Characterizing polynomials with roots in a semialgebraic set

Let p(element of)R[x] be a real-valued polynomial and S(subset of)C a set defined by polynomial inequalities g/sub k/(z,z~)(>)0 for some polynomials g/sub k/ in C[z,z~]. We provide a necessary and sufficient condition on the coefficients of p for all the zeros of p to be in S.