ON G-POLYNOMIALS WITH INTEGER COEFFICIENTS

Abstract

Let $k\ge 2$ be an integer and let $P(X)$ be a monic polynomial of $\mathbb{Z}[X]$ with degree $k-1$. We say that $P(X)$ is a $G$-polynomial if for each $n\in \{1,2,\dots ,k\}$, $P(X)$ divides $P(X^n)$ in the ring $\mathbb{Z}[X]$ if and only if $\text{gcd }(n,k)=1$. We present several approaches on finding necessary and sufficient conditions so that $P(X)$ is a $G$-polynomial. Among other interesting results, we show that $A_k(X):=X^{k-1}+X^{k-2}+\cdots X+1$ is the unique $G$-polynomial of degree $k-1$ if and only if $k$ is a prime number.