Settled polynomials over finite fields
Authors:
Rafe Jones and Nigel Boston
Journal:
Proc. Amer. Math. Soc. 140 (2012), 18491863
MSC (2010):
Primary 11C20, 37P25, 11R32
Published electronically:
October 11, 2011
MathSciNet review:
2888174
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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 postcomposition 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 nonstable polynomials under postcomposition 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.
 1.
Nidal
Ali, Stabilité des polynômes, Acta Arith.
119 (2005), no. 1, 53–63 (French). MR 2163517
(2006h:11125), 10.4064/aa11914
 2.
Mohamed
Ayad and Donald
L. McQuillan, Irreducibility of the iterates of a quadratic
polynomial over a field, Acta Arith. 93 (2000),
no. 1, 87–97. MR 1760091
(2001c:11031)
 3.
Mohamed
Ayad and Donald
L. McQuillan, Corrections to: “Irreducibility of the iterates
of a quadratic polynomial over a field” [Acta Arith.\ {93} (2000),
no. 1, 87–97; MR1760091 (2001c:11031)], Acta Arith.
99 (2001), no. 1, 97. MR 1845367
(2002d:11125), 10.4064/aa9919
 4.
Nigel
Boston and Rafe
Jones, Arboreal Galois representations, Geom. Dedicata
124 (2007), 27–35. MR 2318536
(2009e:11103), 10.1007/s1071100691139
 5.
John
J. Cannon and Derek
F. Holt, The transitive permutation groups of degree 32,
Experiment. Math. 17 (2008), no. 3, 307–314. MR 2455702
(2009j:20003)
 6.
Lynda
Danielson and Burton
Fein, On the irreducibility of the iterates
of 𝑥ⁿ𝑏, Proc. Amer.
Math. Soc. 130 (2002), no. 6, 1589–1596 (electronic). MR 1887002
(2002m:12001), 10.1090/S000299390106258X
 7.
Burton
Fein and Murray
Schacher, Properties of iterates and composites of
polynomials, J. London Math. Soc. (2) 54 (1996),
no. 3, 489–497. MR 1413893
(97h:12007), 10.1112/jlms/54.3.489
 8.
Rafe
Jones, Iterated Galois towers, their associated martingales, and
the 𝑝adic Mandelbrot set, Compos. Math. 143
(2007), no. 5, 1108–1126. MR 2360312
(2008i:11131), 10.1112/S0010437X07002667
 9.
Rafe
Jones, The density of prime divisors in the arithmetic dynamics of
quadratic polynomials, J. Lond. Math. Soc. (2) 78
(2008), no. 2, 523–544. MR 2439638
(2010b:37239), 10.1112/jlms/jdn034
 10.
, An iterative construction of irreducible polynomials reducible modulo every prime, arXiv:1012.2857v1 (2010). To appear in J. Algebra.
 11.
Alina
Ostafe and Igor
E. Shparlinski, On the length of critical orbits of
stable quadratic polynomials, Proc. Amer. Math.
Soc. 138 (2010), no. 8, 2653–2656. MR 2644881
(2011d:11197), 10.1090/S0002993910104043
 12.
E.
Seneta, Nonnegative matrices and Markov chains, Springer
Series in Statistics, Springer, New York, 2006. Revised reprint of the
second (1981) edition [SpringerVerlag, New York; MR0719544]. MR
2209438
 1.
 Nidal Ali, Stabilité des polynômes, Acta Arith. 119 (2005), no. 1, 5363. MR 2163517 (2006h:11125)
 2.
 Mohamed Ayad and Donald L. McQuillan, Irreducibility of the iterates of a quadratic polynomial over a field, Acta Arith. 93 (2000), no. 1, 8797. MR 1760091 (2001c:11031)
 3.
 , Corrections to: ``Irreducibility of the iterates of a quadratic polynomial over a field'' [Acta Arith. 93 (2000), no. 1, 8797], Acta Arith. 99 (2001), no. 1, 97. MR 1845367 (2002d:11125)
 4.
 Nigel Boston and Rafe Jones, Arboreal Galois representations, Geom. Dedicata 124 (2007), no. 1, 2735 (electronic). MR 2318536 (2009e:11103)
 5.
 John J. Cannon and Derek F. Holt, The transitive permutation groups of degree , Experiment. Math. 17 (2008), no. 3, 307314. MR 2455702 (2009j:20003)
 6.
 Lynda Danielson and Burton Fein, On the irreducibility of the iterates of , Proc. Amer. Math. Soc. 130 (2002), no. 6, 15891596 (electronic). MR 1887002 (2002m:12001)
 7.
 Burton Fein and Murray Schacher, Properties of iterates and composites of polynomials,
J. London Math. Soc. (2) 54 (1996), no. 3, 489497. MR 1413893 (97h:12007)
 8.
 Rafe Jones, Iterated Galois towers, their associated martingales, and the adic Mandelbrot set, Compos. Math. 143 (2007), no. 5, 11081126. MR 2360312 (2008i:11131)
 9.
 , The density of prime divisors in the arithmetic dynamics of quadratic polynomials, J. London Math. Soc. (2) 78 (2008), no. 2, 523544. MR 2439638
 10.
 , An iterative construction of irreducible polynomials reducible modulo every prime, arXiv:1012.2857v1 (2010). To appear in J. Algebra.
 11.
 Alina Ostafe and Igor E. Shparlinski, On the length of critical orbits of stable quadratic polynomials, Proc. Amer. Math. Soc. 138 (2010), no. 8, 26532656. MR 2644881
 12.
 E. Seneta, Nonnegative matrices and Markov chains, Springer Series in Statistics, Springer, New York, 2006. Revised reprint of the second (1981) edition [SpringerVerlag, New York; MR0719544]. MR 2209438
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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/S000299392011110542
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 DMS0852826
The second author was partially supported by NSA H982300910116
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.
