A Visualizable, Constructive Proof of the Fundamental Theorem of Algebra, and a Parallel Polynomial Root Estimation Algorithm
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 nth degree polynomial by solving 2n differential equations in parallel.
References
[2] Vyborny, R.: A simple proof of the fundamental theorem of algebra. Mathematica Bohemica 135(1), 5761 (2010).
[3] Arnold, B.H: A topological proof of the fundamental theorem of algebra. The American Mathematical Monthly 56(7), 465466 (1949).
[4] Guillemin, V., Pollack, A.: Dierential Topology. 1st edn. Prentice-Hall (1947).
[5] Feerman, C.: An easy proof of the fundamental theorem of algebra. The American Mathematical Monthly 74(7),854855 (1967).
[6] Rosenbloom, P.C.: An elementary constructive proof of the fundamental theorem of algebra.The American Mathematical Monthly 52(10), 562570 (1945).
[7] Brenner, J.L, Lyndon, R.C.: Proof of the fundamental theorem of algebra. American Mathematical Monthly 88(4), 253256 (1981).
[8] File, D., Miller, S.: Fundamental theorem of algebra lecture notes from the reading classics (euler) working group autumn 2003,
1/ReadingClassics/FundThmAlg_DFile.pdf , https://people.math.osu.edu/sinnott. last accessed 2020/2/1.
[9] Steed, M.: Proofs of the fundamental theorem of algebra, ~may/REU2014/REUPapers/Steed.pdf ,
[10] Linford, K.: http://math.uchicago.edu/last accessed 2020/1/12. An analysis of Charles Feerman's proof of the fundamental theorem of
algebra, http://commons.emich.edu/honors/504, last accessed 2020/2/1. https://faculty. math.illinois.edu/~nmd/classes/2015/418/notes/fund_thm_alg.pdf , last accessed
[11] Dun eld, N.: The fundamental theorem of algebra (class notes), 2020/1/15.
This is an Open Access article distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, adaptation, and reproduction in any medium, provided that the original work is properly cited.
