Counting mod n in pseudofinite fields
Counting mod n in pseudofinite fields
复制标题
计算伪有限域中的 mod n
DOI：
10.1007/s11856-021-2279-x
复制
发表时间：
2021
影响因子：
1
通讯作者：
Johnson, Will
中科院分区：
文献类型：
--
作者：
Johnson, Will
We show that in an ultraproduct of finite fields, the mod-nnonstandard size of definable sets varies definably in families. Moreover, ifKis any pseudofinite field, then one can assign “nonstandard sizes modn” to definable sets inK. Asnvaries, these nonstandard sizes assemble into a definable strong Euler characteristic onK, taking values in the profinite completion \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat {\mathbb{Z}}$$\end{document} 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.
We show that in an ultraproduct of finite fields, the mod-nnonstandard size of definable sets varies definably in families. Moreover, ifKis any pseudofinite field, then one can assign “nonstandard sizes modn” to definable sets inK. Asnvaries, these nonstandard sizes assemble into a definable strong Euler characteristic onK, taking values in the profinite completion \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat {\mathbb{Z}}$$\end{document} 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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