The Largest Roots of the Mandelbrot Polynomials

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 p0(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 


  1. 1.
    Aczél, J.: On history, applications and theory of functional equations. In: Functional Equations: History, Applications and Theory. D. Reidel, Dordrecht (1984)Google Scholar
  2. 2.
    Corless, R.M., Fillion, N.: A graduate introduction to numerical methods. Springer, to appear (2013)Google Scholar
  3. 3.
    Lawrence, P.W., Corless, R.M.: Stability of rootfinding for barycentric Lagrange interpolants. (submitted)Google Scholar
  4. 4.
    Lawrence, P.W., Corless, R.M.: Numerical stability of barycentric Hermite root-finding. In: Proceedings of the 4th International Workshop on Symbolic-Numeric Computation, pp. 147–148. San Jose, USA (2011)Google Scholar
  5. 5.
    Lawrence, P.W., Corless, R.M., Jeffrey, D.J.: Mandelbrot polynomials and matrices (2012) (in preparation)Google Scholar
  6. 6.
    Sloane, N.J.A.: An on-line version of the encyclopedia of integer sequences. Electron. J. Combin. 1, 1–5 (1994). Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

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

Personalised recommendations