The Largest Roots of the Mandelbrot Polynomials

  • Robert M. Corless
  • Piers W. Lawrence
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 50)


This paper gives some details of the experimental discovery and partial proof of a simple asymptotic development for the largest magnitude roots of the Mandelbrot polynomials defined by p 0(z) = 1 and \(p_{n+1}(z) = zp_{n}^{2}(z) + 1\).

Key words

Asymptotic expansion Eigenvalues Mandelbrot polynomials Periodic points Polynomial zeros 

Mathematics Subject Classifications (2010)

Primary 37F10 Secondary 37F45 41A60 



The discussion with Neil Calkin, during the course of the JonFest DownUnder conference, was very helpful. One might even say that had those discussions not taken place, RMC would not have been bitten (by the problem). Peter Borwein pointed out the work of Aczél on functional differential equations. This research work was partially supported by the Natural Sciences and Engineering Research Council of Canada, the University of Western Ontario (and in particular by the Department of Applied Mathematics), the Australian National University (special thanks to the Coffee Gang, Joe Gani, Mike Osborne, David Heath, Ken Brewer, and the other regulars, for their interest), and by the University of Newcastle.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Applied MathematicsThe University of Western OntarioLondonCanada

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