Fixed points and grade of Hilbert polynomial of invariant rings

Tony J. Puthenpurakal

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arXiv · 2512.00811

Fixed points and grade of Hilbert polynomial of invariant rings

Tony J. Puthenpurakal

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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.

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Tony J. Puthenpurakal. 2025-11-30. Fixed points and grade of Hilbert polynomial of invariant rings. https://arxiv.org/abs/2512.00811

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