 

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. 


Abstract
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 f_{c}(z) = z^{2} + c. Combining complexanalytic and arithmetic arguments, Baker and DeMarco showed that this set of parameters is infinite if and only if
a^{2} = b^{2}. Recently, Buff answered a question of theirs, proving that the set of parameters
c ∈ C for which both 0 and 1 are preperiodic for f_{c} 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.


Acknowledgements
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, F31062 Toulouse Cedex 9, France
valentin.huguin@math.univtoulouse.fr

