ﻻ يوجد ملخص باللغة العربية
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.
A rational function of degree at least two with coefficients in an algebraically closed field is post-critically finite (PCF) if all of its critical points have finite forward orbit under iteration. We show that the collection of PCF rational functio
We study the radius of convergence of a differential equation on a smooth Berkovich curve over a non-archimedean complete valued field of characteristic 0. Several properties of this function are known: F. Baldassarri proved that it is continuous and
We explicitly compute the adjoint L-function of those L-packets of representations of the group GSp(4) over a p-adic field of characteristic zero that contain non-supercuspidal representations. As an application we verify a conjecture of Gross-Prasad
Let $n, k$ and $a$ be positive integers. The Stirling numbers of the first kind, denoted by $s(n,k)$, count the number of permutations of $n$ elements with $k$ disjoint cycles. Let $p$ be a prime. In recent years, Lengyel, Komatsu and Young, Leonetti
A variant of Brauers induction method is developed. It is shown that quartic p-adic forms with at least 9127 variables have non-trivial zeros, for every p. For odd p considerably fewer variables are needed. There are also subsidiary new results concerning quintic forms, and systems of forms.