On the polynomial equation $P(Q(x_1,\ldots,x_m))=Q(P(x_1),\ldots,P(x_m))$

Abstract

We consider the equation $P(Q(x_1,\ldots,x_\nu))=Q(P(x_1),\ldots,P(x_\nu))$ in polynomials over the field of complex numbers and prove that if ${\rm deg}(P)>1$, then it is only solvable in polynomials that are affinely conjugate to monomials.