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Uniform bound for points of bounded degree in function fields of positive characteristic

ENS Salle W

I will present a bound for the number of F_q-points of bounded degree in a variety defined over Z, uniform in q. This generalizes work by Sedunova for fixed q. The proof involves model theory of valued fields with algebraic Skolem functions and uniform non-Archimedean Yomdin-Gromov parametrizations. This is joint work with Raf Cluckers and François Loeser.

On differentially large fields.

ENS Salle W

Recall that a field K is large if it is existentially closed in K((t)). Examples of such fields are the complex, the real, and the p-adic numbers. This class of fields has been exploited significantly by F. Pop and others in inverse Galois-theoretic problems. In recent work with M. Tressl we introduced and explored a differential analogue of largeness, that we conveniently call ``differentially large''. I will present some properties of such fields, and use a twisted version of the Taylor morphism to characterise them using formal Laurent series and […]