LINEAR-DIFFERENTIAL EQUATIONS AND PRODUCTS OF LINEAR-FORMS

We show that liouvillian solutions of an nth-order linear differential equation L(y)=0 are related to semi-invariant forms of the differenti al Galois group of L(y)=0 which factor into linear forms. The logarith mic derivative of such a form F, evaluated in the solutions of L(y)=0, is the first coefficient of a polynomial P(u) whose zeros are logarit hmic derivatives of solutions of L(y)=0. Together with the Brill equat ions, this characterization allows one to efficiently test if a semi-i nvariant corresponds to such a coefficient and to compute the other co efficients of P(u) via a factorization of the form F. (C) 1997 Elsevie r Science B.V.