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

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.