The Schmidt–Kolchin Conjecture

In his study of papers by Osgood and Kolchin on rational approximations of algebraic functions, Schmidt observed and proved a theorem which states that the dimension of the vector space of differentially homogeneous differential polynomials is at least 2d. We will show that the dimension is actually equal to 2dand that a basis can be effectively computed. The proof proceeds by developing a term ordering of differential monomials in several variables which ranks monomials relative to a given monomial. This new ordering will be useful in polynomial algorithms in general. The ordering has been implemented in Mathematica.