Asymptotic classes via definable quotients

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Original abstract

We study when finite-dimensional asymptoticity transfers through uniform interpretations and definable quotients. Ambient counting yields finitely many asymptotic alternatives, but the corresponding parameter cells may fail to be definable in the interpreted language; in general, one obtains only weak asymptoticity. We isolate "trace reflection", a one-sided descent condition that removes this obstruction. For interpretations with definable selectors, trace reflection transfers asymptoticity with explicit denominator bounds. For general definable quotients, invariant asymptotic profiles replace definable choice and yield a quotient-transfer theorem. We identify an intrinsic visible denominator and prove that it is the least possible asymptotic denominator, with a coprime-witness criterion for when the interpretation-dependent bound is sharp. We also establish formula-wise, syntactic, and semantic descent criteria, prove composition theorems, and show that trace reflection is strictly weaker than uniform weak bi-interpretability. A framed-profile argument removes the noncanonical coordinate parameters introduced by uniform finite-field reconstruction. As applications, we study projective quotient traces and prove that, for every fixed $k\geq2$, the pure incidence structures arising from 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 $n\geq2$.

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