Absolute reducibility of differential operators and Galois groups
A linear differential operator \(L\) over \(C(x)\) is called absolutely reducible if it admits a factorization over an algebraic extension of \(C(x)\) (equivalently: if the connected component of identity of the Galois group of \(L\) acts reducibly on the space of solutions of \(L\)).The problem of absolute factorization of the operator has an important applied value [see Mat. Zametki 34, No. 4, 481--484 (1983; Zbl 0548.12014)]. Existing methods [see J. Symb. Comput. 28, No. 4-5, 473--494 (1999; Zbl 0997.12009)] of such factorization are very complex and ineffective. The authors offer a more effective approach for the decision of this problem. They give sharp bounds on the degree of the extension that is needed in order to compute an absolute factorization.
scientific article; zbMATH DE number 2105789
scientific article