通过射影次数得到的结式重数及其在张量特征值中的应用
Resultant multiplicity via projective degrees and applications to tensor eigenvalues
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- Ludwig-Maximilians-Universität München（慕尼黑路德维希-马克西米利安大学）
- Inria Paris and Sorbonne University（法国国家数字与信息技术研究所巴黎分部和索邦大学）
- Department of Mathematics, National and Kapodistrian University of Athens（雅典国立卡波季斯特里安大学数学系）
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中文总结 AI 辅助
本文通过射影次数给出齐次多项式系统结式重数的精确公式，将重数估计推广到任意维概型，并应用于张量特征值，解决了相关猜想。
中文摘要 AI 辅助
给定n个变量的n个同次齐次形式构成的系统$\boldsymbol{f}=(f_1,\boldsymbol{\text{...}},f_n)$，Macaulay结式恰好当这些多项式存在公共射影零点时为零，其消失阶衡量了结式超曲面在$\boldsymbol{f}$处的奇异性。本文研究该重数如何反映$\boldsymbol{f}$定义的射影零点概型的几何性质，给出了该重数的精确公式，其用$\boldsymbol{f}$定义的有理映射的射影次数表示。由此得到一个几何下界，涉及射影零点概型不可约分量的次数、维数和重数，将Roy和Ghidelli的重数估计从零维概型推广到任意维数的概型。最后，将该几何估计应用于张量特征值，可直接转化为张量特征值代数重数的下界，该下界由其特征概型的几何性质决定，这解决了Canino等人的一个猜想，进而解决了Qi以及Hu和Ye关于张量特征值的代数重数、几何重数和张量变体重数之间关系的早期猜想。
英文摘要
Given a system $\mathbf{f}=(f_1,\ldots,f_n)$ of $n$ homogeneous forms in $n$ variables of the same degree, Macaulay's resultant vanishes precisely when the polynomials have a common projective zero. Its order of vanishing measures the singularity of the resultant hypersurface at $\mathbf{f}$. In this paper, we study how this multiplicity reflects the geometry of the projective zero scheme defined by $\mathbf{f}$. We give an exact formula for the multiplicity, expressed in terms of the projective degrees of the rational map defined by $\mathbf{f}$. As a consequence, we obtain a geometric lower bound involving the degrees, dimensions, and multiplicities of the irreducible components of the projective zero scheme. This extends the multiplicity estimates of Roy and Ghidelli from zero-dimensional schemes to schemes of arbitrary dimension. Finally, we apply this geometric estimate to tensor eigenvalues. It translates directly into a lower bound for the algebraic multiplicity of a tensor eigenvalue in terms of the geometry of its eigenscheme. This settles a conjecture by Canino et al. and consequently settles earlier conjectures of Qi and of Hu and Ye concerning the relationship between algebraic, geometric, and span multiplicities of tensor eigenvalues.
