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New York Journal of Mathematics
Volume 31 (2025), 633-649

  

Xiao Zhong

Preimages question for surjective endomorphisms on (P1)n

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Published: March 31, 2025.
Keywords: Preimages Question, invariant subvarieties, arithmetic dynamics, rational points on varieties.
Subject [2020]: 37P55, 14G05.

Abstract
Let K be a number field and let f : (P1)n --> (P1)n be a dominant endomorphism defined over K. We show that if V is an f-invariant subvariety (that is, f(V)=V) then there is a positive integer s0 such that (f-s-1(V) \ f-s(V))(K) is empty for every integer s greater than or equal to s0, answering the Preimages Question of Matsuzawa, Meng, Shibata, and Zhang in the case of (P1)n.

Acknowledgements

The author was supported in part by NSERC grant RGPIN-2022-02951.


Author information

Xiao Zhong
University of Waterloo
Department of Pure Mathematics
Waterloo, Ontario N2L 3G1, Canada

x48zhong@uwaterloo.ca