Davenport-Mahler-Mittette界的推广--加权情形
Generalizing The Davenport-Mahler-Mignotte Bound -- The Weighted Case
摘要：
根分离界作为复杂性度量在理解计算代数中各种算法的行为中起着重要作用，例如，根隔离算法。一个经典的结果在单变量设置是达文波特-马勒-马格利特（DMM）界。一种方法是考虑一个有向无环图$(V,E)$在一个度$d$多项式$f(z) \in \mathbb{C}[z]$的根的子集上，其中边从一个较小绝对值的根指向一个较大绝对值的根，并且所有顶点的入度至多为1。则DMM界限是以下乘积的摊销下限：$\prod_{(α,β) \in E}|α-β|$。然而，下界涉及多项式$f$的判别式，并且如果多项式不是无平方的，则下界变得平凡。Eigenwillig（2008）通过使用合适的次判别式而不是判别式解决了这一问题。Escorcielo-Perrucci，2016，通过使用有限差分理论进一步删除了图上的入度约束。Emiris等人，2019年，已经推广了他们的结果，以处理乘积中项$|α-β|$的指数至多是任一根的重数的情况。在本文中，我们推广了这些结果，允许任意的正整数权重的边缘的图，即，对于权函数$w: E \rightarrow \mathbb{Z}_{>0}$，我们导出了$\prod_{(α,β) \in E}|α-β|^{w(α,β)}$的摊销下界。这种乘积出现在一些最近的根聚类算法的复杂度估计中（例如，Becker等人，2016），其中权重通常是根的重数的函数。由于其摊销的性质，我们的界限可以说是优于通过操纵现有的结果，以适应权重所获得的界限。
英文摘要：
Root separation bounds play an important role as a complexity measure in understanding the behaviour of various algorithms in computational algebra, e.g., root isolation algorithms. A classic result in the univariate setting is the Davenport-Mahler-Mignotte (DMM) bound. One way to state the bound is to consider a directed acyclic graph $(V,E)$ on a subset of roots of a degree $d$ polynomial $f(z) \in \mathbb{C}[z]$, where the edges point from a root of smaller absolute value to one of larger absolute, and the in-degrees of all vertices is at most one. Then the DMM bound is an amortized lower bound on the following product: $\prod_{(α,β) \in E}|α-β|$. However, the lower bound involves the discriminant of the polynomial $f$, and becomes trivial if the polynomial is not square-free. This was resolved by Eigenwillig, (2008), by using a suitable subdiscriminant instead of the discriminant. Escorcielo-Perrucci, 2016, further dropped the in-degree constraint on the graph by using the theory of finite differences. Emiris et al., 2019, have generalized their result to handle the case where the exponent of the term $|α-β|$ in the product is at most the multiplicity of either of the roots. In this paper, we generalize these results by allowing arbitrary positive integer weights on the edges of the graph, i.e., for a weight function $w: E \rightarrow \mathbb{Z}_{>0}$, we derive an amortized lower bound on $\prod_{(α,β) \in E}|α-β|^{w(α,β)}$. Such a product occurs in the complexity estimates of some recent algorithms for root clustering (e.g., Becker et al., 2016), where the weights are usually some function of the multiplicity of the roots. Because of its amortized nature, our bound is arguably better than the bounds obtained by manipulating existing results to accommodate the weights.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
