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Matrix equivalence to Smith normal form: new theoretical results for multivariate polynomial matrices

  • Dong Lu
  • Yuanyuan Ruan
  • Dingkang Wang
  • Fanghui Xiao
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Abstract

This paper investigates the Smith normal form equivalence problem for multivariate polynomial matrices. Using methods from matrix theory and polynomial ideal theory, we prove that Frost and Storey's 1978 conjecture holds for a broad class of matrices: such a matrix is equivalent to its Smith normal form if and only if its reduced minors of each order generate the unit ideal. Moreover, by extending the original matrix class via automorphisms of the polynomial ring, we show that our framework applies in a substantially more general setting.

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Cite

BibTeXRIS
BibTeX
@misc{lu2026matrix,
  title = {{Matrix equivalence to Smith normal form: new theoretical results for multivariate polynomial matrices}},
  author = {Dong Lu and Yuanyuan Ruan and Dingkang Wang and Fanghui Xiao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.09286},
  eprint = {2605.09286},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.09286},
}
RIS
TY  - UNPB
TI  - Matrix equivalence to Smith normal form: new theoretical results for multivariate polynomial matrices
AU  - Dong Lu
AU  - Yuanyuan Ruan
AU  - Dingkang Wang
AU  - Fanghui Xiao
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2605.09286
UR  - https://arxiv.org/abs/2605.09286
AB  - This paper investigates the Smith normal form equivalence problem for multivariate polynomial matrices. Using methods from matrix theory and polynomial ideal theory, we prove that Frost and Storey's 1978 conjecture holds for a broad class of matrices: such a matrix is equivalent to its Smith normal form if and only if its reduced minors of each order generate the unit ideal. Moreover, by extending the original matrix class via automorphisms of the polynomial ring, we show that our framework applies in a substantially more general setting.
ER  -