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 = ui1⋯uimvj1⋯vjn 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.
