Characterizing polynomials with roots in a semi-algebraic set

Abstract

Let p /spl epsi/ R[x] be a real-valued polynomial and S /spl sub/ C a semi-algebraic set defined by polynomial inequalities gk(z, z~)/spl ges/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.