Sharp Lower Estimations for Invariants Associated with the Ideal of Antiderivatives of Singularities

IF 0.7 4区数学 Q2 MATHEMATICS

Bulletin of The Iranian Mathematical Society Pub Date : 2024-03-29 DOI:10.1007/s41980-024-00866-z

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引用次数: 0

Abstract

Let (V, 0) be a hypersurface with an isolated singularity at the origin defined by the holomorphic function \(f: (\mathbb {C}^n, 0)\rightarrow (\mathbb {C}, 0)\) . We introduce a new derivation Lie algebra associated to (V, 0). The new Lie algebra is defined by the ideal of antiderivatives with respect to the Tjurina ideal of (V, 0). More precisely, let \(I = (f, \frac{\partial f}{\partial x_1},\ldots , \frac{\partial f}{\partial x_n})\) and \(\Delta (I):= \{g\mid g,\frac{\partial g}{\partial x_1},\ldots , \frac{\partial g}{\partial x_n}\in I\}\) , then \(A^\Delta (V):= \mathcal O_n/\Delta (I)\) and \(L^\Delta (V):= \textrm{Der}(A^\Delta (V),A^\Delta (V))\) . Their dimensions as a complex vector space are denoted as \(\beta (V)\) and \(\delta (V)\) , respectively. \(\delta (V)\) is a new invariant of singularities. In this paper we study the new local algebra \(A^\Delta (V)\) and the derivation Lie algebra \(L^\Delta (V)\) , and also compute them for fewnomial isolated singularities. Moreover, we formulate sharp lower estimation conjectures for \(\beta (V)\) and \(\delta (V)\) when (V, 0) are weighted homogeneous isolated hypersurface singularities. We verify these conjectures for a large class of singularities. Lastly, we provide an application of \(\beta (V)\) and \(\delta (V)\) to distinguishing contact classes of singularities.

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与奇异性反代数理想相关的不变量的尖锐下限估计值

Abstract Let (V, 0) be a hypersurface with an isolated singularity at the origin defined by the holomorphic function \(f: (\mathbb {C}^n, 0)\rightarrow (\mathbb {C}, 0)\) .我们引入一个与 (V, 0) 相关联的新的派生李代数。这个新的李代数是由关于 (V, 0) 的 Tjurina 理想的反理想定义的。更确切地说，让 \(I = (f, \frac{partial f}{partial x_1},\ldots , \frac{partial f}{partial x_n})\) and \(\Delta (I):= \{g\mid g,\frac{partial g}{partial x_1},\ldots , \frac{partial g}{partial x_n}\in I\})then\(A^\Delta (V):= \mathcal O_n/\Delta (I)\) and\(L^\Delta (V):= \textrm{Der}(A^\Delta (V),A^\Delta (V))\) .它们作为复向量空间的维数分别表示为 \(\beta (V)\) 和 \(\delta (V)\) 。分别表示为 \(\delta (V)\) 是一个新的奇点不变量。本文将研究新的局部代数（A^\delta (V)\）和衍生列代数（L^\delta (V)\)，并计算了它们对于少项式孤立奇点的影响。此外，当 (V, 0) 是加权同质孤立超曲面奇点时，我们为 \(\beta (V)\) 和 \(\delta (V)\) 提出了尖锐的下限估计猜想。我们为一大类奇点验证了这些猜想。最后，我们将 \(\beta (V)\) 和 \(\delta (V)\) 应用于区分奇点的接触类。

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来源期刊

Bulletin of The Iranian Mathematical Society Mathematics-General Mathematics

CiteScore

1.40

自引率

0.00%

发文量

期刊介绍： The Bulletin of the Iranian Mathematical Society (BIMS) publishes original research papers as well as survey articles on a variety of hot topics from distinguished mathematicians. Research papers presented comprise of innovative contributions while expository survey articles feature important results that appeal to a broad audience. Articles are expected to address active research topics and are required to cite existing (including recent) relevant literature appropriately. Papers are critically reviewed on the basis of quality in its exposition, brevity, potential applications, motivation, value and originality of the results. The BIMS takes a high standard policy against any type plagiarism. The editorial board is devoted to solicit expert referees for a fast and unbiased review process.