Moreno–Socías conjecture on weakly reverse-lexicographic initial ideals of generic sequences

Establish that for a generic sequence of homogeneous polynomials F in the polynomial ring k[x1, …, xn] with graded reverse lexicographic term order, the leading-term ideal LT(⟨F⟩) is weakly reverse-lexicographic. Concretely, prove that for every minimal generator r of LT(⟨F⟩) and every monomial r′ of the same total degree with r′ larger than r in graded reverse lexicographic order, the monomial r′ also lies in LT(⟨F⟩).

Background

The paper studies generic sequences of homogeneous polynomials and algorithms for predicting the leading monomials of a minimal Gröbner basis without full reductions. A central structural hypothesis is that, under graded reverse lexicographic order (grevlex), the initial ideal of a generic sequence’s ideal is weakly reverse-lexicographic.

This assertion is the Moreno–Socías conjecture, often cited in the literature on generic initial ideals and Gröbner bases. If true, it strongly constrains the shape of the initial ideal, enabling degree-by-degree reconstruction of leading monomials guided by the Hilbert function, which is the foundation of the proposed LGB algorithm.

Throughout the paper, the authors proceed under this conjectural structure to design and analyze their method, highlighting its practical and theoretical impact on Gröbner basis computations for generic systems.