Grobner bases of generic ideals

Let I = (f(1), ... , f(n)) be a homogeneous ideal generated by generic polynomials in the polynomial ring R = K[x(1), ... , x(n)] over a field K, with deg f(i) = d(i). Froberg conjectured a formula for the Hilbert series of R/I, and it was conjectured by Moreno-Socias that the initial ideal of I is almost reverse lexicographic, a property that implies Frob erg's conjecture. We give a description of the initial ideal of I in the case where di >= (Sigma(i=1)(j=1) d(j)) - 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. (c) 2023 Elsevier Inc. All rights reserved.