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The Schmidt-Kolchin Conjecture

G. Reinhart
Journal / Anthology

Journal of Symbolic Computation
Year: 1999
Volume: 28
Page range: 611-630

In his study of papers by Osgood of 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 2^d. We will show that the dimension is actually equal to 2^d and 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.

*Mathematics > Algebra > Polynomials
*Mathematics > Calculus and Analysis > Differential Equations