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 the transformal dimension of \(p\), i.e., to the maximal transformal transcendence degree over \(K\) of a realization of \(p\).
This is joint work with Ehud Hrushovski, Jinhe Ye and Tingxiang Zou.
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