On the BCSS Proof of the Fundamental Theorem of Algebra

TL;DR

This paper provides a detailed, more elementary exposition of the BCSS proof of the Fundamental Theorem of Algebra, highlighting its connection to homotopy continuation algorithms and Be9zout's Theorem, without introducing new results.

Contribution

It offers a clearer, more accessible presentation of the BCSS proof, aiding students and researchers unfamiliar with algebraic geometry.

Findings

Connects the proof to homotopy continuation algorithms

Extends the proof to Be9zout's Theorem

Provides an elementary exposition for educational purposes

Abstract

Section 10.4 of the 1998 Springer-Verlag book {\em Complexity and Real Computation}, by Blum, Cucker, Shub, and Smale, contains a particularly elegant proof of the Fundamental Theorem of Algebra: The central idea of the proof naturally leads to a homotopy continuation algorithm for finding the roots of univariate polynomials, and extends naturally to a proof of B\'{e}zout's Theorem (on the number of roots of systems of $n$ equations in $n$ unknowns). We present a more detailed version of the BCSS Proof which is hopefully useful for students and researchers not familiar with algebraic geometry. So while there are no new results in this paper, the exposition is arguably more elementary. Any errors here are solely the responsibility of the current author.