3. An Application: Positive Polynomials 15

3. Prove that one can choose m ≤

(n+2k−1)

2k

in Proposition 3.3.

Remark: In his solution of Waring’s problem, for all positive integers k and n,

D. Hilbert constructed integer vectors ci and rational numbers γi such that

x

2k

=

m

i=1

γi ci,x

2k

for all x ∈

Rn;

see, for example, Chapter 3 of [N96].

We apply Proposition 3.3 to study positive polynomials.

(3.4) Definition. Let p ∈ H2k,n be a polynomial. We say that p is positive

provided p(x) 0 for all x = 0. Equivalently, p ∈ H2k,n is positive provided

p(x) 0 for all x ∈

Sn−1.

Similarly, a polynomial p ∈ H2k,n is non-negative if

p(x) ≥ 0 for all x.

PROBLEM.

1◦.

Prove that the set of all positive polynomials is a non-empty open convex

set in H2k,n and that the set of all non-negative polynomials is a non-empty closed

convex set in H2k,n.

We apply Proposition 3.3 to prove that a homogeneous polynomial is positive

if and only if it can be multiplied by a suﬃciently high power of x 2 to produce a

sum of even powers of linear functions. The proof below is due to B. Reznick [R95]

and [R00].

(3.5) Proposition. Let p ∈ H2k,n be a positive polynomial. Then there exist a

positive integer s and vectors c1,... , cm ∈ Rn such that

x

2s−2kp(x)

=

m

i=1

ci,x

2s

for all x ∈

Rn.

Sketch of Proof. For a polynomial f ∈ H2k,n,

f(x) =

a=(α1,... ,αn)

λaξ1

α1

. . . ξn

αn

,

let us formally define the differential operator

f(∂) =

a=(α1,... ,αn)

λa

∂α1

∂ξ1

α1

· · ·

∂αn

∂ξn

αn

.

Let us choose a positive integer s 2k and the corresponding identity of Proposition

3.3:

(3.5.1) x

2s

=

m

i=1

ci,x

2s.