Density of Unicritical Polynomials over Local Fields with Rational Periodic Points

2026/07/17 by Anna M. Dietrich
Mathematics · #math.NT

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Abstract

We prove an asymptotic formula for the density of parameters c∈ ℤ_p for which the polynomial x^d+c admits a ℚ_p-rational n-cycle. In the cases where d=2 and p≠ 2,3, we compute these densities explicitly for n=1,2, and 3.

Related

• On rational periodic points of x^d+c
• Uniform bounds on periodic points of polynomials with good reduction
• Local densities of quadratic polynomials over non-dyadic fields and unramified dyadic fields
• Rational Periodic Points of x^d+c and Fermat-Catalan Equations
• The density of polynomials of degree n over \ℤ_p having\n exactly r roots in \ℚ_p