Weighted sums of squares in local rings and their completions, II

Let A be an excellent regular local ring of dimension two, let T be a finitely generated preordering in A, and let [^(T)]{widehat T} be the preordering generated by T in the completion [^(A)]{widehat A} of A. We study the question when the property of being saturated descends from [^(T)]{widehat T} to T, and establish conditions of geometric nature which allow to decide this question. As an application we classify all principal preorderings in A of degree ≤ 3 which are saturated, in the case where A has a real closed residue field. These results have direct implications for nonnegativity certificates for real polynomials on two-dimensional semi-algebraic sets.