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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.
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A dichotomy for T$T$‐convex fields with a monomial group
Abstract We prove a dichotomy for o‐minimal fields , expanded by a ‐convex valuation ring (where is the theory of ) and a compatible monomial group. We show that if is power bounded, then this expansion of is model complete (assuming that is), it has a distal theory, and the definable sets are geometrically tame. On the other hand, if defines an exponential function, then the natural numbers are externally definable in our expansion, precluding any sort of model‐theoretic tameness.
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- Award ID(s):
- 2103240
- PAR ID:
- 10475386
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Mathematical Logic Quarterly
- Volume:
- 70
- Issue:
- 1
- ISSN:
- 0942-5616
- Format(s):
- Medium: X Size: p. 99-110
- Size(s):
- p. 99-110
- Sponsoring Org:
- National Science Foundation
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