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，未知函数f满足方程f^{(p-1)} + f^p = h^p的解与特征p函数域上线性微分算子的结构和分解密切相关。然而，即便对于具有有理系数的算子，在一般代数函数域上求解该方程是必要的，但目前尚未发展出通用的求解方法。本文提出一种算法，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，提出另一算法，可在h的规模的多项式时间与特征p的线性时间内计算规模为h的多项式大小的解。同时，我们讨论了该方法在正特征p下线性微分算子分解中的应用。