On complete intersection real curves
Let C be a curve over \({\mathbb{R}}\) in the sense of \textit{A. Tognoli} [''Algebraic geometry and Nash functions'' (1978; Zbl 0418.14002)]. In this report the following problem is studied: Does there exist an embedding \(\phi\) : \(C\to {\mathbb{R}}^ n\) satisfying (i) \(\Gamma\) (C,\({\mathcal O}_ C)\cong N^{-1}{\mathbb{R}}[\phi (C)]\), where \(N=\{s\in {\mathbb{R}}[\phi (C)]| s(x)\neq 0\forall x\in \phi (C)\}.\) (ii) \({\mathbb{R}}[\phi (C)]\) is an abstract complete intersection. The results are related to Kumar's work on complete intersections in the case of finitely generated algebras over a field [\textit{N. M. Kumar}, J. Math. Kyoto Univ. 17, 533-538 (1977; Zbl 0384.14016)].
scientific article; zbMATH DE number 3922840