The Low Power Theorem for Partial Differential Polynomials

Abstract

In a recent paper of J. F. Ritt' it is shown that every essential irreducible manifold in the manifold of a partial differential polynomial (p.d.p.) is the gene, ral solution of some p.d.p.; or, equivalently, that the prime ideal corresponding to each component has a basic set consisting of a single p.d.p. However, the general solution of a p.d.p. might be an irreducible manifold in the manifold of a given p.d.p. and yet not be essential. This paper presents a method for determining whether or not such a manifold is essential, thereby extending to p.d.p.'s results previously obtained for ordinary d.p.'s by Ritt2 and the author3. Let 9f be a partial differential field of characteristic zero with mn operations of differentiation. Let (f{jyi, ... **, I = WFAy} denote the ring of p.d.p.'s in the unknowns y', *.. , yn with coefficients in St. It is assumed that the yi and their derivatives are completely ordered by means of marks. Unless the contrary is stated, any p.d.p. mentioned below is supposed to belong to Ifly). Let F and A be two algebraically irreducible p.d.p.'s, such that the general solution of A is contained in the manifold of F. Let S be the separant of A. By slightly modifying a result of M.D.P. we shall show that there is an integer t such that StF has a unique representation as a polynomial in A and certain of its partial derivatives Ai