A moment approach to analyze zeros of triangular polynomial sets

作     者：Lasserre, JB 

作者机构：CNRS LAAS F-31077 Toulouse France LAAS Inst Math F-31077 Toulouse France 

出 版 物：《TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY》 (美国数学会汇刊)

年 卷 期：2006年第358卷第4期

页      面：1403-1420页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

主　　题：system of polynomial equations triangular sets moment problem 

摘      要：Let I = 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 root I. We also provide a necessary and sufficient ( numerical) condition for all the zeros of G to be in a given set K subset of 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 G to have (a) only real zeros, (b) to have only real zeros, all contained in a given semi-algebraic set K subset of R-n. In the proof technique, we use a deep result of Curto and Fialkow ( 2000) 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. In addition, the number of distinct real zeros is shown to be the maximal rank of a related moment matrix.