On the derivatives of the integer-valued polynomials

Abstract

In this paper, we study the derivatives of an integer-valued polynomial of a given degree. Denoting by $E_n$ the set of the integer-valued polynomials with degree $\le n$, we show that the smallest positive integer $c_n$ satisfying the property: $\forall P\in E_n,c_n{P}^{\prime }\in E_n$ is $c_n=\mathrm{lcm}(1,2,\dots ,n)$. As an application, we deduce an easy proof of the well-known inequality $\mathrm{lcm}(1,2,\dots ,n)\ge 2^{n-1}$ ($\forall n\ge 1$). In the second part of the paper, we generalize our result for the derivative of a given order $k$ and then we give two divisibility properties for the obtained numbers $c_{n,k}$ (generalizing the $c_n$'s). Leaning on this study, we conclude the paper by determining, for a given natural number $n$, the smallest positive integer ${\lambda }_n$ satisfying the property: $\forall P\in E_n$, $\forall k\in \mathbb{N}$: ${\lambda }_n{P}^{(k)}\in E_n$. In particular, we show that: ${\lambda }_n={\prod }_{p\text{ prime}}{p}^{\lfloor \frac{n}{p}\rfloor }$ ($\forall n\in \mathbb{N}$).

arXiv comment17 pages