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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.more » « less
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We consider existentially closed fields with several orderings, valuations, and p-valuations. We show that these structures are NTP2 of finite burden, but usually have the independence property. Moreover, forking agrees with dividing, and forking can be characterized in terms of forking in ACVF, RCF, and pCF.more » « less
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Let T be a theory. If T eliminates ∃∞, it need not follow that Teq eliminates ∃∞, as shown by the example of the p-adics. We give a criterion to determine whether Teq eliminates ∃∞. Specifically, we show that Teq eliminates ∃∞ if and only if ∃∞ is eliminated on all interpretable sets of “unary imaginaries.” This criterion can be applied in cases where a full description of Teq is unknown. As an application, we show that Teq eliminates ∃∞ when T is a C-minimal expansion of ACVF.more » « less
