We show a way to cut up the complex plane into regions that are mapped 1-1 onto the complex plane by a polynomial. This is done for any finite number of ramification points with any multiplicity. For any polynomial map with only one or two ramification points we can do this explicitly (with minor adjustments). Most of the figures are drawn by approximating solutions to polynomial equations using Newton’s method. However, some of the special cases are computed exactly. At ramification points the plane is cut up by equally spaced arcs and the mapping there acts as if it is a hinge which opens to map to the full plane. In order to show the full extent of possibilities, our last example is a degree 12 polynomial with 5 ramification points of varying degrees.
| Published in | International Journal of Theoretical and Applied Mathematics (Volume 11, Issue 5) |
| DOI | 10.11648/j.ijtam.20251105.12 |
| Page(s) | 78-85 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2025. Published by Science Publishing Group |
Complex Plane, Polynomial Maps, Visualization, Ramification
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APA Style
Meyerson, M. D. (2025). Visualization of Polynomial Maps from the Complex Plane to Itself. International Journal of Theoretical and Applied Mathematics, 11(5), 78-85. https://doi.org/10.11648/j.ijtam.20251105.12
ACS Style
Meyerson, M. D. Visualization of Polynomial Maps from the Complex Plane to Itself. Int. J. Theor. Appl. Math. 2025, 11(5), 78-85. doi: 10.11648/j.ijtam.20251105.12
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Download@article{10.11648/j.ijtam.20251105.12,
author = {Mark Daniel Meyerson},
title = {Visualization of Polynomial Maps from the Complex Plane to Itself},
journal = {International Journal of Theoretical and Applied Mathematics},
volume = {11},
number = {5},
pages = {78-85},
doi = {10.11648/j.ijtam.20251105.12},
url = {https://doi.org/10.11648/j.ijtam.20251105.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20251105.12},
abstract = {We show a way to cut up the complex plane into regions that are mapped 1-1 onto the complex plane by a polynomial. This is done for any finite number of ramification points with any multiplicity. For any polynomial map with only one or two ramification points we can do this explicitly (with minor adjustments). Most of the figures are drawn by approximating solutions to polynomial equations using Newton’s method. However, some of the special cases are computed exactly. At ramification points the plane is cut up by equally spaced arcs and the mapping there acts as if it is a hinge which opens to map to the full plane. In order to show the full extent of possibilities, our last example is a degree 12 polynomial with 5 ramification points of varying degrees.},
year = {2025}
}
TY - JOUR T1 - Visualization of Polynomial Maps from the Complex Plane to Itself AU - Mark Daniel Meyerson Y1 - 2025/12/20 PY - 2025 N1 - https://doi.org/10.11648/j.ijtam.20251105.12 DO - 10.11648/j.ijtam.20251105.12 T2 - International Journal of Theoretical and Applied Mathematics JF - International Journal of Theoretical and Applied Mathematics JO - International Journal of Theoretical and Applied Mathematics SP - 78 EP - 85 PB - Science Publishing Group SN - 2575-5080 UR - https://doi.org/10.11648/j.ijtam.20251105.12 AB - We show a way to cut up the complex plane into regions that are mapped 1-1 onto the complex plane by a polynomial. This is done for any finite number of ramification points with any multiplicity. For any polynomial map with only one or two ramification points we can do this explicitly (with minor adjustments). Most of the figures are drawn by approximating solutions to polynomial equations using Newton’s method. However, some of the special cases are computed exactly. At ramification points the plane is cut up by equally spaced arcs and the mapping there acts as if it is a hinge which opens to map to the full plane. In order to show the full extent of possibilities, our last example is a degree 12 polynomial with 5 ramification points of varying degrees. VL - 11 IS - 5 ER -
Mathematics Department, United States Naval Academy, Annapolis, The United States
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