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The Triviality of Commuting Polynomials in Many Variables.

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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]
Copyright of American Mathematical Monthly is the property of Taylor & Francis Ltd and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Education Research Complete
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DbLabel: Education Research Complete
An: 192252737
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PubType: Academic Journal
PubTypeId: academicJournal
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  Data: The Triviality of Commuting Polynomials in Many Variables.
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  Data: 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]
– Name: AbstractSuppliedCopyright
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  Group: Ab
  Data: <i>Copyright of American Mathematical Monthly is the property of Taylor & Francis Ltd and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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      – Type: doi
        Value: 10.1080/00029890.2025.2594382
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      – Code: eng
        Text: English
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        PageCount: 10
        StartPage: 220
    Subjects:
      – SubjectFull: Polynomials
        Type: general
      – SubjectFull: Functions of several complex variables
        Type: general
      – SubjectFull: Functional equations
        Type: general
      – SubjectFull: Affine transformations
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      – TitleFull: The Triviality of Commuting Polynomials in Many Variables.
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              Text: Mar2026
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              Y: 2026
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