Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

Simultaneously preperiodic points for families of polynomials in normal form


Authors: Dragos Ghioca, Liang-Chung Hsia and Khoa Dang Nguyen
Journal: Proc. Amer. Math. Soc. 146 (2018), 733-741
MSC (2010): Primary 37P05; Secondary 37P30, 37P45
DOI: https://doi.org/10.1090/proc/13762
Published electronically: September 7, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ d>m>1$ be integers, let $ c_1,\dots , c_{m+1}$ be distinct complex numbers, and let $ \mathbf {f}(z):=z^d+t_1z^{m-1}+t_2z^{m-2}+\cdots + t_{m-1}z+t_m$ be an $ m$-parameter family of polynomials. We prove that the set of $ m$-tuples of parameters $ (t_1,\dots , t_m)\in \mathbb{C}^m$ with the property that each $ c_i$ (for $ i=1,\dots , m+1$) is preperiodic under the action of the corresponding polynomial $ \mathbf {f}(z)$ is contained in finitely many hypersurfaces of the parameter space $ \mathbb{A}^m$.


References [Enhancements On Off] (What's this?)

  • [BD11] Matthew Baker and Laura DeMarco, Preperiodic points and unlikely intersections, Duke Math. J. 159 (2011), no. 1, 1-29. MR 2817647, https://doi.org/10.1215/00127094-1384773
  • [BD13] Matthew Baker and Laura De Marco, Special curves and postcritically finite polynomials, Forum Math. Pi 1 (2013), e3, 35. MR 3141413
  • [DeM16] Laura DeMarco, Bifurcations, intersections, and heights, Algebra Number Theory 10 (2016), no. 5, 1031-1056. MR 3531361, https://doi.org/10.2140/ant.2016.10.1031
  • [GH13] Dragos Ghioca and Liang-Chung Hsia, Torsion points in families of Drinfeld modules, Acta Arith. 161 (2013), no. 3, 219-240. MR 3145448, https://doi.org/10.4064/aa161-3-2
  • [GHT13] Dragos Ghioca, Liang-Chung Hsia, and Thomas J. Tucker, Preperiodic points for families of polynomials, Algebra Number Theory 7 (2013), no. 3, 701-732. MR 3095224, https://doi.org/10.2140/ant.2013.7.701
  • [GHT15] D. Ghioca, L.-C. Hsia, and T. J. Tucker, Preperiodic points for families of rational maps, Proc. Lond. Math. Soc. (3) 110 (2015), no. 2, 395-427. MR 3335283, https://doi.org/10.1112/plms/pdu051
  • [GHT16] Dragos Ghioca, Liang-Chung Hsia, and Thomas J. Tucker, Unlikely intersection for two-parameter families of polynomials, Int. Math. Res. Not. IMRN 24 (2016), 7589-7618. MR 3632092, https://doi.org/10.1093/imrn/rnw006
  • [GKN16] Dragos Ghioca, Holly Krieger, and Khoa Nguyen, A case of the dynamical André-Oort conjecture, Int. Math. Res. Not. IMRN 3 (2016), 738-758. MR 3493432, https://doi.org/10.1093/imrn/rnv143
  • [GKNY17] D. Ghioca, H. Krieger, K. D. Nguyen, and H. Ye, The dynamical André-Oort conjecture: unicritical polynomials, Duke Math. J. 166 (2017), no. 1, 1-25. MR 3592687, https://doi.org/10.1215/00127094-3673996
  • [GNT15] Dragos Ghioca, Khoa Nguyen, and Thomas J. Tucker, Portraits of preperiodic points for rational maps, Math. Proc. Cambridge Philos. Soc. 159 (2015), no. 1, 165-186. MR 3349337, https://doi.org/10.1017/S0305004115000274
  • [MZ10] D. Masser and U. Zannier, Torsion anomalous points and families of elliptic curves, Amer. J. Math. 132 (2010), no. 6, 1677-1691. MR 2766181
  • [MZ12] D. Masser and U. Zannier, Torsion points on families of squares of elliptic curves, Math. Ann. 352 (2012), no. 2, 453-484. MR 2874963, https://doi.org/10.1007/s00208-011-0645-4
  • [MZ14] D. Masser and U. Zannier, Torsion points on families of products of elliptic curves, Adv. Math. 259 (2014), 116-133. MR 3197654, https://doi.org/10.1016/j.aim.2014.03.016
  • [Ngu15] Khoa Nguyen, Some arithmetic dynamics of diagonally split polynomial maps, Int. Math. Res. Not. IMRN 5 (2015), 1159-1199. MR 3340351

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37P05, 37P30, 37P45

Retrieve articles in all journals with MSC (2010): 37P05, 37P30, 37P45


Additional Information

Dragos Ghioca
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
Email: dghioca@math.ubc.ca

Liang-Chung Hsia
Affiliation: Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan, ROC
Email: hsia@math.ntnu.edu.tw

Khoa Dang Nguyen
Affiliation: Department of Mathematics and Statistics, University of Calgary, Calgary, AB T2N 4T4, Canada
Email: dangkhoa.nguyen@ucalgary.ca

DOI: https://doi.org/10.1090/proc/13762
Received by editor(s): November 6, 2016
Received by editor(s) in revised form: April 5, 2017
Published electronically: September 7, 2017
Additional Notes: The research of the first author was partially supported by an NSERC Discovery grant.
The second author was supported by MOST grant 105-2918-I-003-006.
The third author was partially supported by a UBC-PIMS fellowship
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society