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Séminaire Calcul Formel
# Fuchs' Theorem on linear differential equations and an exponential function in positive characteristic

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XR203 (XLIM)
### XR203

#### XLIM

Description

In this talk I will define a differential extension of the power series over a field k of positive characteristic p making use of new variables, that behave under differentiation like logarithms in characteristic zero. It turns out that every ordinary linear differential equation with polynomial or power series coefficients over k admits a basis of solutions in this extension; in particular, the exponential differential equation y'=y has a solution exp_p. Such solutions have remarking properties, which we will explore. For example, an analogue of Abel's problem about the algebraicity of logarithmic integrals will be studied over k. This talk is based on joint work with H. Hauser and H. Kawanoue.