License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.AofA.2018.30
URN: urn:nbn:de:0030-drops-89237
URL: https://drops.dagstuhl.de/opus/volltexte/2018/8923/
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Martins, Rodrigo S. V. ; Panario, Daniel ; Qureshi, Claudio ; Schmutz, Eric

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

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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.

BibTeX - Entry

@InProceedings{martins_et_al:LIPIcs:2018:8923,
  author =	{Rodrigo S. V. Martins and Daniel Panario and Claudio Qureshi and Eric Schmutz},
  title =	{{Periods of Iterations of Mappings over Finite Fields with Restricted Preimage Sizes}},
  booktitle =	{29th International Conference on Probabilistic,  Combinatorial and Asymptotic Methods for the Analysis of Algorithms  (AofA 2018)},
  pages =	{30:1--30:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-078-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{110},
  editor =	{James Allen Fill and Mark Daniel Ward},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8923},
  URN =		{urn:nbn:de:0030-drops-89237},
  doi =		{10.4230/LIPIcs.AofA.2018.30},
  annote =	{Keywords: random mappings with indegree restrictions, Brent-Pollard heuristic, periods of mappings}
}

Keywords: random mappings with indegree restrictions, Brent-Pollard heuristic, periods of mappings
Collection: 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)
Issue Date: 2018
Date of publication: 18.06.2018


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