On polynomials which commute with a given polynomial

Author:
William M. Boyce

Journal:
Proc. Amer. Math. Soc. **33** (1972), 229-234

MSC:
Primary 12D99

MathSciNet review:
0291138

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Abstract: By extending a theorem of Jacobsthal, the following result is obtained: if *g* is a nonlinear polynomial, there is an integer such that for each there are either or zero distinct polynomials of degree *m* which commute with *g*. A formula is given for computing from the coefficients of *g*.

**[1]**E. A. Bertram,*Polynomials which commute with a Tchebycheff polynomial*, Amer. Math. Monthly**78**(1971), 650–653. MR**0288104****[2]**H. D. Block and H. P. Thielman,*Commutative polynomials*, Quart. J. Math., Oxford Ser. (2)**2**(1951), 241–243. MR**0045250****[3]**Ernst Jacobsthal,*Über vertauschbare Polynome*, Math. Z.**63**(1955), 243–276 (German). MR**0074373****[4]**J. F. Ritt,*Permutable rational functions*, Trans. Amer. Math. Soc.**25**(1923), no. 3, 399–448. MR**1501252**, 10.1090/S0002-9947-1923-1501252-3**[5]**H. T. Engstrom,*Polynomial substitutions*, Amer. J. Math.**63**(1941), 249–255. MR**0003599****[6]**Howard Levi,*Composite polynomials with coefficients in an arbitrary field of characteristic zero*, Amer. J. Math.**64**(1942), 389–400. MR**0006162**

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DOI:
https://doi.org/10.1090/S0002-9939-1972-0291138-3

Keywords:
Commuting functions,
commuting polynomials,
common fixed point,
Tchebycheff polynomials,
functional composition

Article copyright:
© Copyright 1972
American Mathematical Society