The entropy profiles of a definable set over finite fields

TL;DR

This paper studies the entropy behavior of definable sets over finite fields, showing that their entropy profiles stabilize into finitely many patterns as the field size increases, with computable asymptotic descriptions.

Contribution

It introduces a model-theoretic approach to analyze entropy profiles of definable sets over finite fields, generalizing algebraic matroid interpretations.

Findings

Entropy profiles stabilize into finitely many asymptotic behaviors

Asymptotic entropy profiles and their dominant terms are computable

Generalizes previous algebraic matroid constructions with an information-theoretic perspective

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.