论文标题
在关键轨道上建造具有规定可划分条件的多项式
Construction of Polynomials with prescribed divisibility conditions on the critical orbit
论文作者
论文摘要
我们认为多项式$ f_ {d，c}（x）= x^d+c $在理性字段$ \ q $上。固定整数$ d，n \ ge 2 $，我们表明，对于某些$ c \ in \ q $的$ f_ {d，c}^n（0）$的原始prime除数的密度是正。实际上，在某些假设下，当$ d = 2 $时，我们明确计算后者密度。此外，修复$ d，n \ ge 2 $，我们表明，对于给定的整数$ n> 0 $，\ q $中有$ c \ c \ in \ q $，这样$ \ f^n（0）$至少具有$ n $ primitive Prime除数，每个分隔符至任何预定的功率。这表明，在$ \ f（x）$的关键轨道中的原始素数分隔数的数量上没有统一的上限，这不取决于$ c $。开发的结果提供了一种构建$ \ f（x）$的多项式的方法，该$ \ f（x）$为$ m $ th的迭代的分裂字段（$ m \ ge1 $）具有最大可能的订单。在这项工作的过程中，我们在本地字段上对批判后有限的多项式$ \ f（x）$提供明确的新结果。
We consider the family of polynomials $f_{d,c}(x)=x^d+c$ over the rational field $\Q$. Fixing integers $d, n\ge 2$, we show that the density of primes that can appear as primitive prime divisors of $f_{d,c}^n(0)$ for some $c\in\Q$ is positive. In fact, under certain assumptions, we explicitly calculate the latter density when $d=2$. Furthermore, fixing $d,n\ge 2$, we show that for a given integer $N>0$, there is $c\in \Q$ such that $\f^n(0)$ has at least $N$ primitive prime divisors each of which is appearing up to any predetermined power. This shows that there is no uniform upper bound on the number of primitive prime divisors in the critical orbit of $\f(x)$ that does not depend on $c$. The developed results provide a method to construct polynomials of the form $\f(x)$ for which the splitting field of the $m$-th iteration, $m\ge1$, has Galois group of maximal possible order. During the course of this work, we give explicit new results on post-critically finite polynomials $\f(x)$ over local fields.
