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The arc length of the lemniscate $ \{\vert p(z)\vert =1\}$


Author: Peter Borwein
Journal: Proc. Amer. Math. Soc. 123 (1995), 797-799
MSC: Primary 31A15; Secondary 26D05
DOI: https://doi.org/10.1090/S0002-9939-1995-1223265-3
MathSciNet review: 1223265
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Abstract: We show that the length of the set

$\displaystyle \left\{ {z \in \mathbb{C}:\vert\prod\limits_{i = 1}^n {(z - {\alpha _i})\vert = 1} } \right\}$

is at most $ 8\pi en$. This gives the correct rate of growth in a long-standing open problem of Erdös, Herzog, and Piranian and improves the previous bound of $ 74{n^2}$ due to Pommerenke.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1995-1223265-3
Keywords: Capacity, arclength, Erdös, lemniscate, polynomial
Article copyright: © Copyright 1995 American Mathematical Society

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