Benedetto’s trick and existence of rational preperiodic structures for quadratic polynomials

Faber, Xander

doi:10.1090/S0002-9939-2014-12328-8

Benedetto’s trick and existence of rational preperiodic structures for quadratic polynomials
HTML articles powered by AMS MathViewer

by Xander Faber PDF

Proc. Amer. Math. Soc. 143 (2015), 685-694 Request permission

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.

References

Robert L. Benedetto, Heights and preperiodic points of polynomials over function fields, Int. Math. Res. Not. 62 (2005), 3855–3866. MR 2202175, DOI 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 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 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 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 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 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 10.1155/S1073792894000127
Bjorn Poonen, The classification of rational preperiodic points of quadratic polynomials over $\textbf {Q}$: a refined conjecture, Math. Z. 228 (1998), no. 1, 11–29. MR 1617987, DOI 10.1007/PL00004405
Michael Stoll, Rational 6-cycles under iteration of quadratic polynomials, LMS J. Comput. Math. 11 (2008), 367–380. MR 2465796, DOI 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 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 10.2307/2975624