On open real polynomial maps
It is well known that a polynomial map \(f:\mathbb{C}^n\to \mathbb{C}^p\) is open if and only if its fibers have pure dimension \(n-p\). In this paper we prove the following result: Theorem. Let \(f:\mathbb{R}^n\to \mathbb{R}^n\) be a polynomial map. Denote by \(J(f):\mathbb{R}^n\to \mathbb{R}\) the jacobian of \(f\). The following statements are equivalent: (1) \(f\) is an open map. (2) The fibers of \(f\) are finite and the sign of \(J(f)\) does not change. (3) The fibers of \(f\) are finite and the sign of \(J(f)(x)\) does not change on the fibers of \(f\). Remark. The theorem and its proof remain valid if \(f\) is assumed to be a Nash map. The proof of this theorem will occupy section 1. -- In section 2 we take a first step of open polynomial maps between spaces of different dimension: We prove that an open map \(f:\mathbb{R}^n\to \mathbb{R}^2\), \(n\geq 3\) cannot have zero-dimensional fibers.
scientific article; zbMATH DE number 896576