Some Facts and Conjectures about Mandelbrot Polynomials

Authors

DOI:

https://doi.org/10.5206/mt.v1i1.14037

Keywords:

Mandelbrot polynomials, asymptotic behaviour, Mandelbrot generating function, Online Encyclopedia of Integer Sequences

Abstract

We show here some facts about the Mandelbrot iterates zₖ(c) where z₀(c)=0 and zₙ₊₁(c) = zₙ²(c) + c, which are polynomials in c. Some of the facts have proofs, and some other ``"``facts" only have experimental evidence but no proof. We invite you to try your hand at filling in the gaps.

Edited, and a new version of the paper uploaded: Peter Luschny caught an error, now corrected in Version 5: we had an incorrect OEIS sequence listed.  We also took the opportunity to use unicodeit.net to remove the LaTeX from this abstract.

Versions 1--4 were just administrative changes, while we were learning OJS.

Author Biographies

Neil J. Calkin, Clemson University

Neil J. Calkin studied mathematics as an undergraduate at Trinity College, Cambridge, and received his Ph.D. in Combinatorics and Optimization from the University of Waterloo. In 1994, he co-founded the Electronic Journal of Combinatorics with Herbert S. Wilf.  He is currently a Professor at Clemson University in the Department of Mathematical Sciences. 

Dr. Eunice Y. S. Chan, Western University

Eunice Y. S. Chan did her B.Sc., M.Sc., and Ph.D. in Applied Mathematics at Western University. She is currently a postdoctoral fellow at the Centre for Medical Evidence, Decision Integrity and Clinical Impact (MEDICI Centre),, Department of Anesthesia and Perioperative Medicine, Schulich School of Medicine and Dentistry at Western University in London, Ontario.

Robert M. Corless, Western University

Robert M. Corless did his B.Sc in Mathematics and Computer Science at the University of British Columbia, his M.Math at Waterloo, and his PhD at UBC. He is Emeritus Distinguished University Professor at Western University, a member of the Rotman Institute of Philosophy, the Scientific Director of The Ontario Research Center for Computer Algebra (www.orcca.on.ca) and an Adjunct Professor at the David R. Cheriton School of Computer Science, the University of Waterloo.  He is the Editor-in-Chief of Maple Transactions.

A graph of the complex zeros of a Mandelbrot polynomial of degree 1024. The 1024 scattered points plotted in the complex plain appear as black dots, roughly outlining the famous ``bulbs of the Mandelbrot set.

Downloads

Published

2021-06-02 — Updated on 2023-03-24

Versions

Issue

Vol. 1 No. 1 (2021): Premiere Issue

Section

Communications

Categories

License

Copyright (c) 2021 Neil J. Calkin, Dr. Eunice Y. S. Chan, Robert M. Corless

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.