Let ℝ[X] denote the
real polynomial ring ℝ[X1,…,Xn] and write ∑ℝ[X]2 for the set of sums
of squares in ℝ[X]. Given g1,…,gs∈ ℝ[X] such that the semialgebraic set
K := {x ∈ ℝn∣gi(x) ≥ 0 for all i} is compact, Schmüdgen’s theorem says that if
f ∈ ℝ[X] such that f > 0 on K, then f is in the preordering in ℝ[X] generated by
the gi’s, i.e., f can be written as a finite sum of elements σg1e1…gses, where σ is a
sum of squares in ℝ[X] and each ei∈{0,1}. Putinar’s theorem says that under a
condition on the set of generators {g1,…,gs} (which is a stronger condition than the
compactness of K), any f > 0 on K can be written f = σ0+ σ1g1+⋯+ σsgs, where
σi∈∑ℝ[X]2. Both of these theorems can be viewed as statements about the
existence of certificates of positivity on compact semialgebraic sets. In this note we
show that if the defining polynomials g1,…,gs and polynomial f have coefficients in
ℚ, then in Schmüdgen’s theorem we can find a representation in which the σ’s are
sums of squares of polynomials over ℚ. We prove a similar result for Putinar’s
theorem assuming that the set of generators contains N −∑Xi2 for some
N ∈ ℕ.
Keywords
rational sums of squares, certificates of positivity,
Schmüdgen’s theorem, Putinar’s theorem
