Nearly Optimal Algorithms for the Decomposition of Multivariate Rational Functions and the Extended Lüroth's Theorem

Guillaume Chèze (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

The extended Lüroth's Theorem says that if the transcendence degree of $\KK(\mathsf{f}_1,\dots,\mathsf{f}_m)/\KK$ is 1 then there exists $f \in \KK(\underline{X})$ such that $\KK(\mathsf{f}_1,\dots,\mathsf{f}_m)$ is equal to $\KK(f)$. 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.

Domaines: Calcul formel [cs.SC]; Complexité [cs.CC]

Guillaume Chèze. Nearly Optimal Algorithms for the Decomposition of Multivariate Rational Functions and the Extended Lüroth's Theorem. Journal of Complexity, 2010, 26 (4), pp.344-363. ⟨10.1016/j.jco.2010.05.001⟩. ⟨hal-00477526⟩