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Title: Counting mod n in pseudofinite fields
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
Award ID(s):
1803120
PAR ID:
10313374
Author(s) / Creator(s):
Date Published:
Journal Name:
Israel Journal of Mathematics
ISSN:
0021-2172
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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