On the decomposition of rational functions

  • Authors:

  • Mohamed Ayad;Peter Fleischmann

  • Affiliations:

  • Universite du Littoral, Mathematics, 46 Rue Ferdinand Buisson, 62228 Calais, France;Institute of Mathematics and Statistics, University of Kent, Canterbury CT2 7NF, UK

  • Venue:
  • Journal of Symbolic Computation
  • Year:
  • 2008

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Abstract

Let f:=p/q@?K(x) be a rational function in one variable. By Luroth's theorem, the collection of intermediate fields K(f)@?L@?K(x) is in bijection with inequivalent proper decompositions f=g@?h, with g,h@?K(x) of degrees =2. In [Alonso, Cesar, Gutierrez, Jaime, Recio, Tomas, 1995. A rational function decomposition algorithm by near-separated polynomials. J. Symbolic Comput. 19, 527-544] an algorithm is presented to calculate such a function decomposition. In this paper we describe a simplification of this algorithm, avoiding expensive solutions of linear equations. A MAGMA implementation shows the efficiency of our method. We also prove some indecomposability criteria for rational functions, which were motivated by computational experiments.