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Classification des imaginaires dans VFA

ENS Salle W

(travail en commun avec Silvain Rideau-Kikuchi)Les imaginaires (c'est-à-dire les quotients définissables) dans la théorie ACVF des corpsalgébriquement clos non-trivialement valués sont classifiés par les sortes “géométriques”.Ceci est un résultat fondamental dû à Haskell, Hrushovski et Macpherson. En utilisantl'approche via la densité des types définissables/invariants, nous donnons une réductiondes imaginaires dans des corps valués henséliens, sous des hypothèses assez générales,aux sortes géométriques et à des imaginaires de RV avec des sortes pour certains espacesvectoriels de dimension finie sur le corps résiduel.

Quantitative Fundamental Theorem of Algebra

ENS Salle W

Using subresultants, we modify a recent real-algebraic proof due to Eisermann of the Fundamental Theorem of Algebra () to obtain the following quantitative information: in order to prove the for polynomials of degree d, the Intermediate Value Theorem () is requested to hold for real polynomials of degree at most d^2. We also explain that the classical algebraic proof due to Laplace requires for real polynomials of exponential degree. These quantitative results highlight the difference in nature of these two proofs.

Geometric quadratic Chabauty.

ENS Salle W

Determining all rational points on a curve of genus at least 2 can be difficult. Chabauty's method (1941) is to intersect, for a prime number p, in the p-adic Lie group of p-adic points of the jacobian, the closure of the Mordell-Weil group with the p-adic points of the curve. If the Mordell-Weil rank is less than the genus then this method has never failed. Minhyong Kim's non-abelian Chabauty programme aims to remove the condition on the rank. The simplest case, called quadratic Chabauty, was developed by Balakrishnan, Dogra, Mueller, […]