多元有理函数分解的近优算法及推广的Lüroth定理
Nearly Optimal Algorithms for the Decomposition of Multivariate Rational Functions and the Extended Lüroth's Theorem
摘要：
扩展的Lüroth定理说，如果$\KK(\mathsf{f}_1,\dots,\mathsf{f}_m)/\KK$的超越度为1，则存在$f \in \KK(\underline{X})$使得$\KK(\mathsf{f}_1,\dots,\mathsf{f}_m)$等于$\KK(f)$。在本文中，我们将展示如何计算$f$的概率算法。我们还描述了一个概率和多元有理函数的分解的确定性算法。当$n$固定且$d$趋于无穷大时，本文提出的概率算法是软最优的.我们还给出了一个不可分解性测试的基础上的gcd计算和牛顿的多面体。在最后一节中，我们证明了我们得到了一个多项式时间算法，并对Gutierez-Rubio-Sevilla在2001年提出的指数时间分解算法进行了微小的修改。
英文摘要：
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.
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