Complete intersections in $\textbf {C}^{n}$ and $R^{2n}$

Abstract:

When ${{\mathbf {C}}^n}$ is identified with ${{\mathbf {R}}^{2n}}$ in the usual way, algebraic varieties over the complex numbers give rise to varieties over the reals. We ask when a (strict) complete intersection in ${{\mathbf {C}}^n}$ yields a (strict) complete intersection in ${{\mathbf {R}}^{2n}}$. If the original variety V is connected, a necessary and sufficient condition that its image be a complete intersection is that V be irreducible. We give examples that show that without the connectedness assumption the conclusion is false. In the course of proving this result we give an algebraic analogue of a result by Ephraim on germs of complex and the corresponding real analytic varieties. As our methods apply to varieties over the algebraic closure of an arbitrary real closed field the paper is written in this more general setting.