Collatz多项式：介绍其零点的界限
Collatz polynomials: an introduction with bounds on their zeros
摘要：
Collatz猜想（也被称为3x+1问题）提出以下算法在一定次数的迭代后总是产生数字1：给定一个自然数，乘以3，如果数字是奇数，则加1，将所得数字减半，然后重复。在本文中，对于Collatz猜想成立的每个$N$，我们定义$N^{th}$ Collatz多项式为具有常数项$N$和$k^{th}$项（对于$k > 1$）的首一多项式，该项是$N$在Collatz函数下的$k^{th}$。特别是，我们绑定这些多项式的根的模，证明定理时，他们有合理的整数根，并提出进一步的应用和研究途径。
英文摘要：
The Collatz Conjecture (also known as the 3x+1 Problem) proposes that the following algorithm will, after a certain number of iterations, always yield the number 1: given a natural number, multiply by three and add one if the number is odd, halve the resulting number, then repeat. In this article, for each $N$ for which the Collatz Conjecture holds we define the $N^{th}$ Collatz polynomial to be the monic polynomial with constant term $N$ and $k^{th}$ term (for $k > 1$) the $k^{th}$ iterate of $N$ under the Collatz function. In particular, we bound the moduli of the roots of these polynomials, prove theorems on when they have rational integer roots, and suggest further applications and avenues of research.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
