# Triangles, Ellipses, and Cubic Polynomials

```@article{Minda2008TrianglesEA,
title={Triangles, Ellipses, and Cubic Polynomials},
author={David Minda and Steve Phelps},
journal={The American Mathematical Monthly},
year={2008},
volume={115},
pages={679 - 689}
}```
• Published 1 October 2008
• Mathematics
• The American Mathematical Monthly
(2008). Triangles, Ellipses, and Cubic Polynomials. The American Mathematical Monthly: Vol. 115, No. 8, pp. 679-689.
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## References

SHOWING 1-10 OF 13 REFERENCES
Geometry of Polynomials
During the years since the first edition of this well-known monograph appeared, the subject (the geometry of the zeros of a complex polynomial) has continued to display the same outstanding vitality
An Introduction to Complex Function Theory
This book provides an introduction to the theory of analytic functions of a single complex variable. While presupposing in its readership a degree of mathematical maturity, it insists on no formal
An Elementary Proof of Marden's Theorem
A complete proof of Marden’s Theorem is obtained, requiring very little beyond standard topics from undergraduate mathematics, and spanning analytic geometry, linear algebra, complex analysis, calculus, and properties of polynomials.
MATH
It is still unknown whether there are families of tight knots whose lengths grow faster than linearly with crossing numbers, but the largest power has been reduced to 3/z, and some other physical models of knots as flattened ropes or strips which exhibit similar length versus complexity power laws are surveyed.
edu REFERENCES 1. I. Grattan-Guiness, A limerick retort
• this MONTHLY
• 2005
Prussian Academy of Sciences
• Prussian Academy of Sciences
A limerick retort
• MONTHLY
• 2005