Fixed points and grade of Hilbert polynomial of invariant rings

TL;DR

This paper investigates the structure of the Hilbert polynomial of invariant rings under finite group actions, establishing conditions on its coefficients and providing examples to demonstrate the sharpness of these results.

Contribution

It proves that certain coefficients of the Hilbert polynomial are constant and relates the grade of the polynomial to the dimension of invariants, offering new insights into invariant ring structure.

Findings

Coefficients a_{d-1}(-), ..., a_{d-r}(-) are constant.

The grade of the Hilbert polynomial is at most d - r - 1.

An example demonstrating the sharpness of the result.

Abstract

Let $k$ be a field and let $V$ be a $k$-vector space of dimension $d$. Let $G \subseteq GL(V)$ be a finite group. Let $r = \dim_k (V^*)^G$. Assume $r \geq 1$. Let $R = k[V]^G$ be the ring of invariants of $G$. Let $H_R(n) = a_{d-1}(n)n^{d-1} + \cdots a_1(n)n + a_0(n)$ be the Hilbert polynomial of $R$ where $a_i(-)$ are periodic functions. We show $a_{d-1}(-), \ldots, a_{d-r}(-)$ are constants. In the terminology of Erhart, $\text{grade} H_R \leq d - r-1$. We also give an example which shows that our result is sharp.