The entropy profiles of a definable set over finite fields

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 Matúš which gives an information-theoretic interpretation to algebraic matroids.

arXiv comment22 pages, 3 figures