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.

1