Bounds on the radius of the p-adic Mandelbrot set

TL;DR

This paper investigates bounds on the p-adic radius of the Mandelbrot set for polynomials over nonarchimedean fields, revealing complex structures for primes less than the degree, contrasting with the simpler p ≥ d case.

Contribution

It extends known bounds for post-critically bounded polynomials to primes between d/2 and d and explores the intricate structure of the p-adic Mandelbrot set for smaller primes.

Findings

Critical points have bounded p-adic absolute value when p ≥ d.

The p-adic Mandelbrot set exhibits complex behavior for p < d.

Illustrative example with cubic polynomials over 2-adic numbers.

Abstract

Let f be a degree d polynomial defined over the nonarchimedean field C_p, normalized so f is monic and f(0)=0. We say f is post-critically bounded, or PCB, if all of its critical points have bounded orbit under iteration of f. It is known that if p is greater than or equal to d and f is PCB, then all critical points of f have p-adic absolute value less than or equal to 1. We give a similar result for primes between d/2 and d. We also explore a one-parameter family of cubic polynomials over the 2-adic numbers to illustrate that the p-adic Mandelbrot set can be quite complicated when p is less than d, in contrast with the simple and well-understood p > d case.