Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.08550 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909964270829568 |
|---|---|
| author | Lindy, Etna Noferini, Vanni |
| author_facet | Lindy, Etna Noferini, Vanni |
| contents | 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 Bézout 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. Möller 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 Bézout resultant matrices coincide with certain integers, that we call Möller 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. 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 $x=\infty$, 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 $\mathrm{Res}_x(f,g) \in \mathbb{K}[y]$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_08550 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Smith form of Sylvester and Bézout matrices for zero-dimensional ideals Lindy, Etna Noferini, Vanni Commutative Algebra Algebraic Geometry 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 Bézout 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. Möller 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 Bézout resultant matrices coincide with certain integers, that we call Möller 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. 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 $x=\infty$, 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 $\mathrm{Res}_x(f,g) \in \mathbb{K}[y]$ 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. |
| title | The Smith form of Sylvester and Bézout matrices for zero-dimensional ideals |
| topic | Commutative Algebra Algebraic Geometry |
| url | https://arxiv.org/abs/2512.08550 |