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Permutable polynomials and related topics. (English) Zbl 0897.12001
Contributions to general algebra 9. Proceedings of the conference, Linz, Austria, June 1994. Wien: Hölder-Pichler-Tempsky, 163-182 (1995).
The set of all polynomials with complex coefficients forms a semigroup \(S\) under composition. The authors give a classification (up to conjugation) of commutative subsemigroups of \(S\) not containing a linear polynomial distinct from \(f(X)=X\) and use this classification to describe all commutative subsemigroups of \(S\). They list three infinite sets of such semigroups as well as five explicitly given commutative semigroups (one consisting of all Chebyshev polynomials) and prove that every commutative subsemigroup of \(S\) is conjugated to a subsemigroup of one of the listed. This result is used to give a list of all maximal commutative subsemigroups of \(S\).
For the entire collection see [Zbl 0879.00039].

12D05 Polynomials in real and complex fields: factorization
11T06 Polynomials over finite fields
20M14 Commutative semigroups
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
20M20 Semigroups of transformations, relations, partitions, etc.