arXiv subjects

Rémi Jaoui

Publications and source records attributed to Rémi Jaoui.

10 recordsLinked to original sources

arXiv2025

On the density of strongly minimal algebraic vector fields

Two theorems witnessing the abundance of geometrically trivial strongly minimal autonomous differential equations of arbitrary order are shown. The first one states that a generic algebraic vector field of degree $d\geq 2$ on the affine space of dimension $n \geq 2$ is strongly minimal and geometrically trivial. The second one states that if $X_0$ is the complement of a smooth hyperplane section $H$ of a smooth projective variety $X$ of dimension $n$, then for $d$ large enough, the system of differential equations associated with a generic vector field on $X_0$ with a pole of order at most $d$ along $H$ is strongly minimal and geometrically trivial. This produces the first examples of meromorphic functions that are new in the sense of Painlevé and satisfy autonomous differential equations of order $n \geq 4$.

math.AG↗

arXiv2025

Integration in finite terms and exponentially algebraic functions

We develop techniques at the interface between differential algebra and model theory to study the following problems of exponential algebraicity: Does a given algebraic differential equation admits an exponentially algebraic solution, that is, a holomorphic solution which is definable in the structure of restricted elementary functions? Do solutions of a given list of algebraic differential equations share a nontrivial exponentially algebraic relation, that is, a nontrivial relation definable in the structure of restricted elementary functions? These problems can be traced back to the work of Abel and Liouville on the problem of integration in finite terms. This article concerns generalizations of their techniques adapted to the study of exponential transcendence and independence problems for more general systems of differential equations. As concrete applications, we obtain exponential transcendence and independence statements for several classical functions: the error function, the Bessel functions, indefinite integrals of algebraic expressions involving Lambert's W-function, the equation of the pendulum, as well as corresponding decidability results.

math.LO↗

arXiv2023

Abelian reduction in differential-algebraic and bimeromorphic geometry

A new tool for the model theory of differentially closed fields and of compact complex manifolds is here developed. In such settings, it is shown that a type internal to the field of constants (resp. to the projective line) admits a maximal image whose binding group is an abelian variety. The properties of such "abelian reductions" are investigated in the Galois-theoretic framework provided by stability theory. Several geometric consequences for the birational geometry of algebraic vector fields of characteristic zero are then deduced. 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 are classified up to birational equivalence, and (3) algebraic vector fields whose finite covers admit no nontrivial factors are studied in arbitrary dimension. Analogues of these results in bimeromorphic geometry are also obtained.

math.AG↗

arXiv2022

The degree of nonminimality is at most two

It is shown that if $p$ is a complete type of Lascar rank at least 2 over $A$, in the theory of differentially closed fields of characteristic zero, then there exists a pair of realisations, $a_1$ and $a_2$, such that $p$ has a nonalgebraic forking extension over $A,a_1,a_2$. Moreover, if $A$ is contained in the field of constants then $p$ already has a nonalgebraic forking extension over $A,a_1$. The results are also formulated in a more general setting.

math.LO↗

arXiv2022

On the equations of Poizat and Liénard

We study the structure of the solution sets in universal differential fields of certain differential equations of order two, the Poizat equations, which are particular cases of Liénard equations. We give a necessary and sufficient condition for strong minimality for equations in this class and a complete classification of the algebraic relations for solutions of strongly minimal Poizat equations. We also give an analysis of the non strongly minimal cases as well as applications concerning the Liouvillian and Pfaffian solutions of some Liénard equations.

math.CA↗

arXiv2021

When any three solutions are independent

Given an algebraic differential equation of order greater than one, it is shown that if there is any nontrivial algebraic relation amongst any number of distinct nonalgebraic solutions, along with their derivatives, then there is already such a relation between three solutions. In the autonomous situation when the equation is over constant parameters the assumption that the order be greater than one can be dropped, and a nontrivial algebraic relation exists already between two solutions. These theorems are deduced as an application of the following model-theoretic result: Suppose $p$ is a stationary nonalgebraic type in the theory of differentially closed fields of characteristic zero; if any three distinct realisations of $p$ are independent then $p$ is minimal. If the type is over the constants then minimality (and complete disintegratedness) already follow from knowing that any two realisations are independent. An algebro-geometric formulation in terms of $D$-varieties is given. The same methods yield also an analogous statement about families of compact Kähler manifolds.

math.AG↗

arXiv2020

Generic planar algebraic vector fields are disintegrated

In 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 $d \geq 3$ 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.

math.LO↗

arXiv2020

Relative internality and definable fibrations

We 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 $\mathrm{DCF}_0$ does not have the strong canonical base property, correcting an earlier proof. We also prove that the theory $\mathrm{CCM}$ 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 $\mathrm{DCF}_0$, 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.

math.LO↗

arXiv2019

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

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.

math.AG↗

arXiv2018

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

We study the properties of orthogonality to the constants and disintegration for autonomous algebraic differential equations. We present a criterion of orthogonality to the constants for absolutely irreducible real $D$-varieties relying on the topological dynamic of the associated real analytic flow. More precisely, we prove that if there exists Zariski-dense invariant compact region of the smooth locus of real points of $X$ where the dynamic of the real analytic flow is topologically weakly mixing, then the generic type of $(X,v)$ is orthogonal to the constants. This criterion will be applied in a second part of this article to establish some transcendance properties for the geodesics of a compact algebraically presented compact Riemannian manifold with negative curvature.

math.LO↗