  The paper generalizes Lazarus Fuchs' theorem on the solutions of complex
ordinary linear differential equations with regular singularities to the case
of ground fields of arbitrary characteristic, giving a precise description of
the shape of each solution. This completes partial investigations started by
Taira Honda and Bernard Dwork.
  The main features are the introduction of a differential ring $\mathcal{R}$
in infinitely many variables mimicking the role of the (complex) iterated
logarithms, and the proof that adding these "logarithms" already provides
sufficiently many primitives so as to solve any differential equation with
regular singularity in $\mathcal{R}$. A key step in the proof is the reduction
of the involved differential operator to an Euler operator, its normal form, to
solve Euler equations in $\mathcal{R}$ and to lift their (monomial) solutions
to solutions of the original equation.
  The first (and already very striking) example of this outset is the
exponential function $\exp_p$ in positive characteristic, solution of $y' = y$.
We prove that it necessarily involves all variables and we construct its
explicit (and quite mysterious) power series expansion. Additionally, relations
of our results to the Grothendieck-Katz $p$-curvature conjecture and related
conjectures will be discussed.
