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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].

##### MSC:
 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.