Linear differential equations and products of linear forms Original Research Article

We show that liouvillian solutions of an nth-order linear differential equation L(y) = 0 are related to semi-invariant forms of the differential Galois group of L(y) = 0 which factor into linear forms. The logarithmic 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 logarithmic derivatives of solutions of L(y) = 0. Together with the Brill equations, this characterization allows one to efficiently test if a semi-invariant corresponds to such a coefficient and to compute the other coefficients of P(u) via a factorization of the form F.