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Solving Multivariate Polynomial Matrix Diophantine Equations with Gröbner Basis Method

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Abstract

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.

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Author information

Authors and Affiliations

  1. College of Mathematics and Statistics, Hunan Normal University, Changsha, 410006, China

    Fanghui Xiao

  2. Beijing Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Beijing, 100191, China

    Dong Lu

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

    Dong Lu

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

    Dingkang Wang

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

    Dingkang Wang

Authors
  1. Fanghui Xiao
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  2. Dong Lu
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  3. Dingkang Wang
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Corresponding author

Correspondence to Dong Lu.

Additional information

This research was supported by the National Natural Science Foundation of China under Grant No. 12001030, the CAS Key Project QYZDJ-SSW-SYS022 and the National Key Research and Development Project 2020YFA0712300.

This paper was recommended for publication by Editor MOU Chenqi.

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Xiao, F., Lu, D. & Wang, D. Solving Multivariate Polynomial Matrix Diophantine Equations with Gröbner Basis Method. J Syst Sci Complex 35, 413–426 (2022). https://doi.org/10.1007/s11424-021-0072-x

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  • DOI: https://doi.org/10.1007/s11424-021-0072-x

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