Lang-Weil type bounds in finite difference fields

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Abstract

We establish 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, working in any non-principle ultraproduct K of finite difference fields, the normalized pseudofinite dimension of a quantifier free partial type p is equal to its transformal dimension, i.e., to the maximal transformal transcendence degree over K of a realization of p. The proof uses a strong form of the Lang-Weil estimates (due to Cafure and Matera) 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. This is joint work with Ehud Hrushovski, Jinhe Ye and Tingxiang Zou.

Speakers from the University of Münster

Hils, Martin

Professorship for Mathematical Logic (Prof. Hils)