arXiv 2408.069492024-08-13math.NT

$p$-进商集：具有可约特征多项式的线性递推序列

$p$-Adic quotient sets: linear recurrence sequences with reducible characteristic polynomials

Deepa Antony, Rupam Barman

备注：: To appear at Canadian Mathematical Bulletin. arXiv admin note: text overlap with arXiv:2207.07084

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AI总结：

本文针对特征多项式在有理数域上可约的一类线性递推序列，解答了Sanna提出的判断其$p$-进商集在$p$-进数域中是否稠密的公开问题，补充了此前不可约情形下的相关研究结论。

摘要：

设$(x_n)_{n\geq0}$是阶数$k\geq2$的线性递推序列，对所有整数$n\geq k$满足$x_n=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}$，其中$a_1,\dots,a_k,x_0,\dots,x_{k-1}\in\mathbb{Z}$且$a_k\neq0$。2017年，Sanna提出一个公开问题：对哪些素数$p$，$(x_n)_{n\geq0}$的商集在$\mathbb{Q}_p$中稠密。在近期一篇论文中，我们证明若递推序列的特征多项式有根$\pm\alpha$（$\alpha$为Pisot数），且素数$p$使得该特征多项式在$\mathbb{Q}_p$中不可约，则$(x_n)_{n\geq0}$的商集在$\mathbb{Q}_p$中稠密。本文针对特征多项式在$\mathbb{Q}$上可约的一类线性递推序列，解答了上述问题。

英文摘要：

Let $(x_n)_{n\geq0}$ be a linear recurrence sequence of order $k\geq2$ satisfying $$x_n=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}$$ for all integers $n\geq k$, where $a_1,\dots,a_k,x_0,\dots, x_{k-1}\in \mathbb{Z},$ with $a_k\neq0$. In 2017, Sanna posed an open question to classify primes $p$ for which the quotient set of $(x_n)_{n\geq0}$ is dense in $\mathbb{Q}_p$. In a recent paper, we showed that if the characteristic polynomial of the recurrence sequence has a root $\pm α$, where $α$ is a Pisot number and if $p$ is a prime such that the characteristic polynomial of the recurrence sequence is irreducible in $\mathbb{Q}_p$, then the quotient set of $(x_n)_{n\geq 0}$ is dense in $\mathbb{Q}_p$. In this article, we answer the problem for certain linear recurrence sequences whose characteristic polynomials are reducible over $\mathbb{Q}$.

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