A moment approach to analyze zeros of triangular polynomial sets

Abstract

Let I = ?g1,..., gn? be a zero-dimensional ideal of ?[x1,..., xn] such that its associated set double-struck G sign of polynomial equations gi(x) = 0 for all i = 1,...,n is in triangular form. By introducing multivariate Newton sums we provide a numerical characterization of polynomials in ?I. We also provide a necessary and sufficient (numerical) condition for all the zeros of double-struck G sign to be in a given set double-struck K sign ? ?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 double-struck G sign to have (a) only real zeros, (b) to have only real zeros, all contained in a given semi-algebraic set double-struck K sign ? ?n. In the proof technique, we use a deep result of Curto and Fialkow (2000) on the double-struck K sign-moment problem, and the conditions we provide are given in terms of positive definiteness of some related moment and localizing matrices depending on the gi's via the Newton sums of double-struck G sign. In addition, the number of distinct real zeros is shown to be the maximal rank of a related moment matrix. ? 2005 American Mathematical Society.