  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.
