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DTSTART;TZID=Europe/Paris:20210326T150000
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SUMMARY:Effective isotrivial Mordell-Lang in positive characteristic
DESCRIPTION:The Mordell-Lang conjecture (now a theorem\, proved by Faltings\, Vojta\, McQuillan\,…) asserts that if G is a semiabelian variety G defined over an algebraically closed field of characteristic zero\, X is a subvariety of G\, and Γ is a finite rank subgroup of G\, then X ∩ Γ is a finite union of cosets of Γ. In positive characteristic\, the naive translation of this theorem does not hold\, however Hrushovski\, using model theoretic techniques\, showed that in some sense all counterexamples arise from semiabelian varieties defined over finite fields (the isotrivial case). This was later refined by Moosa and Scanlon\, who showed in the isotrivial case that the intersection of a subvariety of a semiabelian variety G with a finitely generated subgroup Γ of G that is invariant under the Frobenius endomorphism F: G → G is a finite union of sets of the form S+A\, where A is a subgroup of Γ and S is a sum of orbits under the map F. We show how how one can use finite-state automata to give a concrete description of these intersections Γ ∩ X in the isotrivial setting\, by constructing a finite machine that identifies all points in the intersection. In particular\, this allows us to give decision procedures for answering questions such as: is X ∩ Γ empty? finite? does it contain a coset of an infinite subgroup? In addition\, we are able to read off coarse asymptotic estimates for the number of points of height ≤ H in the intersection from the machine. This is joint work with Dragos Ghioca and Rahim Moosa.
URL:https://www.math.ens.psl.eu/evenement/effective-isotrivial-mordell-lang-in-positive-characteristic/
LOCATION:Zoom
CATEGORIES:Séminaire Géométrie et théorie des modèles
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DTSTART;TZID=Europe/Paris:20210326T163000
DTEND;TZID=Europe/Paris:20210326T175000
DTSTAMP:20260407T134222
CREATED:20210326T153000Z
LAST-MODIFIED:20211025T103938Z
UID:8571-1616776200-1616781000@www.math.ens.psl.eu
SUMMARY:Linearization procedures in the semi-minimal analysis of algebraic differential equations
DESCRIPTION:It is well-known that certain algebraic differential equations restrain in an essential way the algebraic relations that their solutions share. For example\, the solutions of the first equation of Painlevé y » = 6y^2 + t are “new” transcendental functions of order two which whenever distinct are algebraically independent (together with their derivatives).I will first describe an account of such phenomena using the language of geometric stability theory in a differentially closed field. I will then explain how linearization procedures and geometric stability theory fit together to study such transcendence results in practice.
URL:https://www.math.ens.psl.eu/evenement/linearization-procedures-in-the-semi-minimal-analysis-of-algebraic-differential-equations/
LOCATION:Zoom
CATEGORIES:Séminaire Géométrie et théorie des modèles
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