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New York Journal of Mathematics
Volume 26 (2020), 526-561

  

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 n-th iterate of a polynomial of the form fr(x) = x2 + r for r ∈ Q. When the number of such factors is bounded independent of n, we call fr(x) eventually stable (over Q). Previous work of Hamblen, Jones, and Madhu [8] shows that fr 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 f1/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 c-values for which the third iterate of f1/c has at least four irreducible factors, and all c-values such that f1/c is irreducible but its third iterate has at least three irreducible factors. This last result requires finding all rational points on a genus-2 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 f1/c, for all c with absolute value at most 109.

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@uni-bayreuth.de

Michael Stoneman:
Department of Mathematics and Statistics
Carleton College
1 North College St, Northfield, MN 55057, USA

mstoneman@google.com