On linear dependence over complete differential algebraic varieties
  We extend Kolchin's results on linear dependence over projective varieties in the constants, to linear dependence over arbitrary complete differential 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. 
James Freitag, Omar Leon Sanchez, William Simmons
math.AG