Initial ideals of generic ideals and variations of Moreno-Socías conjecture
Koichiro Tani
TL;DR
This paper investigates initial ideals of generic homogeneous ideals under arbitrary monomial orders and specifically the degree reverse lexicographic order. It employs exterior algebra to prove that these initial ideals are Borel-fixed, offering a weaker Moreno-Socías-type statement, and introduces a stability-based computation method via comprehensive Gröbner systems to study initial ideals. The lexicographic case is analyzed through Macaulay's theorem and lexsegment ideals to bound the maximal degree of a Gröbner basis and to identify when lexsegment initial ideals arise. The work further provides corollaries and computational criteria that analyze the existence of lexsegment initial ideals under various parameter regimes and ends with open questions forming a lexicographic analogue of Moreno-Socías. These contributions advance understanding of the structure of generic initial ideals and offer practical tools for complexity analysis in Gröbner basis computation.
Abstract
It is known that the initial ideals of generic ideals are the same. Moreno-Socías conjectured that the initial ideal of generic ideals with respect to the degree reverse lexicographic order is weakly reverse lexicographic. In the first half of this paper, we study the initial ideal of generic ideals for arbitrary monomial order and prove that the initial ideal of generic ideals is Borel-fixed. It can be considered as a weakened version of Moreno-Socías conjecture. In the second half, we propose a new method of the computation of the initial ideal of generic ideals using stability condition of Gröbner bases. We apply the method in the case of lexicographic order and study the relationship between the lexsegment ideal and the initial ideal of generic ideals. This study aims to bound the maximal degree of Gröbner basis. At the last, we propose questions that can be considered as a lexicographic analogue of Moreno-Socías conjecture.