关于到判别式及密切簇的距离
ON THE DISTANCE TO DISCRIMINANTS AND OSCULATING VARIETIES
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- Université Côte d’Azur（蔚蓝海岸大学）
- Inria at Université Côte d’Azur（蔚蓝海岸大学国家信息与自动化研究所）
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中文总结 AI 辅助
本文从代数几何角度研究到奇异超曲面簇的距离问题，计算其欧氏距离度，并建立 Chebyshev 多项式的极值性质。
中文摘要 AI 辅助
我们从代数几何的观点研究寻找给定非奇异超曲面的最近奇异超曲面的问题。给定次数的奇异超曲面集合 $\Xi$ 是一个与 Veronese 簇对偶的射影簇。然而，正如 Raffalli [Raf14] 所示，从一般实超曲面到 $\Xi$ 的 Bombieri-Weyl（也称为 apolar）距离可能在 $\Xi$ 的奇异点处取得最小值。我们研究 $\Xi$ 的奇异轨迹及其不可约分量，该分量由具有尖点奇异的超曲面组成。这个簇与 Veronese 簇的切簇是对偶的。我们使用拓扑工具计算其欧氏距离度，并推导出从一般超曲面到 $\Xi$ 的距离函数的临界点数量的公式。我们还开始了对 Veronese 簇的高阶密切簇的类似问题的研究。此外，我们通过刻画使到 $\Xi$ 的距离最大化的实根二元形式，建立了 Chebyshev 多项式的一个新的极值性质（在 [Raf14] 中猜想）。
英文摘要
We study the problem of finding the closest singular hypersurface to a given nonsingular one from algebro-geometric point of view. The set of singular hypersurfaces $Ξ$ of a given degree is a projective variety dual to the Veronese variety. However, as was shown by Raffalli [Raf14], the Bombieri-Weyl (also known as apolar) distance from a general real hypersurface to $Ξ$ could be minimized at a singular point of $Ξ$. We study the singular locus of $Ξ$ and its irreducible component, consisting of hypersurfaces with cuspidal singularities. This variety is projectively dual to the tangential variety of the Veronese variety. We compute its Euclidean Distance Degree using topological tools and deduce a formula for the number of critical points of the distance function to $Ξ$ from a general hypersurface. We also initiate the study of the analogous problem for higher osculating varieties to the Veronese variety. Furthermore, we establish a new extremal property of Chebyshev polynomials (conjectured in [Raf14]) by characterizing real-rooted binary forms that maximize the distance to $Ξ$.
