Involve: A Journal of Mathematics

  • Involve
  • Volume 5, Number 4 (2012), 481-495.

Newly reducible iterates in families of quadratic polynomials

Katharine Chamberlin, Emma Colbert, Sharon Frechette, Patrick Hefferman, Rafe Jones, and Sarah Orchard

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We examine the question of when a quadratic polynomial f(x) defined over a number field K can have a newly reducible n-th iterate, that is, fn(x) irreducible over K but fn+1(x) reducible over K, where fn denotes the n-th iterate of f. For each choice of critical point γ, we consider the family

g γ , m ( x ) = ( x γ ) 2 + m + γ , m K .

For fixed n3 and nearly all values of γ, we show that there are only finitely many m such that gγ,m has a newly reducible n-th iterate. For n=2 we show a similar result for a much more restricted set of γ. These results complement those obtained by Danielson and Fein (Proc. Amer. Math. Soc. 130:6 (2002), 1589–1596) in the higher-degree case. Our method involves translating the problem to one of finding rational points on certain hyperelliptic curves, determining the genus of these curves, and applying Faltings’ theorem.

Article information

Involve, Volume 5, Number 4 (2012), 481-495.

Received: 15 October 2012
Revised: 19 February 2013
Accepted: 4 April 2013
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R09: Polynomials (irreducibility, etc.) 37P05: Polynomial and rational maps 37P15: Global ground fields

polynomial iteration polynomial irreducibility arithmetic dynamics rational points on hyperelliptic curves


Chamberlin, Katharine; Colbert, Emma; Frechette, Sharon; Hefferman, Patrick; Jones, Rafe; Orchard, Sarah. Newly reducible iterates in families of quadratic polynomials. Involve 5 (2012), no. 4, 481--495. doi:10.2140/involve.2012.5.481.

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  • A. F. Beardon, Iteration of rational functions: Complex analytic dynamical systems, Graduate Texts in Mathematics 132, Springer, New York, 1991.
  • W. Bosma, J. Cannon, and C. Playoust, “The Magma algebra system, I: The user language”, J. Symbolic Comput. 24:3–4 (1997), 235–265.
  • N. Boston and R. Jones, “Arboreal Galois representations”, Geom. Dedicata 124 (2007), 27–35.
  • J. E. Cremona, “Elliptic curve data”, online tables, University of Warwick,
  • L. Danielson and B. Fein, “On the irreducibility of the iterates of $x^n-b$”, Proc. Amer. Math. Soc. 130:6 (2002), 1589–1596.
  • X. Faber, B. Hutz, P. Ingram, R. Jones, M. Manes, T. J. Tucker, and M. E. Zieve, “Uniform bounds on pre-images under quadratic dynamical systems”, Math. Res. Lett. 16:1 (2009), 87–101.
  • B. Fein and M. Schacher, “Properties of iterates and composites of polynomials”, J. London Math. Soc. $(2)$ 54:3 (1996), 489–497.
  • D. M. Goldschmidt, Algebraic functions and projective curves, Graduate Texts in Mathematics 215, Springer, New York, 2003.
  • S. Hamblen, R. Jones, and K. Madhu, “The density of primes in orbits of $z^d + c$”, preprint, 2013.
  • M. Hindry and J. H. Silverman, Diophantine geometry: An introduction, Graduate Texts in Mathematics 201, Springer, New York, 2000.
  • R. Jones, “The density of prime divisors in the arithmetic dynamics of quadratic polynomials”, J. Lond. Math. Soc. $(2)$ 78:2 (2008), 523–544.
  • R. Jones, “An iterative construction of irreducible polynomials reducible modulo every prime”, J. Algebra 369 (2012), 114–128.
  • R. Jones and N. Boston, “Settled polynomials over finite fields”, Proc. Amer. Math. Soc. 140:6 (2012), 1849–1863.
  • R. Jones and M. Manes, “Galois theory of quadratic rational functions”, preprint, 2011. To appear in Comment. Math. Helv.
  • R. W. K. Odoni, “On the prime divisors of the sequence $w\sb {n+1}=1+w\sb 1\cdots w\sb n$”, J. London Math. Soc. $(2)$ 32:1 (1985), 1–11.
  • J. H. Silverman, “The difference between the Weil height and the canonical height on elliptic curves”, Math. Comp. 55:192 (1990), 723–743.
  • J. H. Silverman, The arithmetic of dynamical systems, Graduate Texts in Mathematics 241, Springer, New York, 2007.
  • M. Stoll, “Galois groups over ${\mathbb Q}$ of some iterated polynomials”, Arch. Math. $($Basel$)$ 59:3 (1992), 239–244.