A Moment Approach to Analyze Zeros of Triangular Polynomial Sets

Let I = 〈g1, . . . , gn〉 be a zero-dimensional ideal of R[x1, . . . , xn] such that its associated set G 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 G to be in a given set K ⊂ Cn, without explicitly computing the zeros. In addition, we also provide a necessary and sufficient condition on the coefficients of the gi’s for G to have (a) only real zeros, (b) to have only real zeros, all contained in a given semi-algebraic set K ⊂ Rn. 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 gi’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.