Settled polynomials over finite fields
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- by Rafe Jones and Nigel Boston PDF
- Proc. Amer. Math. Soc. 140 (2012), 1849-1863 Request permission
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.References
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Additional Information
- Rafe Jones
- Affiliation: Department of Mathematics and Computer Science, College of the Holy Cross, Worcester, Massachusetts 01610
- MR Author ID: 676504
- ORCID: 0000-0002-4840-4616
- Email: rjones@holycross.edu
- Nigel Boston
- Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
- MR Author ID: 40005
- Email: boston@math.wisc.edu
- 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
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 1849-1863
- MSC (2010): Primary 11C20, 37P25, 11R32
- DOI: https://doi.org/10.1090/S0002-9939-2011-11054-2
- MathSciNet review: 2888174