| Linear Differential Equations and Products of Linear Forms (1997) | |||||||||||||||
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| We show that liouvillian solutions of an n-th 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 characterisation 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 . 1 Introduction In this paper k is a differential field whose field of constants C is algebraically closed of characteristic 0 (e.g. Q(x) with the usual derivation d=dx). For the derivation ffi of k and a 2 k we write ffi k (a) = a (k) and also a (1) = a 0 , a (2) = a 00 ; : : : . Let L(y) = a n d n y dx n + a n\Gamma1 d n\Gamma1 y dx n\Gamma1 + ... | |||||||||||||||
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