### New Results on the Equivalence of Bivariate Polynomial Matrices

ZHENG Xiaopeng1,2, LU Dong3, WANG Dingkang1,2, XIAO Fanghui4

1. 1. KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;
2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China;
3. School of Mathematics, Southwest Jiaotong University, Chengdu 610031, China;
4. MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, China
• Received:2021-08-17 Revised:2021-10-16 Online:2023-01-25 Published:2023-02-09
• Supported by:
This research was supported by the National Natural Science Foundation of China under Grant Nos. 12171469, 12001030 and 12201210, the National Key Research and Development Program under Grant No. 2020YFA0712300, and the Fundamental Research Funds for the Central Universities under Grant No. 2682022CX048.

ZHENG Xiaopeng, LU Dong, WANG Dingkang, XIAO Fanghui. New Results on the Equivalence of Bivariate Polynomial Matrices[J]. Journal of Systems Science and Complexity, 2023, 36(1): 77-95.

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.
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