A moment theoretic approach to estimate the cardinality of certain algebraic varieties

Abstract

For n is an element of N, we consider the algebraic variety V obtained by intersecting n+ 1 algebraic curves of degree n in R-2, when the leading terms of the associated bivariate polynomials are all different. We provide a new proof, based on the Flat Extension Theorem from the theory of truncated moment problems, that the cardinality of V cannot exceed ((2) (n+1)). In some instances, 2 this provides a slightly better estimate than the one given by Bezout's Theorem. Our main result contributes to the growing literature on the interplay between linear algebra, operator theory, and real algebraic geometry.