Rational Certificates of Positivity on Compact Semialgebraic Sets

Given g1, . . . , gs ∈ R[X] = R[X1, . . . , Xn] such that the semialgebraic set K := {x ∈ R | 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 σg1 1 . . . g es s , where σ is a sum of squares in R[X] and each ei ∈ {0, 1}. Putinar’s Theorem says that under a condition stronger than compactness, any f > 0 on K can be written f = σ0 + σ1g1 + · · · + σsgs, where σi ∈ R[X]. 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 − ∑ X i for some N ∈ N.