Regarding an adaptive algorithm for testing multivariate linear dependence

For a finite collection of functions within some differential field of several variables, there exists an adaptive algorithm for calculating a basis of their linear relations. We study the complexity of this algorithm, noting how it compares to some other existing techniques. Also we demonstrate some modifications for improving implementation. In the course of our analysis, we define the marginal set of a Young-like set and show how the size of the former can be bounded in terms of the size of the latter.