Multidimensional structured matrices and polynomial systems

We apply and extend some well known and some recent techniques from algebraic residue theory in order to relate to each other two major subjects of algebraic and numerical computing that is computations with structured matrices and solving a system of polynomial equations 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 struc tured quasi Hankel and quasi Toeplitz matrices naturally associated to systems of multivariate polynomial equations In the second part of the paper we prove some relations between these structured matrices which extend the classical relations of the univariate case Introduction We apply and extend some well known and some recent techniques from al gebraic residue theory in order to relate to each other two major subjects of algebraic and numerical computing that is the computations with structured matrices and solving a system of polynomial equations We also reveal some hid den correlations between these two subjects via the study of the associated op erators of multivariate displacement The latter operators naturally extend the univariate displacement operators which de ne Toeplitz and or Hankel struc ture of matrices cf In our multivariate case we generalize such a matrix structure and arrive at the new classes of operators and structured matrices which include operators and matrices associated to the polynomial systems of equations and which we call quasi Hankel and quasi Toeplitz operators and ma trices since some well known properties of Toeplitz and Hankel operators and matrices can be extended to them see section Due to high importance of computations with structured matrices see e g our study of these matrix classes may be of independent technical interest In section we recall some basic de nitions and facts about algebraic residues in order to extend classical relations between structured matrices to the multivariate case section Next we will state some de nitions R C x xn will denote the polynomial ring in variables x xn over the complex eld C and L C x x n will denote the ring of Laurent s polynomials in the same vari ables We will write x x xn and x x x n n For a vec tor n we will write j j to denote the norm of this vector j j Pni j ij The total degree of a monomial c x with a coe cient c is j j The total degree of a polynomial P c x with coe cients c is the highest total degree of its monomials We will write bSe to denote the cardi nality of a set S ops will stand for arithmetic operations ei will denote the i th unit coordinate vector in C n Our study can be immediately extended from the complex eld C to the case of any number eld of constants having characteristic Furthermore with the exception of the results based on the interpolation techniques of cf proposition our study can be extended to the case of any eld of constants Structured Matrices In this section we propose a generalization of the structure of Toeplitz Hankel and Van der Monde matrices to the case of matrices associated with multivariate polynomials having rows and columns indexed by monomials Quasi Hankel and quasi Toeplitz matrices operators and the associated generating polynomials de nitions and a correlation Definition Let E and F be two subsets of Z and let M m E F be a matrix whose rows and columns are indexed by the elements of E and F respectively M is an E F quasi Hankel matrix i for all E F the en tries m h depend only on For every i n we have m ei ei m provided that ei E ei F such a matrix M is associated with the Laurent polynomial HM x P u E F hux u M is an E F quasi Toeplitz matrix i for all E F the entries m t depend only on For every i n we have m ei ei m provided that ei E ei F such a matrix M is associated with the polynomial TM x P u E F tux u For E m and F n de nition turns into the usual de nition of Hankel resp Toeplitz matrices Definition Let PE L L be the projection map such that PE x x if E otherwise Let E Id PE where Id denotes the identity operator Id e e for all e For any element Q of L let Q L L denote the operator of multiplication by Q For any matrix M m E F let M denote the linear map L L such that M x P E m x if F otherwise The matrix of this linear operator coincides with the matrix M on x x for E F and is null elsewhere We will call this operator an E F quasi Hankel resp an E F quasi Toeplitz operator if the matrix M is an E F quasi Hankel resp an E F quasi Toeplitz matrix Proposition If M is an E F quasi Hankel resp an E F quasi Toeplitz matrix then M P E HM PF resp M PE TM PF To the end of this section we will assume that both sets E and F contain Examples Quasi Toeplitz matrices Let us be given some multivariate polynomials P Pn in the variables x xn and let us consider the matrix associated with the linear map V Vn V Q Qn n X