The Triviality of Commuting Polynomials in Many Variables
Abstract
We consider the equation P(Q(x1,…,xν))=Q(P(x1),…,P(xν)). Here P∈C[x] is a polynomial in one variable over the field of complex numbers and Q∈C[x1,…,xν],ν≥2, is a polynomial in two or more variables, but not in any one of variables alone, i.e., Q∉C[xi],i∈{1,…,ν}. We prove that if deg(P)>1, then all solutions to the functional equation above are characterized by monomials. More specifically, we prove that this equation is solvable if and only if there exists an affine bijection α on C, such that α°P°α−1=xn and α°Q°α−1=cx1p1…xνpν, where cn−1=1.
Fields of Study & Subjects
Algebraic Geometry and Number Theory
Functional Equations Stability Results
Mathematical Dynamics and Fractals
Publication details
- Journal / source
- American Mathematical Monthly
- Volume
- 133
- Issue
- 3
- Pages
- 220–229
- Publication date
- 23 February 2026
- Language
- EN