The number of rational iterated preimages of the origin under unicritical polynomial maps

2025/10/16 by Sano, Kaoru
#14G05 #14Q05 #37P55 #Algebraic Geometry (math.AG) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2510.14397

Abstract

We study rational iterated preimages of the origin under unicritical maps f_d,c(x)=x^d+c. Earlier works of Faber--Hutz--Stoll and Hutz--Hyde--Krause established finiteness and conditional bounds in the quadratic case. Building on this, we prove that for d=2 and c ∈ \mathbb Q∖\0,-1\ there are no rational fourth preimages of the origin, and for all d ≥ 3 there are no rational second preimages outside trivial cases. The proof relies on geometric analysis of preimage curves, the elliptic Chabauty method, and Diophantine reduction. As a result, we determine the number of rational iterated preimages of 0 under f_d,c for all d≥ 2.

Citations

• Perfect powers in elliptic divisibility sequences
• Descent on superelliptic curves
• Pre-images of quadratic dynamical systems
• On the Number of Rational Iterated Pre-images of the Origin Under\n Quadratic Dynamical Systems
• Modular Elliptic Curves and Fermat's Last Theorem

Related

• On the Number of Rational Iterated Pre-images of the Origin Under\n Quadratic Dynamical Systems
• Rational preimages in families of dynamical systems
• Preperiodic points with local rationality conditions in the quadratic unicritical family
• On the pre-image of a point under an isogeny
• Unicritical polynomial maps with rational multipliers