Post-doc position: Factoring linear differential operators using p-adic techniques (M/F)

Descrizione

Post-doc position: Factoring linear differential operators using p-adic techniques (M/F) Sintesi in italiano (traduzione automatica): L'organizzazione internazionale offre una posizione di ricercatore post-dottorato presso il laboratorio di informatica LIX dell'École Polytechnique. Il candidato si occuperà dello sviluppo di algoritmi per la fattorizzazione di operatori differenziali lineari utilizzando tecniche p-adiche. Le mansioni principali includono la risoluzione simbolica di equazioni differenziali e l'implementazione di metodi di fattorizzazione. È richiesta una laurea in matematica o in un campo correlato, con competenze in algebra computazionale e teoria delle equazioni differenziali. Il lavoro si svolgerà in un ambiente di ricerca collaborativo e stimolante, con opportunità di sviluppare ulteriormente tecniche p-adiche per equazioni differenziali non lineari. The main objective of this post-doc proposal is to develop algorithms to factor linear differential operators using p‑adic techniques. The central question of this proposal concerns the efficient symbolic resolution of linear differential equations y^(r)(x) + a_(r‑1)(x) y^(r‑1)(x) + ⋯ + a_0(x) y(x) = 0, (1) where the functions a_i are generally rational functions with rational coefficients. Equations of this type are ubiquitous in the theory of special functions, and appear naturally in physics, but also, for example, in combinatorics (finite differential generating series) or in algebraic geometry (Picard‑Fuchs equations). A particularly important sub‑problem in the research program for solving (1) is to factor the linear differential operator L ≔ ∂^r + a_(r‑1) ∂^(r‑1) + ⋯ + a_0 in the non‑commutative ring ℚ(x)[∂]. Three main approaches have been proposed: traditional computer‑algebra methods [3], reduction of the problem modulo a prime p [6, 2], and symbolic‑numeric methods [4, 1]. This postdoc proposal concerns the second modular approach. The current state of the art [5] shows that there are efficient algorithms to factor the reduction L mod p ∈ 𝔽_p(x)[∂] of the operator L modulo p. A first task is to lift this algorithm into a p‑adic factorization algorithm over ℚ_p(x). Our second goal is to use this to develop a factorization algorithm over ℚ(x) or over ℚ*(x), where ℚ* denotes the algebraic closure of ℚ. We anticipate that the second problem will exhibit certain analogies with the symbolic‑numeric approach of [4, 1]. Beyond the factorization of linear differential operators, the postdoc could be an opportunity to develop p‑adic techniques for solving non-linear differential equations. [1] F. Chyzak, A. Goyer, et M. Mezzarobba. Symbolic-numeric factorization of differential operators. Technical Report https://hal.inria.fr/hal-03580658v1 , HAL, 2022. [2] T. Cluzeau. Algorithmique modulaire des équations différentielles linéaires. PhD thesis, Université de Limoges, 2004. [3] M. van Hoeij. Factorization of linear differential operators. PhD thesis, Univ. of Nijmegen, The Netherlands, 1996. [4] J. van der Hoeven. Around the numeric-symbolic computation of differential Galois groups. JSC, 42:236–264, 2007. [5] R. Pagès. Factorisation of linear differential operators in positive characteristic. PhD thesis, U. of Bordeaux, 2024. [6] M. van der Put. Differential equations modulo p. Compositio Mathematica, 97:227–251, 1995. The candidate will be part of the MAX formal computation team at LIX (the computer‑science laboratory of École Polytechnique) and will benefit from a standard work environment. Annuncio in inglese. Fonte: Euraxess (Commissione europea).