关于多元多项式的可微函数分解和齐次函数分解
On Functional Decomposition of Multivariate Polynomials with Differentiation and Homogenization
摘要：
本文对Ye，Dai，Lam（1999）和Faug $μ$ ere，Perret（2006，2008，2009）提出的基于微分和均匀化的多元多项式函数分解算法进行了理论分析.我们表明，一组随机可分解的四次齐次多项式的度适当的功能分解，可以计算使用该算法具有很高的概率。这解决了Ye，Dai，and Lam（1999）提出的一个猜想。我们还提出了一个猜想，这样一组多项式的分解可以计算出它的均匀化与高概率。最后，我们证明了一组多项式的右分解因子可以从它的右分解因子空间中计算出来。结合这些结果，我们证明了在猜想正确的条件下，当基域特征为零时，该算法能以概率1计算一组随机可分解的四次多项式的度真分解，而当基域是一个具有足够大个数的有限域时，该算法能以概率接近1计算一组随机可分解的四次多项式的度真分解.
英文摘要：
In this paper, we give a theoretical analysis for the algorithms to compute functional decomposition for multivariate polynomials based on differentiation and homogenization which are proposed by Ye, Dai, Lam (1999) and Faug$μ$ere, Perret (2006, 2008, 2009). We show that a degree proper functional decomposition for a set of randomly decomposable quartic homogenous polynomials can be computed using the algorithm with high probability. This solves a conjecture proposed by Ye, Dai, and Lam (1999). We also propose a conjecture such that the decomposition for a set of polynomials can be computed from that of its homogenization with high probability. Finally, we prove that the right decomposition factors for a set of polynomials can be computed from its right decomposition factor space. Combining these results together, we prove that the algorithm can compute a degree proper decomposition for a set of randomly decomposable quartic polynomials with probability one when the base field is of characteristic zero, and with probability close to one when the base field is a finite field with sufficiently large number under the assumption that the conjeture is correct.
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