Algebraic Solutions of Fuchsian Differential Equations

"Algebraic solutions of Fuchsian differential equations" The project studies a topic from the interplay between algebra and analysis. The first studies mathematical objects through polynomial equations and computations with letters, say, variables. As such, algebra has a strong axiomatic character. In contrast, analysis is concerned with convergence and limits, as some parameters of some system vary and one tries to understand how this variation affects the system. In the concrete situation, the differential equations come from analysis, and their solutions are typically differentiable functions. Within Fuchsian equations, called like this in honor of Lazarus Fuchs from the second half of the nineteenth century, there does exist a purely algebraic approach to disclose the analytic features. Therefore the project is both fascinating and difficult. The overall objective is to understand the solutions conceptually.

Algebraic solutions of Fuchsian differential equations The project was situated in the algebraic theory of ordinary differential equations with polynomial coefficients. They are Fuchsian if all singularities are regular, i.e., if the local solutions have at most polynomial growth. In the project, we focused on algebraic aspects of Fuchsian equations, more specifically on the existence of algebraic power series solutions and their specific properties. The study of algebraic power series in several variables and their characterization in terms of recurrences, defining equations, codification is carried out by commutative algebra methods. We applied our earlier work to recent research on generating functions of counting lattice walks, the nested Artin approximation theorem, and algorithmic questions. These investigations produced a second line of research: It starts from classical work and adds new structural insights and methods to differential equations and the existence of algebraic solutions. A normal form theorem for differential operators has been proven which interprets the operator locally at a singularity as a perturbation of its initial form. This was applied to the reduction modulo p and thus related to arithmetic questions and to Eisenstein's theorem about the integrality of algebraic series. We proposed a variant of the p-curvature conjecture of Grothendieck-Katz and tested it computationally. Our technique was a mixture of commutative algebra methods, deformation theory, combinatorics, arithmetic, and experimental studies. With Josef Schicho from Linz and Alin Bostan in Paris two outstanding and very experienced collaborators brought significant input. International partners were, among others, Nick Katz, Princeton, and Michael Singer, North Carolina.