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<p>Based on the primitive factorization theorem, this paper presents an improved algorithm for computing free bases of syzygy modules of bivariate polynomial matrices, which additionally enables efficient computation of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mu \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>-bases for rational parametric surfaces. Experimental results show that the new algorithm outperforms two existing algorithms in terms of computational efficiency. Furthermore, by leveraging this algorithm, we generalize the general matrix factorization theory of full-rank bivariate polynomial matrices to the rank-deficient case for the first time.</p>

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Theory and Algorithms for Bivariate Polynomial Matrix Factorizations

  • 期刊论文

  • 发表日期：

    2026年7月20日

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  • Ligeng Fan,
  • Dong Lu,
  • Dingkang Wang,
  • Xiaopeng Zheng

摘要

Based on the primitive factorization theorem, this paper presents an improved algorithm for computing free bases of syzygy modules of bivariate polynomial matrices, which additionally enables efficient computation of \(\mu \) μ -bases for rational parametric surfaces. Experimental results show that the new algorithm outperforms two existing algorithms in terms of computational efficiency. Furthermore, by leveraging this algorithm, we generalize the general matrix factorization theory of full-rank bivariate polynomial matrices to the rank-deficient case for the first time.