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

Abstract

The extended Lüroth 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.