Rational factors, invariant foliations and algebraic disintegration of compact mixing Anosov flow of dimension 33

In this article, we develop a geometric framework to study the notion of semi-minimality for the generic type of a smooth autonomous differential equation $(X,v)$, based on the study of rational factors of $(X,v)$ and of algebraic foliations on $X$, invariant under the Lie-derivative of the vector field $v$. We then illustrate the effectiveness of these methods by showing that certain autonomous algebraic differential equation of order three defined over the field of real numbers --- more precisely, those associated to mixing, compact, Anosov flows of dimension three --- are generically disintegrated.Comment: The title of the article has been changed in order to reflect better its conten

Abelian reduction in differential-algebraic and bimeromorphic geometry

Several results on the birational geometry of algebraic vector fields in characteristic zero are obtained. In particular, (1) it is shown that if some cartesian power of an algebraic vector field admits a nontrivial rational first integral then already the second power does, (2) two-dimensional isotrivial algebraic vector fields admitting no nontrivial rational first integrals are classified up to birational equivalence, (3) a structural dichotomy is established for algebraic vector fields (of arbitrary dimension) whose finite covers admit no nontrivial factors, and (4) a necessary condition is given on the Albanese map of a smooth projective algebraic variety in order for it to admit an algebraic vector field having no nontrivial rational first integrals. Analogues in bimeromorphic geometry, for families of compact Kaehler manifolds parametrised by Moishezon varieties, are also obtained. These theorems are applications of a new tool here introduced into the model theory of differentially closed fields of characteristic zero (and of compact complex manifolds). In such settings, it is shown that a finite rank type that is internal to the field of constants, C, admits a maximal image whose binding group is an abelian variety in C. The properties of such "abelian reductions" are investigated. Several consequences for types over constant parameters are deduced by combining the use of abelian reductions with the failure of the inverse differential Galois problem for linear algebraic groups over C. One such consequence is that if p is over constant parameters and not C-orthogonal then the second Morley power of p is not weakly C-orthogonal. Statements (1) through (4) are geometric articulations of these consequences.Comment: 32 page

Generic planar algebraic vector fields are strongly minimal and disintegrated

International audienceIn this article, we study model-theoretic properties of algebraic differential equations of order 2, defined over constant differential fields. In particular, we show that the set of solutions of a general differential equation of order 2 and of degree d≥3 in a differentially closed field is strongly minimal and disintegrated. We also give two other formulations of this result in terms of algebraic (non)-integrability and algebraic independence of the analytic solutions of a general planar algebraic vector field

Rational factors, invariant foliations and algebraic disintegration of compact mixing Anosov flows of dimension 3

International audienc

Differential fields and geodesic flows II: Geodesic flows of pseudo-Riemannian algebraic varieties

International audienceWe define the notion of a smooth pseudo-Riemannian algebraic variety (X,g) over a field k of characteristic 0, which is an algebraic analogue of the notion of Riemannian manifold and we study, from a model-theoretic perspective, the algebraic differential equation describing the geodesics on (X,g). When k is the field of real numbers, we prove that if the real points of X are Zariski-dense in X and if the real analytification of (X,g) is a compact Riemannian manifold with negative curvature, then the algebraic differential equation describing the geodesics on (X,g) is absolutely irreducible and its generic type is orthogonal to the constants

Corps différentiels et flots géodésiques I : Orthogonalité aux constantes pour les équations différentielles autonomes

International audienceL'orthogonalité aux constantes est une propriété issue de l’étude modèle-théorique des équations différentielles algébriques et qui traduit des propriétés d’indépendance algébrique remarquables pour ses solutions.Dans cet article, on étudie la propriété d’orthogonalité aux constantes dans un langage algebro-différentiel pour les équations différentielles autonomes ainsi que des méthodes effectives pour établir cette propriété. Le résultat principal est un critère d’orthogonalité aux constantes (et sa version en famille) pour les D-variétés réelles absolument irréductibles (X, v) s’appuyant sur la dynamique du flot réel associé (M,φ). Plus précisément, on montre que s’il existe des parties compactes K de M, Zariski-dense dans X telle que la restriction du flot à K est topologiquement faiblement mélangeante, alors le type générique de (X, v) est orthogonal aux constantes.Ce critère sera appliqué dans [Jao17b] à l’étude modèle-théorique du flot géodésique sur les variétés riemanniennes compactes à courbure strictement négative, présentées algébriquement

Relative internality and definable fibrations

International audienceWe first elaborate on the theory of relative internality in stable theories, focusing on the notion of uniform relative internality (called collapse of the groupoid in an earlier work of the second author), and relating it to orthogonality, triviality of fibrations, the strong canonical base property, differential Galois theory, and GAGA. We prove that DCF0 does not have the strong canonical base property, correcting an earlier proof. We also prove that the theory CCM of compact complex manifolds does not have the strong CBP, and initiate a study of the definable Galois theory of projective bundles. In the rest of the paper we study definable fibrations in DCF0, where the general fibre is internal to the constants, including differential tangent bundles, and geometric linearizations. We obtain new examples of higher rank types orthogonal to the constants

When any three solutions are independent

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On the equations of Poizat and Liénard

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