Regularity index of the generalized minimum distance function

TL;DR

This paper introduces the regularity index of the generalized minimum distance function for graded algebras, demonstrating its properties, computing its stabilization value, and analyzing how it varies with polynomial count to establish bounds.

Contribution

It defines the regularity index for the generalized minimum distance function, computes its stabilization value, and analyzes its variation with the number of polynomials to establish bounds.

Findings

The generalized minimum distance function is non-increasing with degree.

The stabilization value of the function is explicitly computed.

Bounds are established for the regularity index based on polynomial count.

Abstract

We show that the generalized minimum distance function is non-increasing as the degree varies for reduced standard graded algebras over a field. This allows us to define its regularity index and its stabilization value. The stabilization value is computed for every cases. We study how the regularity index varies as the number of polynomial increases, and use this to give bounds for it.