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: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/8923/
Martins, Rodrigo S. V. ;
Panario, Daniel ;
Qureshi, Claudio ;
Schmutz, Eric
Periods of Iterations of Mappings over Finite Fields with Restricted Preimage Sizes
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}
}
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Keywords: |
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random mappings with indegree restrictions, Brent-Pollard heuristic, periods of mappings |
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Collection: |
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29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018) |
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Issue Date: |
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2018 |
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Date of publication: |
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18.06.2018 |
