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Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
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Polynomial $g:\mathbb R^n \rightarrow\mathbb R^n$ with no critical point may have no root
Version 1 (solved): If $g$ : $\mathbb R^n \rightarrow \mathbb R^n$ is a polynomial, $Dg(x)$ is non-degenerate for every $x$, then there exists $x$, such that $g(x)=0$.
Version 2: If $f$ : $\mathbb R^n …