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.

翻译：在特征为p的代数函数域上，以f为未知函数的方程f^{(p-1)} + f^p = h^p的解与特征p函数域上带系数的线性微分算子的结构和因式分解紧密相关。然而，即使对于有理系数的算子，在一般代数函数域上求解该方程也是必要的，但目前尚未发展出通用的求解方法。我们提出一种算法，可在h的“规模”的多项式时间内检验解的存在性；并基于Riemann-Roch空间的计算与除子类群中元素的选择，提出另一种算法，可在h规模的多项式时间（且与特征p呈线性关系）内计算规模为h规模多项式的解。最后，本文讨论该算法在正特征p下线性微分算子因式分解中的应用。