Generalization of semi-regular sequences: Maximal Gröbner basis degree, variants of genericness, and related conjectures

Abstract: Nowadays, the notion of semi-regular sequences, originally proposed by Fr\"oberg, becomes very important not only in Mathematics, but also in Information Science, in particular Cryptology. For example, it is highly expected that randomly generated polynomials form a semi-regular sequence, and based on this observation, secure cryptosystems based on polynomial systems can be devised. In this paper, we deal with a semi-regular sequence and its extension, named a generalized cryptographic semi-regular sequence, and give precise analysis on the complexity of computing a Gr\"obner basis of the ideal generated by such a sequence with help of several regularities of the ideal related to Lazard's bound on maximal Gr\"{o}bner basis degree and other bounds. We also study the genericness of the property that a sequence is semi-regular, and its variants related to Fr\"oberg's conjecture. Moreover, we discuss on the genericness of another important property that the initial ideal is weakly reverse lexicographic, related to Moreno-Soc\'{i}as' conjecture, and show some criteria to examine whether both Fr\"oberg's conjecture and Moreno-Soc\'{i}as' one hold at the same time.