KöMaL 2026 A937

Let $P \in \mathbb{C}[x_1,\ldots,x_n]$ be an irreducible complex polynomial of degree at least 2. Assume that there exists an integer $M>1$ such that $P(x_1,\dots,x_n)\mid P(x_1^M,\dots,x_n^M)$. Prove that there exist a nonzero complex number $c$, a complex root of unity $\zeta$ and non-negative integers $a_1$, $\ldots$, $a_n$, $b_1$, $\dots$, $b_n$ such that $P(x_1,\dots,x_n)=c\left(x_1^{a_1}\cdots x_n^{a_n}-\zeta x_1^{b_1}\cdots x_n^{b_n}\right)$.