Fixed points and grade of Hilbert polynomial of invariant rings
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909934786969600 |
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| author | Puthenpurakal, Tony J. |
| author_facet | Puthenpurakal, Tony J. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_00811 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fixed points and grade of Hilbert polynomial of invariant rings Puthenpurakal, Tony J. Commutative Algebra Primary 13A50 Secondary 13C15 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. |
| title | Fixed points and grade of Hilbert polynomial of invariant rings |
| topic | Commutative Algebra Primary 13A50 Secondary 13C15 |
| url | https://arxiv.org/abs/2512.00811 |