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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.20355 |
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| _version_ | 1866912250801946624 |
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| author | Boege, Tobias |
| author_facet | Boege, Tobias |
| contents | A definable set $X$ in the first-order language of rings defines a family of random vectors: for each finite field $\mathbb{F}_q$, let the distribution be supported and uniform on the $\mathbb{F}_q$-rational points of $X$. We employ results from the model theory of finite fields to show that their entropy profiles settle into one of finitely many stable asymptotic behaviors as $q$ grows. The attainable asymptotic entropy profiles and their dominant terms as functions of $q$ are computable. This generalizes a construction of Matúš which gives an information-theoretic interpretation to algebraic matroids. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_20355 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The entropy profiles of a definable set over finite fields Boege, Tobias Information Theory Logic Number Theory 94A17, 11G25 (primary) 03C98, 14G50, 14Q25, 05B35 (secondary) A definable set $X$ in the first-order language of rings defines a family of random vectors: for each finite field $\mathbb{F}_q$, let the distribution be supported and uniform on the $\mathbb{F}_q$-rational points of $X$. We employ results from the model theory of finite fields to show that their entropy profiles settle into one of finitely many stable asymptotic behaviors as $q$ grows. The attainable asymptotic entropy profiles and their dominant terms as functions of $q$ are computable. This generalizes a construction of Matúš which gives an information-theoretic interpretation to algebraic matroids. |
| title | The entropy profiles of a definable set over finite fields |
| topic | Information Theory Logic Number Theory 94A17, 11G25 (primary) 03C98, 14G50, 14Q25, 05B35 (secondary) |
| url | https://arxiv.org/abs/2502.20355 |