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 $p$.

翻译：在特征为$p$的代数函数域上，方程$f^{(p-1)} + f^p = h^p$关于未知函数$f$的解与系数在该函数域上的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，但目前尚未开发出通用的求解方法。我们提出一种算法，用于在$h$的“大小”的多项式时间内检验解的存在性，并基于黎曼-罗赫空间的计算和除子类群中元素的选择，提出另一种算法，用于在$h$的“大小”的多项式时间（且关于特征$p$呈线性时间）内计算大小关于$h$的“大小”为多项式的解，同时讨论其在正特征$p$下线性微分算子分解中的应用。