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 the 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基如今是解决交换代数与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，这使得我们能够针对后量子密码协议构建代数攻击。因此，确定计算Gröbner基的复杂度在理论与实践层面均至关重要：其中最重要的情形是输入多项式构成（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。第二部分将给出与仿射半正则序列齐次化相关联的（截断）Hilbert-Poincaré级数的显式表达式。基于该公式，我们还将研究由仿射半正则序列及其齐次化生成的理想所对应的（既约）Gröbner基。部分研究结果被认为为计算仿射半正则序列生成理想的Gröbner基方法之正确性提供了数学上的严格证明。