Diagonal polynomials for small dimensions

Abstract

Here R and N denote respectively the real numbers and the nonnegative integers. Also 0 < n ∈ N, and s(x) = x1 + ⋯ + xn when x = (x1 , . . . , xn) ∈ Rn. A diagonal function of dimension n is a map f on Nn (or any larger set) that takes Nn bijectively onto N and, for all x, y in Nn, has f(x) < f(y) whenever s(x) < s(y). We show that diagonal polynomials f of dimension n all have total degree n and have the same terms of that degree, so that the lower-degree terms characterize any such f. We call two polynomials equivalent if relabeling variables makes them identical. Then, up to equivalence, dimension two admits just one diagonal polynomial, and dimension three admits just two. © 1996 Springer-Verlag New York Inc.