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The entropy profiles of a definable set over finite fields

Tobias Boege

TL;DR

This work establishes that entropy profiles derived from uniform distributions on definable sets over finite fields exhibit finitely many stable asymptotic types as the field size grows, with the type determined by the extension degree modulo a period. The analysis combines model-theoretic results on definable sets (via Galois stratification and Lang–Weil-type counts) with information-theoretic tools (entropy profiles, fiber decompositions) to produce computable leading terms and periodic behavior. In particular, irreducible definable sets yield entropy profiles that refine the algebraic matroid of the variety, linking information geometry to algebraic geometry and providing an algebraic matroidal, almost-entropic perspective. The KR configuration illustrates essential conditionality, and the framework extends to toric/linear-congruence cases, enabling explicit, computable entropy profiles and furnishing a suite of post-processing operations to study information inequalities within this definable-algebraic setting.

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.

The entropy profiles of a definable set over finite fields

TL;DR

This work establishes that entropy profiles derived from uniform distributions on definable sets over finite fields exhibit finitely many stable asymptotic types as the field size grows, with the type determined by the extension degree modulo a period. The analysis combines model-theoretic results on definable sets (via Galois stratification and Lang–Weil-type counts) with information-theoretic tools (entropy profiles, fiber decompositions) to produce computable leading terms and periodic behavior. In particular, irreducible definable sets yield entropy profiles that refine the algebraic matroid of the variety, linking information geometry to algebraic geometry and providing an algebraic matroidal, almost-entropic perspective. The KR configuration illustrates essential conditionality, and the framework extends to toric/linear-congruence cases, enabling explicit, computable entropy profiles and furnishing a suite of post-processing operations to study information inequalities within this definable-algebraic setting.

Abstract

A definable set in the first-order language of rings defines a family of random vectors: for each finite field , let the distribution be supported and uniform on the -rational points of . 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 grows. The attainable asymptotic entropy profiles and their dominant terms as functions of are computable. This generalizes a construction of Matúš which gives an information-theoretic interpretation to algebraic matroids.

Paper Structure

This paper contains 11 sections, 5 theorems, 33 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Information inequalities and geometric configurations
  3. Information inequalities
  4. Affine algebraic varieties
  5. The Kaced--Romashchenko configuration
  6. Computing entropy profiles of a definable set
  7. The uniform distribution on a definable set
  8. Measure and decomposition
  9. Entropy profiles and algebraic independence
  10. Linear congruences and monomial maps
  11. Information-theoretic post-processing techniques

Key Result

theorem 3.3

Consider a formula $\varphi(x_1, \dots, x_n, y_1, \dots, y_m)$. There exist finitely many formulas $\psi_k(y_1, \dots, y_m)$, indexed by $k \in K$, with accompanying $\mu_k \in \mathds{Q}$ and $d_k \in \mathds{N}$ such that for every sufficiently large finite field $\mathds{F}_{\mkern-3muq}$ and eve

Figures (3)

  • Figure 1: The inequality $y = 0 \implies x \le 0$ is essentially conditional because the slopes of the purple boundary tangent lines approach zero as $x \to 0$. This can be seen without knowing the exact equation of the boundary: the red dots at $(1/\lambda, 8/9\lambda^2)$ lie inside the region and violate the unconditional version $x \le \lambda y$ for any $\lambda > 0$. They approach the boundary arbitrarily well near the extreme point $(0,0)$.
  • Figure 2: A generic point of the Kaced--Romashchenko variety over $\mathds{R}$.
  • Figure 3: A blue set $X$ and fiber decompositions with respect to its projections onto both coordinate axes. The axis-parallel dotted lines delineate pieces of the projections over which the fibers have constant size.

Theorems & Definitions (27)

  • remark 2.1
  • example 3.1: Roots of a cubic
  • definition 3.2
  • theorem 3.3: Definable
  • definition 3.4
  • example 3.5: Square root of $-1$
  • definition 3.6
  • remark 3.7
  • remark 3.8
  • theorem 3.9: EffectiveCounting
  • ...and 17 more