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Periods of Iterations of Mappings over Finite Fields with Restricted Preimage Sizes

Authors Rodrigo S. V. Martins, Daniel Panario, Claudio Qureshi, Eric Schmutz

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ACM Subject Classification
  • Mathematics of computing → Enumeration
Keywords
  • random mappings with indegree restrictions
  • Brent-Pollard heuristic
  • periods of mappings

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Abstract

Let f be a uniformly random element of the set of all mappings from [n] = {1, ..., n} to itself. Let T(f) and B(f) denote, respectively, the least common multiple and the product of the lengths of the cycles of f. Harris proved in 1973 that log T converges in distribution to a standard normal distribution and, in 2011, Schmutz obtained an asymptotic estimate on the logarithm of the expectation of T and B over all mappings on n nodes. We obtain analogous results for uniform random mappings on n = kr nodes with preimage sizes restricted to a set of the form {0,k}, where k = k(r) >= 2. This is motivated by the use of these classes of mappings as heuristic models for the statistics of polynomials of the form x^k + a over the integers modulo p, where k divides p - 1. We exhibit and discuss our numerical results on this heuristic.

Cite As Get BibTex

Rodrigo S. V. Martins, Daniel Panario, Claudio Qureshi, and Eric Schmutz. Periods of Iterations of Mappings over Finite Fields with Restricted Preimage Sizes. In 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 110, pp. 30:1-30:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.AofA.2018.30

Author Details

Rodrigo S. V. Martins
  • Universidade Tecnológica Federal do Paraná, Apucarana, PR 86812-460, Brazil
Daniel Panario
  • Carleton University, Colonel By Drive, Ottawa, ON K1S 5B6, Canada
Claudio Qureshi
  • Universidade Estadual de Campinas, Campinas, SP 13083-859, Brazil
Eric Schmutz
  • Drexel University, Philadelphia, Pa 19104, USA

Funding

  • Panario, Daniel: This author was partially funded by NSERC of Canada.
  • Qureshi, Claudio: This author was funded by FAPESP under grants 2015/26420-1 and 2013/25977-7.

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