## Algebra & Number Theory

### A finiteness theorem for specializations of dynatomic polynomials

David Krumm

#### Abstract

Let $t$ and $x$ be indeterminates, let $ϕ(x)=x2+t∈ℚ(t)[x]$, and for every positive integer $n$ let $Φn(t,x)$ denote the $n$-th dynatomic polynomial of $ϕ$. Let $Gn$ be the Galois group of $Φn$ over the function field $ℚ(t)$, and for $c∈ℚ$ let $Gn,c$ be the Galois group of the specialized polynomial $Φn(c,x)$. It follows from Hilbert’s irreducibility theorem that for fixed $n$ we have $Gn≅Gn,c$ for every $c$ outside a thin set $En⊂ℚ$. By earlier work of Morton (for $n=3$) and the present author (for $n=4$), it is known that $En$ is infinite if $n≤4$. In contrast, we show here that $En$ is finite if $n∈{5,6,7,9}$. As an application of this result we show that, for these values of $n$, the following holds with at most finitely many exceptions: for every $c∈ℚ$, more than $81%$ of prime numbers $p$ have the property that the polynomial $x2+c$ does not have a point of period $n$ in the $p$-adic field $ℚp$.

#### Article information

Source
Algebra Number Theory, Volume 13, Number 4 (2019), 963-993.

Dates
Revised: 22 January 2019
Accepted: 22 February 2019
First available in Project Euclid: 18 May 2019

https://projecteuclid.org/euclid.ant/1558144826

Digital Object Identifier
doi:10.2140/ant.2019.13.963

Mathematical Reviews number (MathSciNet)
MR3951585

Zentralblatt MATH identifier
07059761

#### Citation

Krumm, David. A finiteness theorem for specializations of dynatomic polynomials. Algebra Number Theory 13 (2019), no. 4, 963--993. doi:10.2140/ant.2019.13.963. https://projecteuclid.org/euclid.ant/1558144826

#### References

• E. Artin, Algebraic numbers and algebraic functions, AMS Chelsea Publishing, Providence, RI, 2006.
• S. Beckmann, “On finding elements in inertia groups by reduction modulo $p$”, J. Algebra 164:2 (1994), 415–429.
• W. Bosma, J. Cannon, and C. Playoust, “The Magma algebra system, I: The user language”, J. Symbolic Comput. 24:3-4 (1997), 235–265.
• T. Bousch, Sur quelques problèmes de dynamique holomorphe, Ph.D. thesis, Université de Paris-Sud, Centre d'Orsay, 1992.
• J. Cannon and D. F. Holt, “Computing maximal subgroups of finite groups”, J. Symbolic Comput. 37:5 (2004), 589–609.
• J. D. Dixon and B. Mortimer, Permutation groups, Graduate Texts in Mathematics 163, Springer, 1996.
• D. S. Dummit and R. M. Foote, Abstract algebra, 3rd ed., John Wiley & Sons, Hoboken, NJ, 2004.
• I. Efrat, Valuations, orderings, and Milnor $K$-theory, Mathematical Surveys and Monographs 124, Amer. Math. Soc., Providence, RI, 2006.
• G. Faltings, “Endlichkeitssätze für abelsche Varietäten über Zahlkörpern”, Invent. Math. 73:3 (1983), 349–366.
• C. Fieker and J. Klüners, “Computation of Galois groups of rational polynomials”, LMS J. Comput. Math. 17:1 (2014), 141–158.
• E. V. Flynn, B. Poonen, and E. F. Schaefer, “Cycles of quadratic polynomials and rational points on a genus-$2$ curve”, Duke Math. J. 90:3 (1997), 435–463.
• R. Frucht, “On the groups of repeated graphs”, Bull. Amer. Math. Soc. 55 (1949), 418–420.
• A. Kerber, Representations of permutation groups, I, Lecture Notes in Mathematics 240, Springer, 1971.
• D. Krumm, “A local-global principle in the dynamics of quadratic polynomials”, Int. J. Number Theory 12:8 (2016), 2265–2297.
• D. Krumm, “code for the computations in the article “A finiteness theorem for specializations of dynatomic polynomials””, 2018, https://github.com/davidkrumm/finiteness_dynatomic. Magma code.
• D. Krumm, “Galois groups in a family of dynatomic polynomials”, J. Number Theory 187 (2018), 469–511.
• D. Krumm and N. Sutherland, “Galois groups over rational function fields and explicit Hilbert irreducibility”, preprint, 2017.
• S. Lang, Algebra, 3rd ed., Graduate Texts in Mathematics 211, Springer, 2002.
• J. S. Leon, “Partitions, refinements, and permutation group computation”, pp. 123–158 in Groups and computation, II (New Brunswick, NJ, 1995), edited by L. Finkelstein and W. M. Kantor, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 28, Amer. Math. Soc., Providence, RI, 1997.
• P. Morton, “Arithmetic properties of periodic points of quadratic maps”, Acta Arith. 62:4 (1992), 343–372.
• P. Morton, “On certain algebraic curves related to polynomial maps”, Compositio Math. 103:3 (1996), 319–350.
• P. Morton, “Arithmetic properties of periodic points of quadratic maps, II”, Acta Arith. 87:2 (1998), 89–102.
• P. Morton and P. Patel, “The Galois theory of periodic points of polynomial maps”, Proc. London Math. Soc. $(3)$ 68:2 (1994), 225–263.
• P. Morton and F. Vivaldi, “Bifurcations and discriminants for polynomial maps”, Nonlinearity 8:4 (1995), 571–584.
• J. Neukirch, Algebraic number theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 322, Springer, 1999.
• B. Poonen, “The classification of rational preperiodic points of quadratic polynomials over ${\bf Q}$: a refined conjecture”, Math. Z. 228:1 (1998), 11–29.
• M. Rosen, Number theory in function fields, Graduate Texts in Mathematics 210, Springer, 2002.
• J. J. Rotman, An introduction to the theory of groups, 4th ed., Graduate Texts in Mathematics 148, Springer, 1995.
• J.-P. Serre, Topics in Galois theory, 2nd ed., Research Notes in Mathematics 1, A K Peters, Ltd., Wellesley, MA, 2008.
• R. P. Stanley, Enumerative combinatorics, Volume 1, 2nd ed., Cambridge Studies in Advanced Mathematics 49, Cambridge University Press, 2012.
• H. Stichtenoth, Algebraic function fields and codes, 2nd ed., Graduate Texts in Mathematics 254, Springer, 2009.
• M. Stoll, “Rational 6-cycles under iteration of quadratic polynomials”, LMS J. Comput. Math. 11 (2008), 367–380.