On surjectivity of smooth maps into Euclidean spaces and the fundamental theorem of algebra

Peng Liu; Shibo Liu

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arXiv · 1706.07281

On surjectivity of smooth maps into Euclidean spaces and the fundamental theorem of algebra

Peng Liu·Shibo Liu

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Abstract

In this note we obtain the surjectivity of smooth maps into Euclidean spaces under mild conditions. As 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 $n\ge2$ has infinitely many critical points.

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Peng Liu, Shibo Liu. 2017-06-16. On surjectivity of smooth maps into Euclidean spaces and the fundamental theorem of algebra. https://arxiv.org/abs/1706.07281

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