Lang-Weil type bounds in finite difference fields

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Abstract

(joint work with Ehud Hrushovski, Jinhe Ye and Tingxiang Zou) We prove Lang-Weil type bounds for the number of rational points of difference varieties over finite difference fields, in terms of the transformal dimension of the variety and assuming the existence of a smooth rational point. It follows that in (certain) non-principle ultraproducts of finite difference fields the course dimension of a quantifier free type equals its transformal transcendence degree. The proof uses a strong form of the Lang-Weil estimates and, as key ingredient to obtain equidimensional Frobenius specializations, the recent work of Dor and Hrushovski on the non-standard Frobenius acting on an algebraically closed non-trivially valued field, in particular the pure stable embeddedness of the residue difference field in this context.

Speakers from the University of Münster

Hils, Martin

Professorship for Mathematical Logic (Prof. Hils)