# A moment approach to analyze zeros of triangular polynomial sets

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\$.

## Citation

Trans. Amer. Math. Soc. 358 (2005), 1403--1420.