A moment approach to analyze zeros of triangular polynomial sets

Let $I=\langle g_1,\ldots, g_n\rangle$ be a zero-dimensional ideal of $\mathbb{R}[x_1,\ldots ,x_n]$ such that its associated set $\mathbb{G}$ of polynomial equations $g_i(x)=0$ for all $i=1,\ldots ,n$ is in triangular form. By introducing multivariate Newton sums we provide a numerical characterization of polynomials in $\sqrt{I}$. We also provide a necessary and sufficient (numerical) condition for all the zeros of $\mathbb{G}$ to be in a given set $\mathbb{K}\subset \mathbb{C}^n$, without explicitly computing the zeros. In addition, we also provide a necessary and sufficient condition on the coefficients of the $g_i$'s for $\mathbb{G}$ to have (a) only real zeros, (b) to have only real zeros, all contained in a given semi-algebraic set $\mathbb{K}\subset\mathbb{R}^n$. In the proof technique, we use a deep result of Curto and Fialkow (2000) on the $\mathbb{K}$-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 $\mathbb{G}$. In addition, the number of distinct real zeros is shown to be the maximal rank of a related moment matrix.