Solutions of algebraic linear ordinary differential equations
Camilo Sanabria Malag\'on

TL;DR
This paper generalizes Klein's classical result by showing that for any finite primitive group in SL_n(C), there exists a standard equation to which all related linear ordinary differential equations can be reduced, extending the hypergeometric case.
Contribution
It extends Klein's result from SL_2(C) to SL_n(C), establishing the existence of standard equations for LODEs with a given differential Galois group, including hypergeometric equations for n=3.
Findings
Existence of a standard equation for any finite primitive group in SL_n(C).
Reduction of LODEs to pullbacks of standard equations over a field extension.
Hypergeometric equations serve as standard equations for n=3 cases.
Abstract
A classical result of F.Klein states that, given a finite primitive group , there exists a hypergeometric equation such that any second order LODE whose differential Galois group is isomorphic to is projectively equivalent to the pullback by a rational map of this hypergeometric equation. In this paper, we generalize this result. We show that, given a finite primitive group , there exist a positive integer and a standard equation such that any LODE whose differential Galois group is isomorphic to is gauge equivalent, over a field extension of degree , to an equation projectively equivalent to the pullback by a map in of this standard equation. For , these standard equations can be chosen to be hypergeometric.
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Taxonomy
TopicsNumerical methods for differential equations · Polynomial and algebraic computation
