Uniform bounds on periodic points of polynomials with good reduction

2025/10/30 by Isaac Rajagopal, Robin Zhang, Rajagopal, Isaac +1
#math.NT #math.DS

paper · pdf · doi:10.1007/s00605-026-02196-0

Abstract

We establish effective bounds on the number of periodic points of degree-d polynomials ϕ defined over p-adic fields and number fields, under a mild reduction hypothesis that is satisfied by all unicritical polynomials X^d + c with c integral at some prime dividing d. As a consequence, we verify the uniform boundedness conjecture for this class of polynomials over number fields K, giving the explicit uniform bound #Per_K(ϕ) ≤ d^[K:ℚ].

Citations

• Gonality of dynatomic curves and strong uniform boundedness of preperiodic points
• Bounds for preperiodic points for maps with good reduction
• Finite orbits for rational functions

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