Solution algebras of differential equations and quasi-homogeneous varieties

Details Solution algebras of differential equations and quasi-homogeneous varieties Solution algebras of differential equations and quasi-homogeneous varieties

2011-07-06

arxiv.org/abs/1107.1179v1

Mathematics

Algebraic Geometry

Scientific paper

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. The spectrum of a solution algebra is an algebraic fiber space over the base variety, with quasi-homogeneous fiber. We also discuss the relevance of this result in Transcendental Number Theory.

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