Chains of Differential Subvarieties in an Algebraic Variety

Let X be a δ-variety over some δ-field . Denote by tdδ(X/ ), or simply tdδ(X) if the ground field is understood, the δ-transcendental degree of X over . Suppose tdδ(X) = d; Johnson [Comment. Math. Helv.44 (1969), 207–216] showed that there is an increasing chain of δ-subvarieties of length ωd in X. The question, also known as the Kolchin Catenary Problem, is: Given a point x X, is there an increasing chain of δ-subvarieties of length ωd starting at x? We will give an affirmative answer to this question if X is an algebraic variety.