Solving special polynomial systems by using structured matrices and algebraic residues

We apply and extend some well known and some recent tech niques from algebraic residue theory in order to relate to each other two major subjects of algebraic and numerical computing that is computa tions with structured matrices and solving a system of polynomial equa tions In the rst part of our paper we extend the Toeplitz and Hankel structures of matrices and some of their known properties to some new classes of structured quasi Hankel and quasi Toeplitz matrices natu rally associated to systems of multivariate polynomial equations In the second part of the paper we apply some results on computations with matrices of these new classes together with some techniques from alge braic residues theory in order to devise an algorithm for approximating a selected solution of a polynomial system of the form