Counting in Uncountably Categorical Pseudofinite Structures
Alexander Van Abel
TL;DR
This work shows that in uncountably categorical pseudofinite structures, the pseudofinite cardinality of any definable set is a rational polynomial in the size of a strongly minimal definable set, with the degree matching the Morley rank $MR$. It extends Pillay's counting result from strongly minimal to uncountably categorical theories by proving a main theorem that partitions parameter space into finitely many definable classes, each yielding a cardinality $|\varphi(M^n,\bar{b})| = F_i(|D|)$ with $F_i \in \mathbb{Q}[X]$ and $\deg F_i = MR(\varphi(M^n,\bar{b}))$. The paper further shows that zero-one classes with such limit theories are polynomial $R$-mecs and $N$-dimensional asymptotic classes, where $N$ is the Morley rank, linking counting in pseudofinite contexts to multidimensional asymptotics. Techniques combine Zil'ber-style stratifications, a two-sorted cardinality framework $L^+$, and a fiber-decomposition approach to reduce counts to polynomial expressions in the size of a strongly minimal set, with explicit considerations of rational coefficients and definability.
Abstract
We show that every definable subset of an uncountably categorical pseudofinite structure has pseudofinite cardinality which is polynomial (over the rationals) in the size of any strongly minimal subset, with the degree of the polynomial equal to the Morley rank of the subset. From this fact, we show that classes of finite structures whose ultraproducts all satisfy the same uncountably categorical theory are polynomial $R$-mecs as well as $N$-dimensional asymptotic classes, where $N$ is the Morley rank of the theory.