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

Abstract

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