 # Séminaires : Géométrie et Théorie des Modèles

 Equipe(s) : lm, Responsables : Zoé Chatzidakis, Raf Cluckers Email des responsables : zoe.chatzidakis@imj-prg.fr Salle : Adresse : Description http://gtm.imj-prg.fr/   Pour recevoir le programme par e-mail, écrivez à : zchatzid_at_dma.ens.fr. Pour les personnes ne connaissant pas du tout de théorie des modèles, des notes introduisant les notions de base (formules, ensembles définissables, théorème de compacité, etc.) sont disponibles ici : http://www.logique.jussieu.fr/~zoe/papiers/MTluminy.dvi. Ces personnes peuvent aussi consulter les premiers chapitres du livre Model Theory and Algebraic Geometry, E. Bouscaren ed., Springer Verlag, Lecture Notes in Mathematics 1696, Berlin 1998. Les notes de quelques-uns des exposés sont disponibles.

 Orateur(s) Thomas Scanlon - UC Berkeley, Titre Skew-invariant curves and algebraic independence Date 13/05/2022 Horaire 14:15 à 15:45 Diffusion Résume A σ-variety over a difference field (K,σ) is a pair (X,φ) consisting of an algebraic variety X over K and φ:X → X^σ is a regular map from X to its transform Xσ under σ. A subvariety Y ⊆ X is skew-invariant if φ(Y) ⊆ Y^σ. In earlier work with Alice Medvedev we gave a procedure to describe skew-invariant varieties of σ-varieties of the form (𝔸^n,φ) where φ(x_1,...,x_n) = (P_1(x_1),...,P_n(x_n)). The most important case, from which the others may be deduced, is that of n = 2. In the present work we give a sharper description of the skew-invariant curves in the case where P_2 = P_1^τ for some other automorphism of K which commutes with σ. Specifically, if P in K[x] is a polynomial of degree greater than one which is not eventually skew-conjugate to a monomial or ± Chebyshev (i.e. P is “nonexceptional”) then skew-invariant curves in (𝔸^2,(P,P^τ)) are horizontal, vertical, or skew-twists: described by equations of the form y = α^{σ^n} ∘ P^{σ^{n-1}} ∘ ⋅⋅⋅ ∘ P^σ ∘ P(x) or x = β^{σ{-1}}∘ P^{τ σ^{-n-2}}∘ P^{τ σ^{-n-3}}∘ ⋅⋅⋅ ∘ P^τ(y) where P = α ∘ β and P^τ = α^{σ^{n+1}}∘ β^{σ^n}} for some integer n. Salle Salle W, ENS, et Zoom Adresse ENS