Gr\"obner Bases of Generic Ideals
Juliane Capaverde, Shuhong Gao

TL;DR
This paper develops an incremental method to compute Gr"obner bases for generic ideals, providing a partial proof that their initial ideals are almost reverse lexicographic under certain degree conditions, advancing understanding of their algebraic structure.
Contribution
It introduces a new incremental approach to determine Gr"obner bases of generic ideals and partially confirms Moreno-Socías conjecture under specific degree constraints.
Findings
Complete description of initial ideal when degrees satisfy a specific inequality.
Partial confirmation that initial ideals are almost reverse lexicographic under certain conditions.
Improves previous results by Cho and Park.
Abstract
Let be a homogeneous ideal in the polynomial ring over a field generated by generic polynomials. Using an incremental approach based on a method by Gao, Guan and Volny, and properties of the standard monomials of generic ideals, we show how a Gr\"obner basis for the ideal can be obtained from that of . If , we are able to give a complete description of the initial ideal of in the case where . It was conjectured by Moreno-Soc\'ias that the initial ideal of is almost reverse lexicographic, which implies a conjecture by Fr\"oberg on Hilbert series of generic algebras. As a result, we obtain a partial answer to Moreno-Soc\'ias Conjecture: the initial ideal of is almost reverse lexicographic if the degrees of generators…
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