Benedetto’s trick and existence of rational preperiodic structures for quadratic polynomials
Author:
Xander Faber
Journal:
Proc. Amer. Math. Soc. 143 (2015), 685-694
MSC (2010):
Primary 37P35; Secondary 37P05, 37P40
DOI:
https://doi.org/10.1090/S0002-9939-2014-12328-8
Published electronically:
September 19, 2014
MathSciNet review:
3283655
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We refine a result of R. Benedetto in $p$-adic analysis in order to exhibit infinitely many quadratic polynomials over $\mathbb {Q}$ with a specified graph of rational preperiodic points.
- Robert L. Benedetto, Heights and preperiodic points of polynomials over function fields, Int. Math. Res. Not. 62 (2005), 3855–3866. MR 2202175, DOI https://doi.org/10.1155/IMRN.2005.3855
- Robert L. Benedetto, Preperiodic points of polynomials over global fields, J. Reine Angew. Math. 608 (2007), 123–153. MR 2339471, DOI https://doi.org/10.1515/CRELLE.2007.055
- Gregory S. Call and Susan W. Goldstine, Canonical heights on projective space, J. Number Theory 63 (1997), no. 2, 211–243. MR 1443758, DOI https://doi.org/10.1006/jnth.1997.2099
- John R. Doyle, Xander Faber, and David Krumm, Preperiodic points for quadratic polynomials over quadratic fields, New York J. Math. 20 (2014), 507–605.
- E. V. Flynn, Bjorn Poonen, and Edward F. Schaefer, Cycles of quadratic polynomials and rational points on a genus-$2$ curve, Duke Math. J. 90 (1997), no. 3, 435–463. MR 1480542, DOI https://doi.org/10.1215/S0012-7094-97-09011-6
- Benjamin Hutz and Patrick Ingram, On Poonen’s conjecture concerning rational preperiodic points of quadratic maps, Rocky Mountain J. Math. 43 (2013), no. 1, 193–204. MR 3065461, DOI https://doi.org/10.1216/RMJ-2013-43-1-193
- Patrick Morton, On certain algebraic curves related to polynomial maps, Compositio Math. 103 (1996), no. 3, 319–350. MR 1414593
- Patrick Morton, Arithmetic properties of periodic points of quadratic maps. II, Acta Arith. 87 (1998), no. 2, 89–102. MR 1665198, DOI https://doi.org/10.4064/aa-87-2-89-102
- Patrick Morton and Joseph H. Silverman, Rational periodic points of rational functions, Internat. Math. Res. Notices 2 (1994), 97–110. MR 1264933, DOI https://doi.org/10.1155/S1073792894000127
- Bjorn Poonen, The classification of rational preperiodic points of quadratic polynomials over ${\bf Q}$: a refined conjecture, Math. Z. 228 (1998), no. 1, 11–29. MR 1617987, DOI https://doi.org/10.1007/PL00004405
- Michael Stoll, Rational 6-cycles under iteration of quadratic polynomials, LMS J. Comput. Math. 11 (2008), 367–380. MR 2465796, DOI https://doi.org/10.1112/S1461157000000644
- J. G. van der Corput, Über Summen von Primzahlen und Primzahlquadraten, Math. Ann. 116 (1939), no. 1, 1–50 (German). MR 1513216, DOI https://doi.org/10.1007/BF01597346
- Ralph Walde and Paula Russo, Rational periodic points of the quadratic function $Q_c(x)=x^2+c$, Amer. Math. Monthly 101 (1994), no. 4, 318–331. MR 1270956, DOI https://doi.org/10.2307/2975624
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37P35, 37P05, 37P40
Retrieve articles in all journals with MSC (2010): 37P35, 37P05, 37P40
Additional Information
Xander Faber
Affiliation:
Department of Mathematics, University of Hawaii at Manoa, Honolulu, Hawaii 96822
Received by editor(s):
May 1, 2013
Published electronically:
September 19, 2014
Communicated by:
Matthew A. Papanikolas
Article copyright:
© Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.