Counting mod n in pseudofinite fields

TL;DR

This paper develops a framework for assigning definable nonstandard sizes to sets in pseudofinite fields, leading to a definable Euler characteristic valued in the profinite integers, and explores the decidability of finite field theories with parity quantifiers.

Contribution

It introduces a nonstandard size measure for definable sets in pseudofinite fields and constructs a definable Euler characteristic valued in the profinite integers.

Findings

Nonstandard sizes vary definably in families within ultraproducts of finite fields.

A definable strong Euler characteristic can be assigned to pseudofinite fields, depending on a nonstandard Frobenius.

The theory of finite fields with added parity quantifiers remains decidable, contingent on a deferred algebraic geometry proof.

Abstract

We show that in an ultraproduct of finite fields, the mod-$n$ nonstandard size of definable sets varies definably in families. Moreover, if $K$ is any pseudofinite field, then one can assign "nonstandard sizes mod $n$" to definable sets in $K$. As $n$ varies, these nonstandard sizes assemble into a definable strong Euler characteristic on $K$, taking values in the profinite completion $\hat{\mathbb{Z}}$ of the integers. The strong Euler characteristic is not canonical, but depends on the choice of a nonstandard Frobenius. When $\operatorname{Abs}(K)$ is finite, the Euler characteristic has some funny properties for two choices of the nonstandard Frobenius. Additionally, we show that the theory of finite fields remains decidable when first-order logic is expanded with parity quantifiers. However, the proof depends on a computational algebraic geometry statement whose proof is deferred to a later paper.