Asymptotic classes via definable quotients

Abstract

In this paper, we study when uniform interpretations and definable quotients preserve finite-dimensional asymptotic classes. Counting in the ambient structures gives finitely many asymptotic alternatives. However, the corresponding parameter sets need not be definable in the interpreted language, so in general the interpreted class is only weakly asymptotic. We define trace reflection, a one-sided condition which ensures this definability. We first prove a transfer theorem for interpretations with definable selectors and give explicit bounds on the denominator. We then use invariant profiles to prove a transfer theorem for general definable quotients, without assuming definable choice. We define an intrinsic visible denominator and prove that it is the least asymptotic denominator. We also give a coprime-witness criterion for the interpretation-dependent bound to be sharp. We prove formula-wise, syntactic, and semantic criteria for descent, as well as composition theorems, and show that trace reflection is strictly weaker than uniform weak bi-interpretability. Using profiles of finite frames, we remove the noncanonical parameters used to reconstruct finite-field coordinates. As applications, we study projective quotient traces and show that, for every fixed $k\ge 2$, the pure incidence structures given by graphs of polynomials of degree less than $k$ over finite fields form a full $k$-dimensional asymptotic class. The denominator $k$ is minimal, and every member is $K_{k,n}$-free for all $n\ge 2$.

arXiv comment46 pages