Course #3: An overview of difference Galois theory

When studying special functions of the complex variable, one would like to determine whether a function is algebraic over the field of rational functions or not. To refine this classification, one could also be interested in the differential dependence of the function. Starting from the holonomic or D-finite functions, that is the ones that satisfy a linear differential equation, we consider a new class of complexity, called differential transcendence, that corresponds to the functions that do not satisfy a polynomial relation with their derivatives. A celebrated example is the Gamma function, which is differentially transcendental by a result of Hölder. When the special function is given by a linear functional equation, the difference Galois theory provides powerful and systematic tools that allows to determine such kind of relations. In this talk, I will try to give an overview of this Galois theory by focusing on examples and applications.Non UBCUnreviewedAuthor affiliation: Institut de mathématiques de ToulouseFacult