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Preprint Number 1705
1705. Will Johnson Counting mod n in pseudofinite fields E-mail: Submission date: 16 December 2019 Abstract: 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. Mathematics Subject Classification: Keywords and phrases: |

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