Characterizing polynomials with roots in a semi-algebraic set

Consider a real polynomial $p$ and a semi-algebraic subset $S$ of the complex plane, defined by finitely many polynomial inequalities $g_k(z,\bar{z}) \geq 0$ for some complex polynomials $\{g_k\}$. We provide necessary and sufficient conditions on the coefficients of $p$ for the zeros of $p$ to be in $S$.


IEEE Trans. Automatic Control 49 (2004), pp. 727--730.