Foundations of invariant theory for the down operator

This paper lays out the basic theory of the down operator D of the infinite polynomial ring R = k[x0, x1, x2, ...], defined by Dxi = xi−1 (i ≥ 1) and Dx0 = 0. Here, k is any field of characteristic zero. The only linear invariant is x0, and the quadratic invariants are well known and easily described. One of the paper’s main results, Thm. 6.2, gives a complete description of the cubic invariants, ordered according to bi-degree and the number of variables involved. The distinction between core and compound invariants is introduced, and quartic and quintic invariants are studied relative to this property. As an application of the theory, Thm. 8.2 gives a new family of counterexamples to Hilbert’s Fourteenth Problem; the proof of non-finite generation is much simpler than for previously known examples.