A Root Parametrized Differential Equation for the Special Linear Group

TL;DR

This paper constructs an explicit linear differential equation over a differential field with parameters, having the special linear group as its Galois group, and demonstrates its generic property for realizing all such Galois groups.

Contribution

It introduces a new explicit parameterized differential equation with Galois group SL_{l+1}(C) that is generic for all Picard-Vessiot extensions with subgroups of this group.

Findings

Explicit construction of the differential equation with Galois group SL_{l+1}(C)

Proof of the equation's generic property for realizing all subgroups as Galois groups

Demonstration of the equation's role as a universal object in the differential Galois theory context.

Abstract

Let $C \langle \boldsymbol{t} \rangle$ be the differential field generated by $l$ differential indeterminates $\boldsymbol{t}=(t_1, \dots, t_l)$ over an algebraically closed field $C$ of characteristic zero. In this article we present an explicit linear parameter differential equation over $C \langle \boldsymbol{t} \rangle$ with differential Galois group $\mathrm{SL}_{l+1}(C)$ and show that it is a generic equation in the following sense: If $F$ is an algebraically closed differential field with constants $C$ and $E/F$ is a Picard-Vessiot extension with differential Galois group $H(C) \subseteq \mathrm{SL}_{l+1}(C)$, then a specialization of our equation defines a Picard-Vessiot extension differentially isomorphic to $E/F$.