3 problems

Let $f\colon\mathbb R^n\to\mathbb R^n$ be a polynomial map with nowhere zero Jacobian determinant. Let $S_f$ denote the set of points where $f$ is not proper, and write…

Let $F:\mathbb R^n\to\mathbb R^n$ be a local diffeomorphism. Assume that for every affine hyperplane $H\subset\mathbb R^n$, the pre-image $F^{-1}(H)$ is connected. Hyperplane-preim…

Let $Q^n$ be an $n$-rectangle in $\mathbb R^n$, and let $F:Q^n\to Q^n$ be a local diffeomorphism. Assume that every principal minor of $DF(x)$ is nonzero on $\partial Q^n$. Gale–Ni…