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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Settled polynomials over finite fields


Authors: Rafe Jones and Nigel Boston
Journal: Proc. Amer. Math. Soc. 140 (2012), 1849-1863
MSC (2010): Primary 11C20, 37P25, 11R32
Published electronically: October 11, 2011
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Abstract: We study the factorization into irreducibles of iterates of a quadratic polynomial $ f$ over a finite field. We call $ f$ settled when the factorization of its $ n$th iterate for large $ n$ is dominated by ``stable'' polynomials, namely those that are irreducible under post-composition by any iterate of $ f$. We prove that stable polynomials may be detected by their action on the critical orbit of $ f$ and that the critical orbit also gives information about the splitting of non-stable polynomials under post-composition by iterates of $ f$. We then define a Markov process based on the critical orbit of $ f$ and conjecture that its limiting distribution describes the full factorization of large iterates of $ f$. This conjecture implies that almost all quadratic $ f$ defined over a finite field are settled. We give several types of evidence for our conjecture.


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Additional Information

Rafe Jones
Affiliation: Department of Mathematics and Computer Science, College of the Holy Cross, Worcester, Massachusetts 01610
Email: rjones@holycross.edu

Nigel Boston
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email: boston@math.wisc.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11054-2
PII: S 0002-9939(2011)11054-2
Received by editor(s): June 11, 2010
Received by editor(s) in revised form: February 1, 2011
Published electronically: October 11, 2011
Additional Notes: The first author was partially supported by NSF DMS-0852826
The second author was partially supported by NSA H98230-09-1-0116
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.