Algebraic Solutions of Fuchsian Differential Equations

Wider research context / theoretical framework:
The project is 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 will focus on algebraic aspects of Fuchsian equations, more specifically on the existence of algebraic power series solutions and their specific properties.

Hypotheses / research questions / objectives:
We plan to go in two directions. The study of algebraic power series in several variables and their characterization in terms of recurrences, defining equations, codification and by commutative algebra methods. We wish to apply our earlier work to recent research on generating functions of counting lattice walks, the nested Artin approximation theorem, and algorithmic questions. These investigations shall prepare the second line of research: It starts from classical work and aims at adding new structural insights and methods to differential equations and the existence of algebraic solutions. We propose a normal form theorem for differential operators which interprets the operator locally at a singularity as a perturbation of its initial form. This will be applied to reduction modulo p and relates to arithmetic questions and to Eisenstein's theorem about the integrality of algebraic series. We will also propose a variant of the p-curvature conjecture of Grothendieck-Katz and test it computationally.

Approach / methods:
Our techniques will be a mixture of commutative algebra methods, deformation and perturbation theory, combinatorics, arithmetic, and experimental studies.

Level of originality / innovation:
Our experience with singularity theory of algebraic varieties will bring a new flavour to the study of the singularities of differential equations. New conjectures are formulated and will be investigated.

Primary researchers involved:
With Josef Schicho from Linz, Alin Bostan from INRIA Paris, Michael Singer from North Carolina and Nicholas Katz from Princeton University we have outstanding and very experienced collaborators who already agreed to take part in the project and will provide theoretical input as well as computational skills. Two finishing PhD students of the PI will enter the project as postdocs and will collaborate applying the results and techniques of their theses to the new research field, while two starting PhD students will be the youngsters of the project.