The Smith form of Sylvester and B\'ezout matrices for zero-dimensional ideals

TL;DR

This paper characterizes the Smith form of Sylvester and Bézout matrices for zero-dimensional ideals in bivariate polynomials, linking their invariant factors to dual spaces and intersection multiplicities, with computational applications.

Contribution

It provides a novel characterization of the Smith form of Sylvester and Bézout matrices using dual spaces and Möller indices, extending to cases with common factors and infinite intersections.

Findings

Smith form characterized via dual spaces

Möller indices determine partial multiplicities

Results include algebraic and computational implications

Abstract

Let $\mathbb{K}$ be a field and let $f,g \in \mathbb{K}[x,y]$ be such that the ideal $\langle f,g \rangle$ is zero-dimensional. We study the Sylvester and B\'{e}zout resultant polynomial matrices, built by interpreting $f$ and $g$ as univariate polynomials in $x$ with coefficients in $\mathbb{K}[y]$. We characterize their Smith forms over $\mathbb{K}[y]$ in terms of the dual spaces of differential operators, that were defined and studied by H. M. M\"{o}ller et al. In particular, if $\mathbb{K}$ is algebraically closed we show that, if the leading coefficients of $f$ and $g$ are coprime over $\mathbb{K}[y]$, then the partial multiplicities of the Sylvester and B\'{e}zout resultant matrices coincide with certain integers, that we call M\"{o}ller indices. These indices are uniquely determined by $\langle f,g \rangle$, and can be easily computed from a Gauss basis, as defined in [M. G. Marinari, H. M. M\"{o}ller, 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 $x=\infty$, again describing all the invariant factors of Sylvester and B\'{e}zout resultant matrices. As a corollary, this fully characterizes the algebraic multiplicity of all the roots of the resultant $\mathrm{Res}_x(f,g) \in \mathbb{K}[y]$ in terms of the intersection multiplicities for $f$ and $g$, including those arising from infinite intersections. We discuss both algebraic and computational implications of our results.