DIAGONAL POLYNOMIALS FOR SMALL DIMENSIONS

Here R and N denote respectively the real numbers and the nonnegative integers. Also 0 < n is an element of N, and s(x) = x(l) +...+ x(n) wh en x = (x(l)..., x(n)) is an element of R(n). A diagonal function of d imension n is a map f on N-n (or any larger set) that takes N-n biject ively onto N and, for all x, y in N-n, has f(x) < f(y) whenever s(x) < s(y). We show that diagonal polynomials f of dimension n all have tot al degree n and have the same terms of that degree, so that the lower- degree terms characterize any such f. We call two polynomials equivale nt if relabeling variables makes them identical. Then, up to equivalen ce, dimension two admits just one diagonal polynomial, and dimension t hree admits just two.