Real algebraic curves as complete intersections (Q1077484)

It is known that for each \(n\geq 3\) there exists a nonsingular algebraic curve C in \({\mathbb{R}}^ n\) such that the ideal \(I_ p(C)\) of \({\mathbb{R}}[x_ 1,...,x_ n]\) of polynomials vanishing on C cannot be generated by n-1 elements. Let \(O({\mathbb{R}}^ n)\) be the ring of regular functions on \({\mathbb{R}}^ n\). The ring \(O({\mathbb{R}}^ n)\) is naturally isomorphic to the localization of \({\mathbb{R}}[x_ 1,...,x_ n]\) with respect to the multiplicatively closed subset consisting of polynomials which do not vanish on \({\mathbb{R}}^ n\). It is shown in the paper that for any nonsingular algebraic curve C in \({\mathbb{R}}^ n\), the ideal I(C) of \(O({\mathbb{R}}^ n)\) of regular functions vanishing on C can be generated by n-1 elements.