(a.k.a. ANR-19-CE40-0018)

Deciding irrationality and transcendence. Classification algorithms for number theory and combinatorics.

Classifying the objects of a mathematical theory requires to make its predicates effective and to automate its computations efficiently so that they are feasible on concrete instances. This is what we propose to do in order to solve problems in relation to numbers, analytic functions, and generating series. We want indeed to make effective, automatic, and efficient the classification of certain objects in number theory (E-functions, G-functions, Mahler functions) and in combinatorics (constrained walks). We tackle these topics in a single project as the study of the underlying functional equations will benefit from the same algorithmic tools (Galois theory, integration, Gröbner bases, explicit formula reconstruction). Beside striking results on numbers and walks, we expect that the general scope of the tools developed will have a broad and lasting impact.

Listed in the project proposal:

- Saclay: Alin Bostan, Mireille Bousquet-Mélou, Frédéric Chyzak (project coordinator, in charge of partner), Stéphane Fischler, Pierre Lairez, Kilian Raschel
- Lyon: Boris Adamczewski, Éric Delaygue, Charlotte Hardouin, Tanguy Rivoal, Julien Roques (in charge of partner), Bruno Salvy, Jacques-Arthur Weil
- Paris: Jérémy Berthomieu (in charge of partner), Lucia Di Vizio, Thomas Dreyfus, Marc Mezzarobba, Mohab Safey El Din

Our (current and past) doctoral students:

- Pierre Bonnet (Bordeaux and Toulouse)
- Hadrien Brochet (Palaiseau)
- Alexandre Goyer (Palaiseau)
- Gaétan Guillot (Orsay)
- Elio Joseph (Orsay)
- Gabriel Lepetit (Grenoble)
- Hana Melánová (Vienna and Paris)
- Andreas Nessmann (Tours and Vienna)
- Hadrien Notarantonio (Palaiseau and Paris)
- Raphaël Pagès (Bordeaux and Palaiseau)
- Marina Poulet (Lyon)
- Sergey Yurkevich (Palaiseau and Vienna)

Other collaborators affiliated with the project:

- Andrew Elvey Price (Tours, CR)
- Gwladys Fernandes (Versailles, post-doc)
- Sandro Franceschi (Orsay, post-doc)
- Vincent Neiger (Paris, MdC)

- 2024/06/09–14: Joint Conference by
*De rerum natura*and Functional Equations and Interactions, on the occasion of the closing of the project, in Anglet. - 2023: Recent Trends in Computer Algebra 2023 (several weeks of events).
- 2023/03/13–17: Journées ALEA 2023, at CIRM, Luminy.
- 2023/01/09–11: Atelier E-fonctions, G-fonctions et périodes, in Paris.
- 2022/09/15: General meeting, in Toulouse. After an annual meeting of GDR EFI.
- 2022/04/11–15: Périodes, motifs et équations différentielles : entre arithmétique et géométrie, in Paris.
- 2022/06/07: Differential Seminar, in Palaiseau.
- 2021/06/03–04: General meeting (virtual), in Strasbourg. After an annual meeting of GDR EFI.
- 2020/02/25: Differential Seminar, in Palaiseau.
- 2020/02/24: First general meeting (kick-off), in Palaiseau.

(This list gathers those Hal publications referencing the ANR project with proper id in their metadata.)

- gfun, a Maple package that provides tools for guessing a sequence or a series from its first terms, and for manipulating rigorously solutions of linear differential or recurrence equations, by using the equation as a data-structure.
- msolve, an open-source C library for solving multivariate polynomial systems.
- ore_algebra.analytic, a Sagemath package for the rigorous computation of values of univariate D-finite functions and connection matrices between regular singular points of univariate differential operators.

A project funded by ANR (2020–2023). AAP: CE40 - Instrument: PRC - Edition: 2019.