3. An Application: Positive Polynomials 15
3. Prove that one can choose m ≤
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
for all x ∈
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 ∈
Similarly, a polynomial p ∈ H2k,n is non-negative if
p(x) ≥ 0 for all x.
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]
(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
for all x ∈
Sketch of Proof. For a polynomial f ∈ H2k,n,
. . . ξn
let us formally define the differential operator
· · ·
Let us choose a positive integer s 2k and the corresponding identity of Proposition