p-adic Mandelbrot集半径的界
Bounds on the radius of the p-adic Mandelbrot set
摘要：
设f是定义在非阿基米德域C_p上的d次多项式，正规化后f是幺半群且f（0）=0.如果f的所有临界点在f的迭代下都有有界轨道，则称f是后临界有界的。已知如果p大于或等于d且f是PCB，则f的所有临界点具有小于或等于1的p-adic绝对值。对于d/2和d之间的素数，我们给出了类似的结果。我们还探讨了2-adic数上的单参数三次多项式族，以说明当p小于d时，p-adic Mandelbrot集可能非常复杂，与简单且易于理解的p > d情况相反。
英文摘要：
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.
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