Riemannian Geometry and the Fundamental Theorem of Algebra

TL;DR

This paper uses Riemannian geometry to provide a geometric proof of the Fundamental Theorem of Algebra, showing that every non-constant polynomial must have a zero.

Contribution

It introduces a novel geometric approach using Riemannian metrics on the sphere to prove the Fundamental Theorem of Algebra.

Findings

Constructs a Riemannian metric on the sphere assuming no zeros for a polynomial.

Derives contradictions from geometric arguments, confirming the existence of polynomial zeros.

Provides a new geometric perspective on a classical algebraic theorem.

Abstract

If a (non-constant) polynomial has no zero, then a certain Riemannian metric is constructed on the two dimensional sphere. Several geometric arguments are then shown to contradict this fact.