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Bounds on the radius of the p-adic Mandelbrot set

Jacqueline Anderson — 2013

Acta Arithmetica

Let f ( z ) = z d + a d - 1 z d - 1 + . . . + a 1 z p [ z ] 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...

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