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

TL;DR

This paper investigates polynomial equations of a specific form over complex numbers and proves that solutions with degree greater than one are limited to polynomials conjugate to monomials, revealing a structural rigidity.

Contribution

It characterizes solutions to the polynomial functional equation, showing they must be affinely conjugate to monomials when the degree exceeds one.

Findings

Solutions are only affinely conjugate to monomials for degree > 1

The equation admits no other polynomial solutions beyond conjugates of monomials

Structural rigidity of polynomial solutions in complex field

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.