Solutions of algebraic linear ordinary differential equations

2021/04/24 by Malagón, Camilo Sanabria
Mathematics · Computer Science · #Numerical methods for differential equations #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2104.12007

Abstract

A classical result of F.Klein states that, given a finite primitive group G⊆ SL₂(ℂ), there exists a hypergeometric equation such that any second order LODE whose differential Galois group is isomorphic to G 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 G⊆ SL_n(ℂ), there exist a positive integer d=d(G) and a standard equation such that any LODE whose differential Galois group is isomorphic to G is gauge equivalent, over a field extension F of degree d, to an equation projectively equivalent to the pullback by a map in F of this standard equation. For n=3, these standard equations can be chosen to be hypergeometric.

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