For a bivariate polynomial square matrix with the determinant being the product of two distinct and irreducible univariate polynomials, we prove that it is equivalent to its Smith form. We design an algorithm to reduce this class of bivariate polynomial matrices to their Smith forms, and an example is given to illustrate the algorithm. Furthermore, we extend the above class of matrices to a more general case, and derive a necessary and sufficient condition for the equivalence of a matrix and one of its all possible existing Smith forms. 

　　Publication: 

　　Journal of Symbolic Computation, Volume 118, September–October 2023, Pages: 1-16. 

　　DOI: 10.1016/j.jsc.2023.01.001 

　　Author: 

　　Dong Lu 

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

　　Dingkang Wang 

　　KLMM, Academy of Mathematics and Systems Science, CAS, 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 

　　Xiaopeng Zheng 

　　KLMM, Academy of Mathematics and Systems Science, CAS, Beijing 100190, China 

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