Factoring Linear Differential Operators in Positive Characteristic by means of Solving a Norm Equation

The solutions of the equation $f^{(p-1)} + f^p = h^p$ in the unknown function $f$over an algebraic function field of characteristic $p$ are very closely linked to the structure and factorisations 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 necessary even 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 of elements in the divisor class group, for computing solutions of size polynomial in the ``size'' of h in polynomial time in the size of h and linear in the characteristic $p$, and discuss its applications to the factorisation of linear differential operators in positive characteristic $p$.