Nearly optimal algorithms for the decomposition of multivariate rational functions and the extended Lüroth Theorem

  • Authors:

  • Guillaume Chèze

  • Affiliations:

  • Institut de Mathématiques de Toulouse, Université Paul Sabatier Toulouse 3, MIP Bít 1R3, 31 062 Toulouse cedex 9, France

  • Venue:
  • Journal of Complexity
  • Year:
  • 2010

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Abstract

The extended Luroth Theorem says that if the transcendence degree of K(f"1,...,f"m)/K is 1 then there exists f@?K(X@?) such that K(f"1,...,f"m) is equal to K(f). In this paper we show how to compute f with a probabilistic algorithm. We also describe a probabilistic and a deterministic algorithm for the decomposition of multivariate rational functions. The probabilistic algorithms proposed in this paper are softly optimal when n is fixed and d tends to infinity. We also give an indecomposability test based on gcd computations and Newton's polytope. In the last section, we show that we get a polynomial time algorithm, with a minor modification in the exponential time decomposition algorithm proposed by Gutierez-Rubio-Sevilla in 2001.