Hartogs-type theorems in real algebraic geometry, I

IF 1.2 1区数学 Q1 MATHEMATICS

Journal fur die Reine und Angewandte Mathematik Pub Date : 2022-03-01 DOI:10.1515/crelle-2022-0037

Marcin Bilski, J. Bochnak, W. Kucharz

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引用次数: 2

Abstract

Abstract Let f : X → ℝ {f:X\rightarrow\mathbb{R}} be a function defined on a connected nonsingular real algebraic set X in ℝ n {\mathbb{R}^{n}} . We prove that regularity of f can be detected by controlling the restrictions of f to either algebraic curves or algebraic surfaces in X. If dim ⁡ X ≥ 2 {\operatorname{dim}X\geq 2} and k is a positive integer, then f is a regular function whenever the restriction f | C {f|_{C}} is a regular function for every algebraic curve C in X that is a 𝒞 k {\mathcal{C}^{k}} submanifold homeomorphic to the unit circle and is either nonsingular or has precisely one singularity. Moreover, in the latter case, the singularity of C is equivalent to the plane curve singularity defined by the equation x p = y q {x^{p}=y^{q}} for some primes p < q {p

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实代数几何中的hartogs型定理，1

摘要设f:X→{f:X\rightarrow\mathbb{R}}是定义在连通的非奇异实代数集合X上的函数，该函数定义在一个连通的非奇异实代数集合X上，并定义在∈n {\mathbb{R} ^{n}}上。我们证明了f的正则性可以通过控制f对X中的代数曲线或代数曲面的限制来检测。如果dim (X)≥2 {\operatorname{dim} X \geq 2}且k是正整数，则只要限制f| C f|{＿C{是X中与单位圆同态的 k }}{\mathcal{C} ^{k}}子流形的每一个代数曲线C的正则函数，且该曲线非奇异或恰好有一个奇异，f就是正则函数。在后一种情况下，对于{某些素数p

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来源期刊

Journal fur die Reine und Angewandte Mathematik 数学-数学

CiteScore

2.50

自引率

6.70%

发文量

审稿时长

6-12 weeks

期刊介绍： The Journal für die reine und angewandte Mathematik is the oldest mathematics periodical still in existence. Founded in 1826 by August Leopold Crelle and edited by him until his death in 1855, it soon became widely known under the name of Crelle"s Journal. In the almost 180 years of its existence, Crelle"s Journal has developed to an outstanding scholarly periodical with one of the worldwide largest circulations among mathematics journals. It belongs to the very top mathematics periodicals, as listed in ISI"s Journal Citation Report.

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