 

David DeMark,
Wade Hindes,
Rafe Jones,
Moses Misplon,
Michael Stoll, and
Michael Stoneman
Eventually stable quadratic polynomials over Q
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Published: 
May 26, 2020. 
Keywords: 
Iterated polynomials, irreducible polynomials, rational points, hyperelliptic curves, arboreal Galois representation. 
Subject: 
37P15, 11R09, 37P05, 12E05, 11R32. 


Abstract
We study the number of irreducible factors (over Q) of the nth iterate of a polynomial of the form f_{r}(x) = x^{2} + r for r ∈ Q. When the number of such factors is bounded independent of n, we call f_{r}(x) eventually stable (over Q). Previous work of Hamblen, Jones, and Madhu [8] shows that f_{r} is eventually stable unless r has the form 1/c for some c ∈ Z\{0,1}, in which case existing methods break down. We study this family, and prove that several conditions on c of various flavors imply that all iterates of f_{1/c} are irreducible. We give an algorithm that checks the latter property for all c up to a large bound B in time polynomial in log B. We find all cvalues for which the third iterate of f_{1/c} has at least four irreducible factors, and all cvalues such that f_{1/c} is irreducible but its third iterate has at least three irreducible factors. This last result requires finding all rational points on a genus2 hyperelliptic curve for which the method of Chabauty and Coleman does not apply; we use the more recent variant known as elliptic Chabauty. Finally, we apply all these results to completely determine the number of irreducible factors of any iterate of f_{1/c}, for all c with absolute value at most 10^{9}.


Acknowledgements
We thank Jennifer Balakrishnan for conversations related to the proof of Theorem 1.6, and the anonymous referee for many helpful suggestions.


Author information
David DeMark:
School of Mathematics
University of Minnesota
206 Church Street SE, Minneapolis, MN 55455, USA
demar180@umn.edu
Wade Hindes:
Department of Mathematics
Texas State University
601 University Drive, San Marcos, TX 78666, USA
wmh33@txstate.edu
Rafe Jones:
Department of Mathematics and Statistics
Carleton College
1 North College St, Northfield, MN 55057, USA
rfjones@carleton.edu
Moses Misplon:
Department of Mathematics and Statistics
Carleton College
1 North College St, Northfield, MN 55057, USA
mzrmisplon@gmail.com
Michael Stoll:
Mathematisches Institut
Universität Bayreuth
95440 Bayreuth, Germany
Michael.Stoll@unibayreuth.de
Michael Stoneman:
Department of Mathematics and Statistics
Carleton College
1 North College St, Northfield, MN 55057, USA
mstoneman@google.com

