ABSTRACT: Functional equations with one catalytic variable naturally appear in enumerative combinatorics (e.g. when counting planar maps, walks,...). The relevant solution of such an equation is a formal power series with polynomial coefficients in what is called the catalytic variable. Classifying the nature of this solution (e.g. algebraic, D-finite,...) has been an important topic of research since the 60's, starting with the works of Brown and Tutte. In 2006, Bousquet-Mélou and Jehanne obtained a general theorem giving the algebraicity of those solutions. In this talk, I will first briefly reintroduce the combinatorial context and Bousquet-Mélou and Jehanne's result and I will then present links with Artin's approximation theory and Popescu's theorem. I will finally state and prove a recent effective result by Buchacher and Kauers for the algebraicity of the solutions of linear systems of DDEs.
Alin Bostan, Lucia Di Vizio, Kilian Raschel