Hartogs-type theorems in real algebraic geometry, I

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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