Algebraic Solutions of Fuchsian Differential Equations II

The project is situated in the algebraic theory of ordinary differential equations with polynomial coefficients. In the complex setting, an equation is called Fuchsian if all its singularities are regular, i.e., if the local solutions have at most polynomial growth at the singular points. There exist purely algebraic characterizations of regularity, and thus the concept lends itself also to differential equations in positive characteristic. On the other hand, the existence of algebraic solutions in characteristic zero is still not completely clarified, and represents a wide area of recent activity. Objectives: Already the exponential function exp in characteristic p is an interesting example. Namely, to solve the respective equation, one has to introduce infinitely many variables which mimic the iterates of the complex logarithm. Surprisingly enough, setting then all but finitely many equal to zero gives a projection of exp which turns out to be an algebraic function. This can be shown to hold for all first order equations with regular singularities. Extending the observation to higher order equations is one punch-line of the new project. The second direction concerns differential Artin approximation (in characteristic zero), aiming at extending techniques of Artin, Popescu, Néron desingularization and Denef-Lipshitz to obtain a strong approximation result for (certain) differential equations. The goal is to find effective criteria which ensure the algebraicity of the solutions. This will be related to the higher curvatures of the plane curve defined by the minimal polynomial of algebraic functions. Both directions of research continue and enforce current studies in the ongoing project of the PI of which the solicited one will be a follow-up. Approach: Our techniques will be a mixture of commutative algebra methods, differential algebra and differential Galois theory, deformation and perturbation theory, combinatorics, arithmetic, and experimental studies. Innovation: Our experience with singularity theory of algebraic varieties is expected to bring a new flavor to the study of the singularities of differential equations. Differential equations in positive characteristic are only partially explored but seem to present, as our studies show, highly-structured patterns.