  The solutions of the equation f^{ (p--1) }+ f^p = h^p in the unknown function
f overan algebraic function field of characteristic p are very closely linked
to the structure and fac-torisations of linear differential operators with
coefficients in function fields of characteristic p.However, while being able
to solve this equation over general algebraic function fields is necessaryeven
for operators with rational coefficients, no general resolution method has been
developed.We present an algorithm for testing the existence of solutions in
polynomial time in the ``size''of h and an algorithm based on the computation
of Riemann-Roch spaces and the selection ofelements in the divisor class group,
for computing solutions of size polynomial in the ``size'' of hin polynomial
time in the size of h and linear in the characteristic p, and discuss its
applicationsto the factorisation of linear differential operators in positive
characteristic p.
