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Preprint Number 1705

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1705. Will Johnson
Counting mod n in pseudofinite fields

Submission date: 16 December 2019


Comments: Expanded version of thesis chapter; 45 pages

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 ℤ of the integers. The strong Euler characteristic is not canonical, but depends on the choice of a nonstandard Frobenius. When 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.

Mathematics Subject Classification:

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Full text arXiv 1912.07223: pdf, ps.

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