Gr\"obner Bases of Generic Ideals

TL;DR

This paper develops an incremental method to compute Gr"obner bases for generic ideals, providing a partial proof that their initial ideals are almost reverse lexicographic under certain degree conditions, advancing understanding of their algebraic structure.

Contribution

It introduces a new incremental approach to determine Gr"obner bases of generic ideals and partially confirms Moreno-Socías conjecture under specific degree constraints.

Findings

Complete description of initial ideal when degrees satisfy a specific inequality.

Partial confirmation that initial ideals are almost reverse lexicographic under certain conditions.

Improves previous results by Cho and Park.

Abstract

Let $I = ( f_1, \dots, f_n )$ be a homogeneous ideal in the polynomial ring $K[x_1, \dots,x_n]$ over a field $K$ generated by generic polynomials. Using an incremental approach based on a method by Gao, Guan and Volny, and properties of the standard monomials of generic ideals, we show how a Gr\"obner basis for the ideal $(f_1, \dots, f_i)$ can be obtained from that of $(f_1, \dots, f_{i-1})$. If $deg f_i = d_i$, we are able to give a complete description of the initial ideal of $I$ in the case where $d_i \geq \left(\sum_{j=1}^{i-1}d_j\right) - i -1$. It was conjectured by Moreno-Soc\'ias that the initial ideal of $I$ is almost reverse lexicographic, which implies a conjecture by Fr\"oberg on Hilbert series of generic algebras. As a result, we obtain a partial answer to Moreno-Soc\'ias Conjecture: the initial ideal of $I$ is almost reverse lexicographic if the degrees of generators satisfy the condition above. This result improves a result by Cho and Park. We hope this approach can be strengthened to prove the conjecture in full.