论文标题
关于与多项式递归相关的生长常数的超越性
On the transcendence of growth constants associated with polynomial recursions
论文作者
论文摘要
令$ p（x）：= = a_d x^d+\ cdots+a_0 \ in \ mathbb {q} [x] $，$ a_d> 0 $，成为度量$ d \ geq 2 $的多项式。令$（x_n）$为满足\ begin {qore*} x_ {n+1} = p（x_n）\ mbox {for all} \ quad n = 0,1,1,2 \ ldots，\ quad Quad \ mbox {和quad x_n \ quad as quad a as quad a as quad a as quad， n \ to \ infty。 \ end {equation*} 设置$α：= \ lim_ {n \ to \ infty} x^{d^{ - n}} _ n $。然后，在假设$ a_d^{1/（d-1）} \ in \ mathbb {q} $中，在dubickas \ cite \ cite {dubickas}的最新结果中，要么$α$是跨性别的，要么$α$可以是整数，或者是二式Pisot pisot单位，或者是$ late $ late $ late $ lationcat $ \ mathbb {q} $。 In this paper, we study the nature of such $α$ without the assumption that $a_d^{1/(d-1)}$ is in $\mathbb{Q}$, and we prove that either the number $α$ is transcendental, or $α^h$ is a Pisot number with $h$ being the order of the torsion subgroup of the Galois closure of the number field $ \ mathbb {q}（α，a_d^{ - \ frac {1} {d-1}}}）$。本文提出的其他结果研究了不等式的解决方案$ ||q_1α_1^n+\ cdots+q_kα_k^n+β|| <θ^n $ in $（n，q_1，q_1，\ ldots，q_k）在这里，$ k $代表一个数字字段，$θ\ in（0,1）$。符号$ || x || $表示$ x $及其最近的整数之间的距离，$ \ mathbb {z} $。
Let $P(x):=a_d x^d+\cdots+a_0\in\mathbb{Q}[x]$, $a_d>0$, be a polynomial of degree $d\geq 2$. Let $(x_n)$ be a sequence of integers satisfying \begin{equation*} x_{n+1}=P(x_n)\mbox{for all}\quad n=0,1,2\ldots,\quad\mbox{and} \quad x_n\to\infty\quad\mbox{as}\quad n\to\infty. \end{equation*} Set $α:=\lim_{n\to\infty} x^{d^{-n}}_n$. Then, under the assumption $a_d^{1/(d-1)}\in\mathbb{Q}$, in a recent result by Dubickas \cite{dubickas}, either $α$ is transcendental, or $α$ can be an integer, or a quadratic Pisot unit with $α^{-1}$ being its conjugate over $\mathbb{Q}$. In this paper, we study the nature of such $α$ without the assumption that $a_d^{1/(d-1)}$ is in $\mathbb{Q}$, and we prove that either the number $α$ is transcendental, or $α^h$ is a Pisot number with $h$ being the order of the torsion subgroup of the Galois closure of the number field $\mathbb{Q}(α, a_d^{-\frac{1}{d-1}})$. Other results presented in this paper investigate the solutions of the inequality $||q_1 α_1^n+\cdots+q_k α_k^n +β||<θ^n$ in $(n,q_1,\ldots,q_k)\in \mathbb{N}\times(K^\times)^k$, considering whether $β$ is rational or irrational. Here, $K$ represents a number field, and $θ\in (0,1)$. The notation $||x||$ denotes the distance between $x$ and its nearest integer in $\mathbb{Z}$.
