The Betti polynomials of powers of an ideal

TL;DR

This paper investigates the asymptotic behavior of Betti numbers of powers of an ideal in regular local or polynomial rings, revealing that their limiting behavior is governed by specific polynomial coefficients and establishing bounds and properties of these polynomials.

Contribution

It introduces the concept of Kodiyalam polynomials to describe the limiting Betti number behavior and proves that this behavior depends only on their highest degree coefficients.

Findings

Betti numbers of ideal powers become polynomial functions for large exponents

The limiting behavior is determined by the coefficients of the highest degree terms in Kodiyalam polynomials

Lower bounds are established for these coefficients in special cases

Abstract

For an ideal $I$ in a regular local ring or a graded ideal $I$ in the polynomial ring we study the limiting behavior of the Betti numbers of S/I^k as k goes to infinity. By Kodiyalam's result it is known that in each homological degree the Betti number is a polynomial for large k. We call these polynomials the Kodiyalam polynomials and encode the limiting behavior in their generating polynomial. It is shown that the limiting behavior depends only on the coefficients on the Kodiyalam polynomials in the highest possible degree. For these we exhibit lower bounds in special cases and conjecture that the bounds are valid in general. We also show that the Kodiyalam polynomials have weakly descending degrees and identify a situation where the polynomials have all highest possible degree.