1. Introduction and overview
An n-ary m-ic form is a polynomial p(x,,...,x) which is homogeneous of
degree m. In this paper we study the representations of real n-ary m-ic forms
of even degree as sums of m-th powers of linear forms. Our work can be
regarded as a partial generalization to higher degree of the familiar theory of
real psd quadratic forms and their representations as a sum of squares. (A
companion paper [R5] will discuss forms of odd degree and, more generally,
forms with complex coefficients.) To each n-ary m-ic p, we associate a
quadratic form H in
(n m'_
.~ ) variables so that, if p is a sum of m-th
powers of linear forms, then H is psd. The converse is false in general, and,
in a rather precise way, is dual to Hilbert's classical result that for
"sufficiently large" n and m, there are psd n-ary m-ic forms which cannot be
written as a sum of squares of forms.
Ve use the familiar multinomial notation in describing real n-ary d-ic
forms, where d is not assumed to be even. Let
ln
denote the set of n-tuples of
non-negative integers. For i = (i.9...,i ) e
Z?n
and x = (x..,...,x )
IRn,
we
i i
write x for
Etak
and define |i|= Si, and c(i) = |i|!/II(ii !), the associated
multinomial coefficient. (For small values of n, the variables will be called
x, y, z, w, ) For 1 d 6 2, let
(1.1) l(n,d) = {i 2n : |i| = d},
so |l(n,d)| = N(n,d) = (n + * ~ * ). If d is even, we write d = m = 2s.
Throughout the paper, the symbols "d", "i","m", "n" and "s" are reserved for
Author supported in part by the National Science Foundation
Received by editor September 30, 1988. Received in revised form October 29,
1990.
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