On Linear Dependence Over Complete Differential Algebraic Varieties

Abstract

We extend Kolchin's results from [12] on linear dependence over the constant points of projective algebraic varieties to linear dependence over arbitrary complete differential algebraic varieties. We show that in this more general setting, the notion of linear dependence still has necessary and sufficient conditions given by the vanishing of a certain system of differential-polynomials equations. We also discuss some conjectural questions around completeness and the catenary problem.

Key Words:

• Differential algebraic geometry
• Model theory

2010 Mathematics Subject Classification:

• 12H05
• 03C98

ACKNOWLEDGMENTS

The authors began this work during a visit to the University of Waterloo, which was made possible by a travel grant from the American Mathematical Society through the Mathematical Research Communities program. We gratefully acknowledge this support which made the collaboration possible.