论文标题
二项式多项式的绝对不可约性
Absolute irreducibility of the binomial polynomials
论文作者
论文摘要
在本文中，我们研究了二项式多项式的分解行为$ \ binom {x} {n} = \ frac {x（x-1）\ cdots（x-n+1）} {n！} $及其在整数valued polynomials $ \ operatoNornOrnOrnOrnOrnOrnOrneorn $ \ z} $ z} $ {int}（in int}（Intbb）中的势力。尽管众所周知，二项式多项式是$ \ operatatorName {int}（\ mathbb {z}）$中的不可约性元素，但其权力的分解行为尚未完全理解。我们填补了这一空白，并表明二项式多项式在$ \ operatotorname {int}（\ m athbb {z}）$中，也就是说，$ \ binom {x} {x} {n} {n}^m $ rable因$ \ operatornameame {int}的不可置述的元素而言是$ \ operatornameame {int}的不可修复的元素{ \ mathbb {n} $。通过根据线性代数和数字理论来重新解决该问题，我们表明可以将问题降低到确定$ n $的评估矩阵的排名，所谓的估计矩阵。计算此排名的主要成分是以下数字理论结果，我们还提供了一个证明：如果$ n> 10 $和$ n $，$ n-1 $，\ ldots，$ n-（k-1）$是复合整数，那么存在一个质量$ p> 2k $ p> 2k $，将这些整数划分为其中一个。
In this paper we investigate the factorization behaviour of the binomial polynomials $\binom{x}{n} = \frac{x(x-1)\cdots (x-n+1)}{n!}$ and their powers in the ring of integer-valued polynomials $\operatorname{Int}(\mathbb{Z})$. While it is well-known that the binomial polynomials are irreducible elements in $\operatorname{Int}(\mathbb{Z})$, the factorization behaviour of their powers has not yet been fully understood. We fill this gap and show that the binomial polynomials are absolutely irreducible in $\operatorname{Int}(\mathbb{Z})$, that is, $\binom{x}{n}^m$ factors uniquely into irreducible elements in $\operatorname{Int}(\mathbb{Z})$ for all $m\in \mathbb{N}$. By reformulating the problem in terms of linear algebra and number theory, we show that the question can be reduced to determining the rank of, what we call, the valuation matrix of $n$. A main ingredient in computing this rank is the following number-theoretical result for which we also provide a proof: If $n>10$ and $n$, $n-1$, \ldots, $n-(k-1)$ are composite integers, then there exists a prime number $p > 2k$ that divides one of these integers.
