On \(d\)-solvability for linear differential equations
Let \(F\) be an ordinary differential field of characteristic zero, \(L(y)=0\) be a linear differential equation of degree \(n\) and \(d\) be a natural number less then \(n\). In the paper the following question, originally posed by \textit{L. Fuchs} [Act. Math. 1, 321--362 (1883; JFM 15.0256.01)] is discussed. Is it possible to express all solutions of the equation \(L(y)=0\) in terms of solutions of homogeneous linear differential equations of order less than or equal to \(d\)? The author specifies the statement of the problem within the frames of differential Galois theory. For this purpose he introduces the concept of \(d\)-solvability for Picard-Vessiot extensions and linear algebraic groups. Then he proves the equivalence of these concepts within differential Galois theory. The main result of the paper is a theorem connecting the \(d\)-solvability of a connected semisimple algebraic group with the minimal dimensions of representations of simple subalgebras of its Lie algebra. The results of the paper can be useful, for example, in number theory in the proof of algebraic independence of values of \(E\)-functions [see the reviewer, Mat. Zametki 34, No. 4, 481--484 (1983; Zbl 0548.12014)].
scientific article; zbMATH DE number 5543812