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

DOI:
https://doi.org/10.1090/S0002-9939-2011-11054-2

Published electronically:
October 11, 2011

MathSciNet review:
2888174

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Abstract | References | Similar Articles | Additional Information

Abstract: We study the factorization into irreducibles of iterates of a quadratic polynomial over a finite field. We call settled when the factorization of its th iterate for large is dominated by ``stable'' polynomials, namely those that are irreducible under post-composition by any iterate of . We prove that stable polynomials may be detected by their action on the critical orbit of and that the critical orbit also gives information about the splitting of non-stable polynomials under post-composition by iterates of . We then define a Markov process based on the critical orbit of and conjecture that its limiting distribution describes the full factorization of large iterates of . This conjecture implies that almost all quadratic 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:
https://doi.org/10.1090/S0002-9939-2011-11054-2

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.