The entropy profiles of a definable set over finite fields

Tobias Boege

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arXiv · 2502.20355

The entropy profiles of a definable set over finite fields

Tobias Boege

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Abstract

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 Mat\'u\v{s} which gives an information-theoretic interpretation to algebraic matroids.

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Tobias Boege. 2025-02-27. The entropy profiles of a definable set over finite fields. https://arxiv.org/abs/2502.20355

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