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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

A moment approach to analyze zeros of triangular polynomial sets


Author: Jean B. Lasserre
Journal: Trans. Amer. Math. Soc. 358 (2006), 1403-1420
MSC (2000): Primary 12D10, 26C10, 30E05
DOI: https://doi.org/10.1090/S0002-9947-05-03972-3
Published electronically: November 1, 2005
MathSciNet review: 2186979
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Abstract: 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.


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Additional Information

Jean B. Lasserre
Affiliation: LAAS-CNRS and Institute of Mathematics, LAAS, 7 Avenue du Colonel Roche, 31077 Toulouse Cédex, France
Email: lasserre@laas.fr

DOI: https://doi.org/10.1090/S0002-9947-05-03972-3
Keywords: System of polynomial equations, triangular sets, moment problem
Received by editor(s): April 10, 2002
Published electronically: November 1, 2005
Article copyright: © Copyright 2005 American Mathematical Society