Let be a degree d polynomial. 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 ≥ 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 1/2d ≤ p < d. We also explore a one-parameter family of cubic polynomials over ℚ₂ to illustrate that the p-adic Mandelbrot set can be quite complicated when p < d, in contrast with the simple and well-understood p ≥ d case.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa158-3-5,
author = {Jacqueline Anderson},
title = {Bounds on the radius of the p-adic Mandelbrot set},
journal = {Acta Arithmetica},
volume = {161},
year = {2013},
pages = {253-269},
zbl = {1300.11123},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa158-3-5}
}
Jacqueline Anderson. Bounds on the radius of the p-adic Mandelbrot set. Acta Arithmetica, Tome 161 (2013) pp. 253-269. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa158-3-5/