Solution algebras of differential equations and quasi-homogeneous varieties: a new differential Galois correspondence

TL;DR

This paper establishes a novel correspondence between solution algebras of linear differential equations and affine quasi-homogeneous varieties, linking Differential Algebra with Geometric Invariant Theory and impacting Transcendental Number Theory.

Contribution

It introduces a new anti-equivalence of categories connecting solution algebras to affine quasi-homogeneous varieties, expanding the theoretical framework of Differential Galois theory.

Findings

Solution algebras correspond to affine quasi-homogeneous varieties.

Spectrum of a solution algebra forms an algebraic fiber space with quasi-homogeneous fibers.

Relevance to Transcendental Number Theory is discussed.

Abstract

We develop a new connection between Differential Algebra and Geometric Invariant Theory, based on an anti-equivalence of categories between solution algebras associated to a linear differential equation (i.e. differential algebras generated by finitely many polynomials in a fundamental set of solutions), and affine quasi-homogeneous varieties (over the constant field) for the differential Galois group of the equation. Solution algebras can be associated to any connection over a smooth affine variety. It turns out that he spectrum of a solution algebra is an algebraic fiber space over the base variety, with quasi-homogeneous fiber. We discuss the relevance of this result to Transcendental Number Theory.