Gröbner Bases of Generic Ideals

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 Gröbner 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 $d_i\ge \left({\sum }_{j=1}^{i-1}{d}_j\right)-i-1$. It was conjectured by Moreno-Socías that the initial ideal of $I$ is almost reverse lexicographic, which implies a conjecture by Fröberg on Hilbert series of generic algebras. As a result, we obtain a partial answer to Moreno-Socías 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.

19 pages