Gröbner bases of generic ideals

Abstract

Let I=(f1,…,fn) be a homogeneous ideal generated by generic polynomials in the polynomial ring R=K[x1,…,xn] over a field K, with deg⁡fi=di. Fröberg conjectured a formula for the Hilbert series of R/I, and it was conjectured by Moreno-Socías that the initial ideal of I is almost reverse lexicographic, a property that implies Fröberg's conjecture. We give a description of the initial ideal of I in the case where di≥(∑j=1i−1dj)−i−1, and show that the initial ideal of I is almost reverse lexicographic if the degrees of generators satisfy the inequality for each i. This improves a result by Cho and Park, and we hope this approach can be strengthened to prove the conjecture in full.