A Bézoutian-Based Method for Solving Overdetermined Systems of Polynomial Equations

The solution of overdetermined systems of polynomial equations shows promise for solving complicated problems in signal processing and related fields. Polynomials can represent more complicated interactions than their linear counterparts, while overdeterminedness enables the mitigation of noise. In the overdetermined case, the number of common roots can correspond to the number of physically meaningful solutions (which can be less than the Bézout number from the non-overdetermined case). This article extends an algebraic method based on Bézoutian tensors to the non-square case to find all (approximate) roots of an overdetermined system of polynomial equations. The problem is shown to be a partially coupled canonical polyadic decomposition, computable via a rank-1 map. The practical utility of solving overdetermined systems of polynomial equations is shown in an underdetermined blind signal separation problem with constant modulus.