Algebraic solutions of linear differential equations: An arithmetic approach

Authors:
Open Access
Publisher website
Google Scholar
Abstract
Given a linear differential equation with coefficients in Q ( x ) \mathbb {Q}(x) , an important question is to know whether its full space of solutions consists of algebraic functions, or at least if one of its specific solutions is algebraic. After presenting motivating examples coming from various branches of mathematics, we advertise in an elementary way a beautiful local-global arithmetic approach to these questions, initiated by Grothendieck in the late sixties. This approach has deep ramifications and leads to the still unsolved Grothendieck–Katz p p -curvature conjecture.
Funding Information
  • Agence Nationale de la Recherche (ANR-18-CE40-0026)
  • Agence Nationale de la Recherche (ANR-19-CE40-0018)
  • Agence Nationale de la Recherche (ANR-22-CE91-0007 & FWF I6130-N)
  • Agence Nationale de la Recherche (DeRerumNatura (ANR-19-CE40-0018))
  • Agence Nationale de la Recherche (ANR-18-CE40-0026)
  • Agence Nationale de la Recherche (ANR-19-CE40-0018)
  • Agence Nationale de la Recherche (ANR-22-CE91-0007 & FWF I6130-N)
  • Agence Nationale de la Recherche (DeRerumNatura (ANR-19-CE40-0018))