zbMATH — the first resource for mathematics

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.

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
Full Text: DOI
[1] Deimling, K., Nonlinear Functional Analysis, (1985), Berlin: Springer-Verlag, Berlin · Zbl 0559.47040
[2] Dugundji, J., Topology, (1966), Boston, MA: Allyn and Bacon, Inc, Boston, MA
[3] Lazer, A. C.; Leckband, M., The fundamental theorem of algebra via the Fourier inversion formula, Amer. Math. Monthly, 117, 5, 455-457, (2010) · Zbl 1208.30008
[4] Sen, A., Fundamental theorem of algebra—yet another proof, Amer. Math. Monthly, 107, 9, 842-843, (2000) · Zbl 1028.12001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.