论文标题

几乎是Brieskorn-pham类型的半加权同质性多项式的Bernstein-Sato多项式

Bernstein-Sato polynomials of semi-weighted-homogeneous polynomials of nearly Brieskorn-Pham type

论文作者

Saito, Morihiko

论文摘要

令$ f $为一个半加权的多项式,在0时具有孤立的奇异性。令$α_{f,k} $是0。Malgrangeand varchenko的频谱数量是$ f $的频谱数字,而Malgrange and varchenko则有非阴性整数$ r_k $ r_k $ r_k $,以便$ r_k $ a $α__{多项式$ b_f(s)$除以$ s+1 $。但是,很难在加权均质多项式的$μ$ constant变形的参数空间上明确确定这些变化$ r_k $。假设后者几乎是Brieskorn-pham类型,我们可以获得一种非常简单的算法来确定这些偏移,可以通过使用单数(甚至C)而无需使用Gröbner碱基来实现。这意味着在两种可变情况下,Kato M. Kato和P. cassou-noguès的经典工作的完善,表明参数空间的分层可以通过使用参数重量的(部分)添加剂半群结构来控制。作为推论,例如,我们获得了所有可移动$ b_f(s)$的可移位根的足够条件。我们还可以制作示例,其中$ b_f(s)$的最小根与其他词相距甚远,以及具有$ b_f(s)$ noncencectect的词根的半均匀多项式示例。

Let $f$ be a semi-weighted-homogeneous polynomial having an isolated singularity at 0. Let $α_{f,k}$ be the spectral numbers of $f$ at 0. By Malgrange and Varchenko there are non-negative integers $r_k$ such that the $α_{f,k}-r_k$ are the roots up to sign of the local Bernstein-Sato polynomial $b_f(s)$ divided by $s+1$. However, it is quite difficult to determine these shifts $r_k$ explicitly on the parameter space of $μ$-constant deformation of a weighted homogeneous polynomial. Assuming the latter is nearly Brieskorn-Pham type, we can obtain a very simple algorithm to determine these shifts, which can be realized by using Singular (or even C) without employing Gröbner bases. This implies a refinement of classical work of M. Kato and P. Cassou-Noguès in two variable cases, showing that the stratification of the parameter space can be controlled by using the (partial) additive semigroup structure of the weights of parameters. As a corollary we get for instance a sufficient condition for all the shiftable roots of $b_f(s)$ to be shifted. We can also produce examples where the minimal root of $b_f(s)$ is quite distant from the others as well as examples of semi-homogeneous polynomials with roots of $b_f(s)$ nonconsecutive.

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