Generalizing The Davenport-Mahler-Mignotte Bound -- The Weighted Case

TL;DR

This paper extends the Davenport-Mahler-Mignotte bound to include weighted edges, providing a more general lower bound on products of root differences that is useful for analyzing the complexity of root clustering algorithms.

Contribution

It introduces a generalized bound for weighted root difference products, improving upon previous unweighted and special case bounds in computational algebra.

Findings

Derived an amortized lower bound for weighted root difference products.

Applicable to root clustering algorithms with multiplicity-based weights.

Improves the robustness of root separation bounds in complexity analysis.

Abstract

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.