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On the surjectivity of smooth maps into Euclidean spaces and the fundamental theorem of algebra. (English) Zbl 1402.30006
Summary: In this note, we obtain the surjectivity of smooth maps into Euclidean spaces under mild conditions. As an application, we give a new proof of the fundamental theorem of algebra. We also observe that any \(C^{1}\)-map from a compact manifold into Euclidean space with dimension \(n\geq 2\) has infinitely many critical points.

MSC:
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
58C25 Differentiable maps on manifolds
57R70 Critical points and critical submanifolds in differential topology
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