Bibliographic Details
| Title: |
The Triviality of Commuting Polynomials in Many Variables. |
| Authors: |
Mingajev, Arseny (AUTHOR) amingaje@trinity.edu |
| Source: |
American Mathematical Monthly. Mar2026, Vol. 133 Issue 3, p220-229. 10p. |
| Subject Terms: |
Polynomials, Functions of several complex variables, Functional equations, Affine transformations |
| Abstract: |
We consider the equation P (Q (x 1 , ... , x ν)) = Q (P (x 1) , ... , P (x ν)) . Here P ∈ C [ x ] is a polynomial in one variable over the field of complex numbers and Q ∈ C [ x 1 , ... , x ν ] , ν ≥ 2 , is a polynomial in two or more variables, but not in any one of variables alone, i.e., Q ∉ C [ x i ] , 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 = x n and α ° Q ° α − 1 = c x 1 p 1 ... x ν p ν , where c n − 1 = 1 . [ABSTRACT FROM AUTHOR] |
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| Database: |
Education Research Complete |
