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 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.
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Acknowledgment
The author would like to thank Tianyi Xu, for helpful discussions about recursive ind-definability, Tom Scanlon, who read an earlier version of this paper appearing in the author’s dissertation, and the anonymous referee, who offered countless helpful comments and introduced the author to some of the important related papers.
This material is based upon work supported by the National Science Foundation under Grant No. DGE-1106400 and Award No. DMS-1803120. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.
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Johnson, W. Counting mod n in pseudofinite fields. Isr. J. Math. 247, 697–739 (2022). https://doi.org/10.1007/s11856-021-2279-x
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DOI: https://doi.org/10.1007/s11856-021-2279-x