Differential equations, difference equations and algebraic relations: An extension to a theorem of Compoint

TL;DR

This paper extends Compoint's theorem to show that for certain differential and difference equations over algebraic curves, the algebraic relations among solutions are generated by G-invariants, under specific Galois group conditions.

Contribution

It generalizes Compoint's theorem to include differential and difference equations over algebraic curves with reductive Galois groups and finite determinant groups.

Findings

Algebraic relations are generated by G-invariants in the specified context.

Extension of Compoint's theorem to differential and difference equations.

Applicable to equations over projective curves with reductive Galois groups.

Abstract

Let C be an algebraically closed field and X a projective curve over C. Consider an ordinary linear differential equation, or a linear differ- ence equation, with coefficients in the field of rational functions of X, and assume that its Galois Group G has finite determinant group and is reductive. In this context, the ideal of algebraic relations satisfied by a full system of solutions is generated by the G-invariants it contains. This result extends a theorem of E. Compoint.