A Visualizable, Constructive Proof of the Fundamental Theorem of Algebra, and a Parallel Polynomial Root Estimation Algorithm

2020/02/09 by Christopher Thron, Thron, Christopher, Jordan T. Barry +1
Computer Science · Mathematics · #12D10 #65H04 #FOS: Mathematics #General Mathematics (math.GM) #Mathematical and Theoretical Analysis #Numerical Methods and Algorithms #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2002.04418

openalex publication_date 2020/02/09 · openalex created_date 2020/02/24 · openalex updated_date 2026/07/28

Abstract

This paper presents an alternative proof of the Fundamental Theorem of Algebra that has several distinct advantages. The proof is based on simple ideas involving continuity and differentiation. Visual software demonstrations can be used to convey the gist of the proof. A rigorous version of the proof can be developed using only single-variable calculus and basic properties of complex numbers, but the technical details are somewhat involved. In order to facilitate the reader's intuitive grasp of the proof, we first present the main points of the argument, which can be illustrated by computer experiments. Next we fill in some of the details, using single-variable calculus. Finally, we give a numerical procedure for finding all roots of an n'th degree polynomial by solving 2n differential equations in parallel.

Related

• On the BCSS Proof of the Fundamental Theorem of Algebra
• Symbolic computation of the roots of any polynomial with integer coefficients
• Iterative polynomial-root-finding procedure with enhanced accuracy
• The LC Method: A parallelizable numerical method for approximating the roots of single-variable polynomials
• A One-Line Proof of the Fundamental Theorem of Algebra with Newton's Method as a Consequence