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 $Int_n(\mathbb{Z})$ the set of the integer-valued polynomials with degree $\leq n$, we show that the smallest positive integer $c_n$ satisfying the property: $\forall P \in Int_n(\mathbb{Z}), c_n P' \in Int_n(\mathbb{Z})$ is $c_n = \lcm(1 , 2 , \dots , n)$. As an application, we deduce an easy proof of the well-known inequality $\lcm(1 , 2 , \dots , n) \geq 2^{n - 1}$ ($\forall n \geq 1$). In the second part of the paper, we generalize our result for the derivative of a given order. 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: $\forall P \in Int_n(\mathbb{Z})$, $\forall k \in \mathbb{N}$: $\lambda_n P^{(k)} \in Int_n(\mathbb{Z})$.