Continuation of root functionals of a system of polynomial equations and the reduction of polynomials modulo its ideal

The notion of a root functional of polynomials is a generalization of the notion of a root for a multiple root. A root functional is a linear functional that is defined on a polynomial ring and annuls the ideal of a system of polynomials. A bounded root functional is a functional that annuls d-th component of the ideal in some filtration in this ideal. It was constructed the operation of continuation of root functionals and the operation of reduction of polynomials modulo the ideal on the basis of the extension operation for bounded root functionals when the number of polynomials is equal to the number of variables and the ideal of polynomials is 0-dimensional. The extension operation has connection with the multivariate Bezoutian construction.