A moment approach to analyze zeros of triangular polynomial sets

Jean B. Lasserre (lasserrelaas.fr)

Abstract: 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 $G$.

Keywords: zeros of systems of polynomial equations; K-moment problem

Category 1: Other Topics (Other )

Category 2: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Category 3: Nonlinear Optimization

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

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Entry Submitted: 03/16/2004
Entry Accepted: 03/19/2004
Entry Last Modified: 09/18/2006

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