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.

翻译：Gröbner基是当前解决交换代数和代数几何中各类问题的核心工具。其典型应用之一是多变量多项式方程组求解，这使我们能够构建针对后量子密码协议的攻击方法。因此，从理论与应用角度而言，确定Gröbner基计算的复杂度具有重要意义：其中最重要的情形之一，是输入多项式构成（超定）仿射半正则序列。本文第一部分旨在综述Gröbner基计算及其复杂度。第二部分将给出仿射半正则序列齐次化对应的（截断）Hilbert-Poincaré级数的显式公式。基于该公式，我们还研究了仿射半正则序列生成的理想及其齐次化对应的（约化）Gröbner基。部分结果可视为对仿射半正则序列生成理想的Gröbner基计算方法正确性的严格数学证明。