Archiv der Mathematik

, Volume 97, Issue 2, pp 115–124 | Cite as

A proof of a theorem by Fried and MacRae and applications to the composition of polynomial functions

  • Erhard AichingerEmail author
  • Stefan Steinerberger


Fried and MacRae (Math. Ann. 180, 220–226 (1969)) proved that for univariate polynomials \({p,q, f, g \in \mathbb{K}[t]}\) (\({\mathbb{K}}\) a field) with p, q nonconstant, p(x) − q(y) divides f(x) − g(y) in \({\mathbb{K}[x,y]}\) if and only if there is \({h \in \mathbb{K}[t]}\) such that f = h(p(t)) and g = h(q(t)). Schicho (Arch. Math. 65, 239–243 (1995)) proved this theorem from the viewpoint of category theory, thereby providing several generalizations to multivariate polynomials. In the present note, we give a new proof of one of these generalizations. The theorem by Fried and MacRae yields a way to prove the following fact for nonconstant functions f, g from \({\mathbb{C}}\) to \({\mathbb{C}}\) : if both the composition \({f \circ g}\) and g are polynomial functions, then f has to be a polynomial function as well. We give an algebraic proof of this fact and present a generalization to multivariate polynomials over algebraically closed fields. This provides a way to prove a generalization of a result by Carlitz (Acta Sci. Math. (Szeged) 24, 196–203 (1963)) that describes those univariate polynomials over finite fields that induce bijective functions on all of their finite extensions.

Mathematics Subject Classification (2010)

13B25 (12E05) 


Polynomial composition Multivariate Fried-MacRae-theorems Injective polynomials 


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  1. 1.
    Alonso C., Gutierrez J., Recio T.: A note on separated factors of separated polynomials. J. Pure Appl. Algebra 121, 217–222 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Barton D.R., Zippel R.: Polynomial decomposition algorithms. J. Symbolic Comput. 1, 159–168 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    F. Binder, Fast computations in the lattice of polynomial rational function fields, in: Y. N. Lakshman (ed.), Proceedings of the 1996 international symposium on symbolic and algebraic computation, ISSAC ’96, Zuerich, Switzerland, July 24–26, 1996. New York, NY: ACM Press. 43–48, 1996.Google Scholar
  4. 4.
    Carlitz L.: Permutations in finite fields. Acta Sci. Math. (Szeged) 24, 196–203 (1963)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Cox D., Little J., O’Shea D.: Ideals, varieties, and algorithms, Undergraduate Texts in Mathematics. Springer-Verlag, New York (1992)Google Scholar
  6. 6.
    Eisenbud D.: Commutative algebra. Springer-Verlag, New York (1995)zbMATHGoogle Scholar
  7. 7.
    Fried M.D., MacRae R.E.: On curves with separated variables. Math. Ann. 180, 220–226 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    R. Lidl and H. Niederreiter, Finite fields, Encyclopedia of Mathematics and its Applications 20, Addison-Wesley Publishing Company Advanced Book Program, Reading, MA, 1983.Google Scholar
  9. 9.
    Prager W., Schwaiger J.: Generalized polynomials in one and in several variables. Math. Pannon. 20, 189–208 (2009)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Rudin W.: Real and complex analysis. McGraw-Hill Book Co., New York (1966)zbMATHGoogle Scholar
  11. 11.
    Schicho J.: A note on a theorem of Fried and MacRae. Arch. Math. (Basel) 65, 239–243 (1995)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Institut für AlgebraJohannes Kepler Universität LinzLinzAustria
  2. 2.Mathematisches InstitutRheinische Friedrich-Wilhelms-Universität BonnBonnGermany

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