Let $I=(g_1,..., g_n)$ be a zero-dimensional ideal of $ \R[x_1,...,x_n]$ such that its associated set $G$ of polynomial equations $g_i(x)=0$ for all $i=1,...,n$, is in triangular form. By introducing multivariate Newton sums we provide a numerical characterization of polynomials in the radical ideal of $I$. We also provide a necessary and sufficient (numerical) condition for all zeros of $G$ to be in a given set $K\subset C^n$ without computing explicitly the zeros. The technique that we use relies on a deep result of Curto and Fialkow on the $K$-moment problem and the conditions we provide are given in terms of positive definiteness of some related moment and localizing matrices depending on the $g_i$'s via the Newton sums of $G$.
Trans. Amer. Math. Soc. 358 (2005), 1403--1420.