Vol. 17, No. 1, 1966

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On a problem of J. F. Ritt

Kathleen B O’Keefe

Vol. 17 (1966), No. 1, 149–157

In the Ritt algebra R{u,v} = R(u0,v0,u1,v1,u2,v2,) where the derivation is such that yi= yi+1 for y = u or v, consider the differential ideal Ω = [uv] = ((uv),(uv)1,(uv)2,). Let P = ui1uimvj1vjn be a power product in u, v and their derivatives. For sufficiently large q, it is known that Pq 0[uv]. Power products of the form uivj are of particular interest; one of J. F. Ritt’s unsolved problems is to find the smallest q such that (uivj)q 0[uv]. The purpose of this paper is to solve this problem in the special case i = 1. The main theorem is: The smallest q such that (u1vj)q 0[uv] is 2 + j. Part of the solution involves generalizing some results of D. G. Mead and part is an application of the well-known reduction process of H. Levi.

Mathematical Subject Classification
Primary: 12.80
Received: 19 June 1964
Revised: 2 December 1964
Published: 1 April 1966
Kathleen B O’Keefe