Conditions for Generic Initial Ideals to be Almost Reverse Lexicographic

TL;DR

This paper investigates conditions under which the generic initial ideal of a homogeneous Artinian ideal is almost reverse lexicographic, linking conjectures in algebra and analyzing specific cases like complete intersections.

Contribution

It establishes a condition for the generic initial ideal to be almost reverse lexicographic, connects Moreno-Socias and Fröberg conjectures, and analyzes the strong Lefschetz property for certain ideals.

Findings

Moreno-Socias conjecture implies Fröberg conjecture.

For codimension ≤ 3, the strong Lefschetz property is equivalent to the generic initial ideal being almost reverse lexicographic.

Specific conditions on degrees of monomial complete intersections ensure the generic initial ideal is almost reverse lexicographic.

Abstract

Let $I$ be a homogeneous Artinian ideal in a polynomial ring $R=k[x_1,...,x_n]$ over a field $k$ of characteristic 0. We study an equivalent condition for the generic initial ideal $\gin(I)$ with respect to reverse lexicographic order to be almost reverse lexicographic. As a result, we show that Moreno-Socias conjecture implies Fr\"{o}berg conjecture. And for the case $\Codim I \le 3$, we show that $R/I$ has the strong Lefschetz property if and only if $\gin(I)$ is almost reverse lexicographic. Finally for a monomial complete intersection Artinian ideal $I=(x_1^{d_1},...,x_n^{d_n})$, we prove that $\gin(I)$ is almost reverse lexicographic if $d_i > \sum_{j=1}^{i-1} d_j - i + 1$ for each $i \ge 4$. Using this, we give a positive partial answer to Moreno-Socias conjecture, and to Fr\"{o}berg conjecture.