New York Journal of Mathematics
Volume 27 (2021), 363-378


Valentin Huguin

Simultaneously preperiodic integers for quadratic polynomials

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Published: February 25, 2021.
Keywords: Preperiodic points, quadratic polynomials, unlikely intersections.
Subject: Primary 37P05; Secondary 37F45, 37P35.

In this article, we study the set of parameters c ∈ C for which two given complex numbers a and b are simultaneously preperiodic for the quadratic polynomial fc(z) = z2 + c. Combining complex-analytic and arithmetic arguments, Baker and DeMarco showed that this set of parameters is infinite if and only if a2 = b2. Recently, Buff answered a question of theirs, proving that the set of parameters c ∈ C for which both 0 and 1 are preperiodic for fc is equal to {-2, -1, 0}. Following his approach, we complete the description of these sets when a and b are two given integers with |a| not equal to |b|.


The author would like to thank his Ph.D. advisors, Xavier Buff and Jasmin Raissy, for helpful discussions without which this paper would not exist and the anonymous referee for his comments.

Author information

Valentin Huguin:
Institut de Mathématiques de Toulouse, UMR 5219
Université de Toulouse
CNRS, UPS, F-31062 Toulouse Cedex 9, France