Rational certificates of positivity on compact semialgebraic sets
Victoria Powers · Pacific Journal of Mathematics · 2011
Let [ޒX] denote the real polynomial ring [ޒX 1 , . . ., X n ] and write [ޒX] 2 for the set of sums of squares in s , where σ is a sum of squares in [ޒX] and each e i ∈ {0, 1}.Putinar's theorem says that under a condition on the set of generators {g 1 , . . ., g s } (which is a stronger condition than the compactness of K ), any f > 0 on K can be writtenBoth 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 g 1 , . . ., g s and polynomial f have coefficients in ,ޑ then in Schmüdgen's theorem we can find a representation in which the σ 's are sums of squares of polynomials over .ޑWe prove a similar result for Putinar's theorem assuming that the set of generators contains N -X 2 i for some N ∈ .ގ