Determination of the basis of the space of all root functionals of a system of polynomial equations and of the basis of its ideal by the operation of the extension of bounded root functionals

Determination of the basis of the space of all root functionals of a system of polynomial equations and of the basis of its ideal by the operation of the extension of bounded root functionals (Presented by Corresponding Member of the NAS of Ukraine A. A. Letichevsky) The notion of a root functional of a system of polynomials or an ideal of polynomials is a generalization of the notion of a root, in particular, for a multiple root. A basis of the space of all root functionals and a basis of the ideal are found by using the operation of extension of bounded root functionals when the number of equations is equal to the number of unknowns and if it is known that the number of roots is finite. The asyptotic complexity of these methods is d O(n) operations, where n is the number of equations and unknowns, d is the maximal degree of polynomials. Presence of roots at infinity leads to large degrees of polynomials in Buchberger algorithm for construction of a Gröbner basis of the ideal of polynomials [8]. Therefore the complexity of Buchberger algorithm such large, in the case of the 0-dimensional variety of roots it is equal to d O(n 2) for the number of operations [9], where d is the maximal degree of polynomials, n is the number of variables. In the paper [10] it is shown the exactness of this estimation. For a system of polynomial equations, in which the number of polynomials is equal to the number of variables, the application of extension operations to bounded root functionals [6], [7] gradually cuts components of functionals, lying at infinity, not exiting over the limits of degrees ≤ (d 1 − 1) +. .. + (d n − 1), where d 1 ,. .. , d n are degrees of polynomials. This allows, in the case, if it is known, that the variety of roots is 0-dimensional, to find a basis of the space of all root functionals of the system of polynomials and a basis of the ideal of polynomials in O(D 4) operations, where D = C n d1+...+dn. A similar complexity is had by the method, based on the use of a multivariate resultant, that find all isolated roots of polynomials in d O(n) operations, even in the case of the infinite number of roots at affine domain and at infinity [11].