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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.07062 |
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| _version_ | 1866915842151677952 |
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| author | Katz, Nicholas M. Villegas, Fernando Rodriguez |
| author_facet | Katz, Nicholas M. Villegas, Fernando Rodriguez |
| contents | In this short note we prove a couple of facts about polynomial count varieties, answering natural questions that they raise. A polynomial count $X$ variety is essentially one for which its number of points over finite fields is given by a polynomial in the field size. Well-known examples include affine or projective space (or more generally the Grassmanian) and other standard varieties.
The two questions we address are the following.
1) If $X$ is smooth, polynomial count with $\#X(q)=q^n$ for some $n$, is $X$ isomorphic to $n$-dimensional affine space?
2) If $X$ is a polynomial count, is it true that its Hodge numbers in a given graded piece of fixed weight satisfy~$h^{p,q}=0$ unless $p=q$?
We show that in both cases the answer is no. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_07062 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Remarks on polynomial count varieties Katz, Nicholas M. Villegas, Fernando Rodriguez Number Theory Algebraic Geometry In this short note we prove a couple of facts about polynomial count varieties, answering natural questions that they raise. A polynomial count $X$ variety is essentially one for which its number of points over finite fields is given by a polynomial in the field size. Well-known examples include affine or projective space (or more generally the Grassmanian) and other standard varieties. The two questions we address are the following. 1) If $X$ is smooth, polynomial count with $\#X(q)=q^n$ for some $n$, is $X$ isomorphic to $n$-dimensional affine space? 2) If $X$ is a polynomial count, is it true that its Hodge numbers in a given graded piece of fixed weight satisfy~$h^{p,q}=0$ unless $p=q$? We show that in both cases the answer is no. |
| title | Remarks on polynomial count varieties |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2603.07062 |