Representations of positive real polynomials

13 June 2008
Alex Prestel
We consider finite sequences $h = (h_1, . . . h_s)$ of real polynomials in $X_1, . . . ,X_n$ and assume that the semi-algebraic subset $S(h)$ of $R^n$ defined by $h1(a1, . . . , an) \leq 0$, . . . , $hs(a1, . . . , an) \leq 0$ is bounded. We call $h$ (quadratically) archimedean if every real polynomial $f$, strictly positive on $S(h)$, admits a representation $f = \sigma_0 + h_1\sigma_1 + \cdots + h_s\sigma_s$ with each $\sigma_i$ being a sum of squares of real polynomials. If every $h_i$ is linear, the sequence h is archimedean. In general, h need not be archimedean. There exists an abstract valuation theoretic criterion for h to be archimedean. We are, however, interested in an effective procedure to decide whether h is archimedean or not. In dimension n = 2, E. Cabral has given an effective geometric procedure for this decision problem. Recently, S. Wagner has proved decidability for all dimensions using among others model theoretic tools like the Ax-Kochen-Ershov Theorem.