Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Generalizing The Davenport-Mahler-Mignotte Bound -- The Weighted Case   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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The categories of this paper are: chao-dyn
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The categories of this paper are: astro-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: stat
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The categories of this paper are: hep-ex
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The categories of this paper are: physics
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The categories of this paper are: acc-phys
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The categories of this paper are: atom-ph
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The categories of this paper are: supr-con
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The categories of this paper are: nucl-th
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The categories of this paper are: nlin
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The categories of this paper are: chem-ph
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The categories of this paper are: math
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The categories of this paper are: adap-org
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The categories of this paper are: nucl-ex
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The categories of this paper are: mtrl-th