Asymptotic stabilization of Betti diagrams of generic initial systems

TL;DR

This paper studies the long-term behavior of Betti diagrams of ideal systems, showing that for ideals with generators of the same degree, the diagrams stabilize and their entries follow polynomial patterns asymptotically.

Contribution

It extends previous work by analyzing Betti diagrams of initial and generic initial ideals, demonstrating asymptotic polynomial behavior and stabilization of their shape.

Findings

Betti diagram entries are asymptotically polynomial functions.

The shape of Betti diagrams stabilizes for ideals with generators of the same degree.

Stabilization occurs in reverse lexicographic generic initial systems.

Abstract

Several authors investigating the asymptotic behaviour of the Betti diagrams of the graded system obtained by taking powers of an ideal have shown that the shape of the nonzero entries in the diagrams stabilizes when $I$ is a homogeneous ideal with generators of the same degree. In this paper, we study the Betti diagrams of graded systems of ideals built by taking the initial ideals or generic initial ideals of powers, and discuss the stabilization of additional collections of Betti diagrams. Our main result shows that when I has generators of the same degree, the entries in the Betti diagrams of the reverse lexicographic generic initial system are given asymptotically by polynomials and that the shape of the diagrams stabilizes.