Hartogs-type theorems in real algebraic geometry, I

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2022-03-01 DOI:10.1515/crelle-2022-0037

Marcin Bilski, J. Bochnak, W. Kucharz

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

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.

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