Galois action on solutions of a differential equation
To a linear differential equation \(L\) with coefficients in \(\mathbb Q (x)\), one can associate a Riccati equation \(R\) whose solutions are the logarithmic derivatives of solutions of \(L\). The aim of this paper is, under the hypothesis that \(R\) has a rational (resp. algebraic) solution, to give a bound for the degree of an extension \(C\) of \(\mathbb Q\) such that \(R\) has a solution in \(C(x)\) (resp. has an algebraic solution whose coefficients of the monic irreducible polynomial are in \(C(x)\)). The answer is complete for second-order equations and almost complete for third-order equations (in that case no example showing the bounds to be sharp is available). The first part of the paper is devoted to stating basic facts in differential Galois theory over the differential field \(k(x)\) or \(k((x))\) when \(k\) is a non necessarily algebraically closed field of characteristic 0. Bounds for the degree of \(C\) are given in the second part. They rely on a classification due to \textit{J. Kovačić} for second-order equations [J. Symb. Comput. 2, 3--43 (1986; Zbl 0603.68035)] and to Singer and Ulmer for third-order equations. Examples showing that bounds are sharp are given in the third part.
scientific article; zbMATH DE number 845990