Linear differential equations and products of linear forms

[摘要] 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. (C) 1997 Elsevier Science B.V.

[发布日期] 1997-05-01

[效力级别]  Proceedings Paper