On Liouvillian Solutions of Third Order Homogeneous Linear Differential Equations

Noura Okko

Journal of Mathematics Research, 2019, vol. 11, issue 3, 14-22

Abstract: In this article we will consider third order homogeneous differential equations: L(y)=y'''+a_1y'+a_0y(a_0,a_1 ∈k) whose Galois group G（L） is imprimitive. This case is characterised by the fact that the third symmetric power equation L ^ⓢ3(y)=0 has an exponential solution whose square is rational (Singer & Ulmer 1993). If L(y)=0 has a Liouvillian solution z whose logarithmic derivative u=z'/z is algebraic over a differential field (k,') ,we will give an algorithm to find the relation between a_0, a_1 , the semi-invariant S=Y_1Y_2Y_3 which is unique up to multiplication by a constant, the coefficients C_0, C_1 of the minimal polynomial P(u) of u and their derivatives. The aim of this work is to diminutize the number of constants C_m stated in the algorithm of Singer & Ulmer (Singer & Ulmer 1993 Algorithm p. 31) whose determination is not easy to do, and we will achieve this by using Groebner Basis.

Keywords: differential Galois Theory; Liouvillian solutions; Groebner Basis; invariant theory; representation theory (search for similar items in EconPapers)
Date: 2019
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