On algebraic solutions of linear differential equations with primitive unimodular Galois group

Abstract

The known algorithms for computing a liouvillian solution of an ordinary homogeneous linear differential equation L(y) = 0 use the fact that, if there is a liouvillian solution, then there is a solution z whose logarithmic derivative z′/z is algebraic over the field of coefficients. Their result is a minimal polynomial for z′/z. In this paper we show that, if there is no logarithmic derivative of a solution of small algebraic degree, then the solution z itself must be algebraic and the algebraic degree of z can be bounded. This can be used to improve algorithms computing liouvillian solutions and allows a direct computation of the minimal polynomial Q(θ) of z. In order to improve the computation of the minimal polynomial Q(θ), we get a criterion, in terms of the differential Galois group, from which the sparsity of Q(θ) can be derived.