Rational functions with linear relations
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- by Ariane M. Masuda and Michael E. Zieve
- Proc. Amer. Math. Soc. 136 (2008), 1403-1408
- DOI: https://doi.org/10.1090/S0002-9939-07-09246-5
- Published electronically: December 7, 2007
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Abstract:
We find all polynomials $f,g,h$ over a field $K$ such that $g$ and $h$ are linear and $f(g(x))=h(f(x))$. We also solve the same problem for rational functions $f,g,h$, in case the field $K$ is algebraically closed.References
- Gunnar af Hällström, Über halbvertauschbare Polynome, Acta Acad. Aboensis 21 (1957), no. 2, 20 (German). MR 84595
- Gunnar af Hällström, Über Halbvertauschbarkeit zwischen linearen und allgemeineren rationalen Funktionen, Math. Japon. 4 (1957), 107–112 (German). MR 98740
- I. N. Baker and A. Erëmenko, A problem on Julia sets, Ann. Acad. Sci. Fenn. Ser. A I Math. 12 (1987), no. 2, 229–236. MR 951972, DOI 10.5186/aasfm.1987.1205
- R. M. Beals and M. E. Zieve, Decompositions of polynomials, preprint, 2007.
- Günther Eigenthaler and Wilfried Nöbauer, Über die mit einem Polynom vertauschbaren linearen Polynome, Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 199 (1990), no. 4-7, 143–153 (German). MR 1119733
- A. È. Erëmenko, Some functional equations connected with the iteration of rational functions, Algebra i Analiz 1 (1989), no. 4, 102–116 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 4, 905–919. MR 1027462
- P. Fatou, Sur l’iteration analytique et les substitutions permutables, J. Math. Pures Appl. (9), 2, 1923, 343–384
- Gaston Julia, Mémoire sur la permutabilité des fractions rationnelles, Ann. Sci. École Norm. Sup. (3) 39 (1922), 131–215 (French). MR 1509242
- G. Levin and F. Przytycki, When do two rational functions have the same Julia set?, Proc. Amer. Math. Soc. 125 (1997), no. 7, 2179–2190. MR 1376996, DOI 10.1090/S0002-9939-97-03810-0
- Gary L. Mullen, Polynomials over finite fields which commute with linear permutations, Proc. Amer. Math. Soc. 84 (1982), no. 3, 315–317. MR 640221, DOI 10.1090/S0002-9939-1982-0640221-5
- Hong Goo Park, Polynomials satisfying $f(x+a)=f(x)+c$ over finite fields, Bull. Korean Math. Soc. 29 (1992), no. 2, 277–283. MR 1180621
- J. F. Ritt, On the iteration of rational functions, Trans. Amer. Math. Soc. 21 (1920), no. 3, 348–356. MR 1501149, DOI 10.1090/S0002-9947-1920-1501149-6
- J. F. Ritt, Prime and composite polynomials, Trans. Amer. Math. Soc. 23 (1922), no. 1, 51–66. MR 1501189, DOI 10.1090/S0002-9947-1922-1501189-9
- J. F. Ritt, Permutable rational functions, Trans. Amer. Math. Soc. 25 (1923), no. 3, 399–448. MR 1501252, DOI 10.1090/S0002-9947-1923-1501252-3
- Charles Wells, Polynomials over finite fields which commute with translations, Proc. Amer. Math. Soc. 46 (1974), 347–350. MR 347785, DOI 10.1090/S0002-9939-1974-0347785-5
Bibliographic Information
- Ariane M. Masuda
- Affiliation: School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, ON K1S 5B6, Canada
- Address at time of publication: Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON K1N 5B6, Canada
- MR Author ID: 791815
- Email: amasuda@uottawa.ca
- Michael E. Zieve
- Affiliation: Center for Communications Research, 805 Bunn Drive, Princeton, New Jersey 08540
- MR Author ID: 614926
- Email: zieve@math.rutgers.edu
- Received by editor(s): February 15, 2007
- Published electronically: December 7, 2007
- Additional Notes: The authors thank Bob Beals, Alan Beardon, Alex Erëmenko, and Patrick Ng for useful correspondence.
- Communicated by: Wen-Ching Winnie Li
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 1403-1408
- MSC (2000): Primary 39B12; Secondary 12E05, 30D05
- DOI: https://doi.org/10.1090/S0002-9939-07-09246-5
- MathSciNet review: 2367113