Weighted Sums of Squares in Local Rings and Their Completions, I

Let A be an excellent local ring of real dimension ≤ 2, let T be a finitely generated preordering in A, and let T̂ be the preordering generated by T in the completion Â. Under a weak condition on the residue field k = A/m we show that T saturated implies T̂ saturated, and that a weak version of the converse holds as well. We also prove a transfer result between different real closed fields in the case where A is henselian and k is real closed. These results have direct implications for nonnegativity certificates for real polynomials which are nonnegative on suitable 2-dimensional semi-algebraic sets.