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

TL;DR

This paper proves conditions under which smooth maps into Euclidean spaces are surjective and offers a new proof of the Fundamental Theorem of Algebra, also discussing critical points of certain maps.

Contribution

It establishes surjectivity criteria for smooth maps and provides a novel proof of the Fundamental Theorem of Algebra, connecting differential topology with algebra.

Findings

Smooth maps into Euclidean spaces are surjective under mild conditions.

A new proof of the Fundamental Theorem of Algebra is presented.

Any $C^1$-map from a compact manifold into Euclidean space with dimension ≥2 has infinitely many critical points.

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.