The Smith form of Sylvester and Bézout matrices for zero-dimensional ideals

Etna Lindy; Vanni Noferini

Paper

The Smith form of Sylvester and Bézout matrices for zero-dimensional ideals

Abstract

Let

be a field and let

be such that the ideal

is zero-dimensional. We study the Sylvester and Bézout resultant polynomial matrices, built by interpreting

and

as univariate polynomials in

with coefficients in

. We characterize their Smith forms over

in terms of the dual spaces of differential operators, that were defined and studied by H. M. Möller et al. In particular, if

is algebraically closed we show that, if the leading coefficients of

and

are coprime over

, then the partial multiplicities of the Sylvester and Bézout resultant matrices coincide with certain integers, that we call Möller indices. These indices are uniquely determined by

, and can be easily computed from a Gauss basis, as defined in [M. G. Marinari, H. M. Möller, T. Mora, Trans. Amer. Math. Soc. 348(8):3283--3321, 1996], of the dual spaces. We then generalize this result to the case of common factors in the leading coefficients, which correspond to intersections at

, again describing all the invariant factors of Sylvester and Bézout resultant matrices. As a corollary, this fully characterizes the algebraic multiplicity of all the roots of the resultant

in terms of the intersection multiplicities for

and

, including those arising from infinite intersections. We discuss both algebraic and computational implications of our results.