# 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.

## Key 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.

---
Source: https://tomesphere.com/paper/1704.05601