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arXiv · 2401.07768

On Hilbert-Poincar\'{e} series of affine semi-regular polynomial sequences and related Gr\"{o}bner bases

Momonari Kudo·Kazuhiro Yokoyama
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Abstract

Gr\"{o}bner bases are nowadays central tools for solving various problems in commutative algebra and algebraic geometry. A typical use of Gr\"{o}bner bases is the multivariate polynomial system solving, which enables us to construct algebraic attacks against post-quantum cryptographic protocols. Therefore, the determination of the complexity of computing Gr\"{o}bner bases is very important both in theory and in practice: One of the most important cases is the case where input polynomials compose an (overdetermined) affine semi-regular sequence. The first part of this paper aims to present a survey on Gr\"{o}bner basis computation and its complexity. In the second part, we shall give an explicit formula on the (truncated) Hilbert-Poincar\'{e} series associated to the homogenization of an affine semi-regular sequence. Based on the formula, we also study (reduced) Gr\"{o}bner bases of the ideals generated by an affine semi-regular sequence and its homogenization. Some of our results are considered to give mathematically rigorous proofs of the correctness of methods for computing Gr\"{o}bner bases of the ideal generated by an affine semi-regular sequence.

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Momonari Kudo, Kazuhiro Yokoyama. 2024-01-15. On Hilbert-Poincar\'{e} series of affine semi-regular polynomial sequences and related Gr\"{o}bner bases. https://arxiv.org/abs/2401.07768

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