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: $\prod_{(\alpha,\beta) \in E}|\alpha-\beta|$. 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 $|\alpha-\beta|$ 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 $\prod_{(\alpha,\beta) \in E}|\alpha-\beta|^{w(\alpha,\beta)}$. 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.

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概括 Davenport-Mahler-Mignotte 界——加权案例

根分离边界在理解计算代数中各种算法（例如，根分离算法）的行为方面作为复杂性度量起着重要作用。单变量设置中的一个经典结果是 Davenport-Mahler-Mignotte (DMM) 界限。陈述界限的一种方法是考虑在阶数 $d$ 多项式 $f(z) \in \mathbb{C}[z]$ 的根子集上的有向无环图 $(V,E)$，其中边从绝对值较小的根指向绝对值较大的根，所有顶点的入度最多为1。那么 DMM 界限是以下乘积的摊销下界：$\prod_{(\alpha,\beta) \in E}|\alpha-\beta|$。然而，下界涉及多项式 $f$ 的判别式，如果多项式不是无平方的，则下界变得微不足道。这是由 Eigenwillig (2008) 解决的，通过使用合适的子判别式而不是判别式。Escorcielo-Perrucci，2016 年，通过使用有限差分理论进一步删除了图中的入度约束。Emiris 等人，2019 年，将他们的结果推广到处理产品中 $|\alpha-\beta|$ 项的指数至多是任一根的重数的情况。在本文中，我们通过在图的边缘上允许任意正整数权重来概括这些结果，即，对于权重函数 $w：E \rightarrow \mathbb{Z}_{>0}$，我们推导出了一个摊销的下限绑定在 $\prod_{(\alpha,\beta) \in E}|\alpha-\beta|^{w(\alpha,\beta)}$ 上。这种产品出现在一些最近的根聚类算法的复杂度估计中（例如，Becker et al., 2016），其中权重通常是根的多重性的某种函数。由于其摊销性质，我们的界限可以说比通过操纵现有结果以适应权重而获得的界限更好。