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

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