Absolute reducibility of differential operators and Galois groups

Abstract

A differential operator L∈ C(x)[d/dx] is called absolutely reducible if it admits a factorization over an algebraic extension of C(x) . In this paper, we give sharp bounds on the degree of the extension that is needed in order to compute an absolute factorization. Algorithms to characterize and compute absolute factorizations are then elaborated. The ingredients are differential Galois theory, a group-theoretic study of absolute factorization, and a descent technique for differential operators with coefficients in C(x) .