Rémi Jaoui

Rémi Jaoui

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Publications and preprints

Preprints

  • On the density of strongly minimal algebraic vector fields, arXiv , 51 pages (submitted)

It is shown that ‘‘most’’ complex algebraic vector fields define strongly minimal and geometrically trivial differential equations.

Publications

  • Abelian reduction in differential-algebraic and bimeromorphic geometry with Rahim Moosa, arXiv , to appear in Annales de l’institut Fourier (2024).

We prove Galois-theoretic generalizations of several results of Fujiki concerning the biremeromorphic geometry of compact complex manifolds.

  • On the equations of Poizat and Liénard with James Freitag, Dave Marker and Joel Nagloo arXiv. Published in International Mathematical Research Notices 2023, no. 19, pp. 16478–16539.

We study the equations of Poizat from the perspective of geometric stability theory and obtain a full classification of the algebraic relations between solutions of such equations

  • The degree of nonminimality is at most two with James Freitag and Rahim Moosa. arXiv Published in Journal of Mathematical Logic, Vol. 23 (2023), no. 3, Paper No. 2250031, 6 pp.

We show that in the theory DCF_0, if p is a type of Lascar rank greater or equal to two, then there are two realisations of p such that p has a forking nonalgebraic extension over these two realisations.

  • Relative internality and definable fibrations with Léo Jimenez and Anand Pillay. arXiv. Published in Advances in Mathematics Vol. 415 (2023), Paper No. 108870, 38 pp.

We prove that in a theory of finite rank, given a minimal C-internal fibration of types with a base orthogonal to the constants then the fibration is uniformly internal to the constants if and only if it splits as a direct product.

  • When any three solutions are independent with James Freitag and Rahim Moosa. arXiv Published in Inventiones Mathematicae Vol. 230 (2022) no.3, pp. 1249-1265.

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.

  • Generic planar algebraic vector fields are strongly minimal and disintegrated. arXiv Published in Algebra and Number Theory Vol. 15 (2021) no. 10, pp. 2449–2483.

It is shown that complex planar (polynomial) vector fields of degree greater or equal to three are almost always strongly minimal and geometrically trivial.

  • Rational factors, invariant foliations and algebraic disintegration of compact mixing Anosov flows of dimension 3. arXiv Published in Confluentes Mathematici Vol. 12 (2020) no. 2, pp. 49–78.

It is shown that a complex differential equation of dimension three which induces on some compact connected component of the real locus a mixing Anosov flow of dimension three is semiminimal and geometrically trivial.

  • Differential fields and geodesic flows II - Geodesic flows of pseudo-Riemannian algebraic varieties arXiv Published in Israel Journal Math. Vol 230 (2019) no. 2, pp. 527–561.

It is shown that the generic type of the geodesic differential equations describing a compact real-algebraic manifold of negative curvature is orthogonal to the constants. Existence of such real-algebraic manifolds is established.

  • Differential fields and geodesic flows I - Orthogonality to the constants for autonomous differential equations (in french) arXiv Published in Bulletin de la Société Mathématiques de France Vol 148 (2020) no. 3, pp. 529–595.

It is shown that a complex differential equation of dimension three which induces on some compact connected component of the real locus a weakly mixing topological flow is orthogonal to the constants.