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The numerical range of limits of real rational functions
Abstract
Given two nonzero polynomials f, g ∈ ℝ[x1,…,xn] vanishing at the origin 0 ∈ ℝn, we study the numerical range of the limit limx → 0 f(x)/g(x), denoted by ℛf,g, which is defined as the set of all possible limits of the rational function f/g as x tends to $0$. When n = 2 and g has a non-isolated zero at $0$, we describe all possible forms of ℛf,g and provide an effective algorithm to compute this set based on Puiseux expansions. This is a follow-up to our recent results~\cite{Dinh2025}, in which we only considered the case when $0$ is an isolated zero of g.
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BibTeX
@misc{dinh2026numerical,
title = {{The numerical range of limits of real rational functions}},
author = {Si Tiep Dinh and Feng Guo and Hong Duc Nguyen and Tien Son Pham},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36842},
eprint = {2610.36842},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36842},
}
RIS
TY - UNPB
TI - The numerical range of limits of real rational functions
AU - Si Tiep Dinh
AU - Feng Guo
AU - Hong Duc Nguyen
AU - Tien Son Pham
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36842
UR - https://arxiv.org/abs/2610.36842
AB - Given two nonzero polynomials f, g ∈ ℝ[x_1,…,x_n] vanishing at the origin 0 ∈ ℝ^n, we study the numerical range of the limit lim_x → 0 f(x)/g(x), denoted by ℛ_f,g, which is defined as the set of all possible limits of the rational function f/g as x tends to $0$. When n = 2 and g has a non-isolated zero at $0$, we describe all possible forms of ℛ_f,g and provide an effective algorithm to compute this set based on Puiseux expansions. This is a follow-up to our recent results~\cite{Dinh2025}, in which we only considered the case when $0$ is an isolated zero of g.
ER -