When any three solutions are independent

TL;DR

This paper proves that for certain differential equations, algebraic relations among solutions are determined by relations among just three solutions, using model theory and algebraic geometry.

Contribution

It establishes that algebraic relations among solutions of differential equations are already present among three solutions, extending to the autonomous case with two solutions, via model-theoretic methods.

Findings

Relations among solutions are determined by three solutions in general.

In autonomous cases, two solutions suffice for algebraic relations.

Model-theoretic and algebraic geometric methods underpin the results.

Abstract

Given an algebraic differential equation of order greater than one, it is shown that if there is any nontrivial algebraic relation amongst any number of distinct nonalgebraic solutions, along with their derivatives, then there is already such a relation between three solutions. In the autonomous situation when the equation is over constant parameters the assumption that the order be greater than one can be dropped, and a nontrivial algebraic relation exists already between two solutions. These theorems are deduced as an application of the following model-theoretic result: Suppose $p$ is a stationary nonalgebraic type in the theory of differentially closed fields of characteristic zero; if any three distinct realisations of $p$ are independent then $p$ is minimal. If the type is over the constants then minimality (and complete disintegratedness) already follow from knowing that any two realisations are independent. An algebro-geometric formulation in terms of $D$-varieties is given. The same methods yield also an analogous statement about families of compact K\"ahler manifolds.