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Communications in Mathematical Physics

, Volume 252, Issue 1–3, pp 535–541 | Cite as

A Lower Bound for Periods of Matrices

  • Pietro CorvajaEmail author
  • Zéev Rudnick
  • Umberto Zannier
Article

Abstract

For a nonsingular integer matrix A, we study the growth of the order of A modulo N. We say that a matrix is exceptional if it is diagonalizable, and a power of the matrix has all eigenvalues equal to powers of a single rational integer, or all eigenvalues are powers of a single unit in a real quadratic field. For exceptional matrices, it is easily seen that there are arbitrarily large values of N for which the order of A modulo N is logarithmically small. In contrast, we show that if the matrix is not exceptional, then the order of A modulo N goes to infinity faster than any constant multiple of  log N.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Pietro Corvaja
    • 1
    Email author
  • Zéev Rudnick
    • 2
  • Umberto Zannier
    • 3
  1. 1.Dipartimento di Mathematica e InfUdineItaly
  2. 2.Raymond and Beverly Sackler School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael
  3. 3.I.U.A.V. -DCAVeneziaItaly

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