Bounds for Representations of Polynomials Positive on Compact Semi-algebraic Sets 1. Statement of the Results

By Schm udgen's Theorem, polynomials f 2 RX1;::: ; Xn] strictly positive on a bounded basic semialgebraic subset of R n , admit a certain representation involving sums of squares i from RX1 We show the existence of eeective bounds on the degrees of the i by proving rst a suitable version of Schm udgen's Theorem over non-archimedean real closed elds, and then applying the Compactness and Completeness Theorem from Model Theory. In this paper we deal withèeectivity problems' in connection with the following theorem proved by K. Schm udgen in Sch]. (a) 0g is a basic closed semi-algebraic subset of R n. Schm udgen's Theorem states that if S(h) is bounded, then every f 2 RX] that is strictly positive on S(h) admits a representation (?) f = X 2f0;1g m h 1 1 : : : h m m with from RX] 2 , the set of sums of squares of polynomials from RX]. Hilbert's 17-th Problem dealt with the non-compact case h 1 = = h m = 1.