On the $h$-vectors of the powers of graded ideals

TL;DR

This paper investigates the asymptotic behavior of the $h$-vectors and Hilbert coefficients of powers of graded ideals in polynomial rings, revealing linear bounds and polynomial patterns under certain generation conditions.

Contribution

It establishes that for large powers, the postulation number and Hilbert coefficients of ideal powers follow predictable linear and polynomial patterns, especially when the ideal is generated in a single degree.

Findings

Postulation number of $I^k$ is linearly bounded for large $k$.

Hilbert coefficients $e_i(I^k)$ are polynomial functions of $k$ for large $k$ when $I$ is generated in a single degree.

Postulation number is exactly linear in $k$ if $I$ is generated in a single degree.

Abstract

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over the field $K$, and let $I\subset S$ be a graded ideal. It is shown that for $k \gg0$ the postulation number of $I^k$ is bounded by a linear function of $k$, and it is a linear function of $k$, if $I$ is generated in a single degree. By using the relationship of the $h$-vector with the higher iterated Hilbert coefficients of $I^k$ it is shown that the Hilbert coefficients $e_i(I^k)$ of $I^k$ are polynomials for $k \gg 0$, whenever $I$ is generated in a single degree.