The numerical range of limits of real rational functions

Source abstract

Given two nonzero polynomials $f, g \in \mathbb{R}[x_1,\dots,x_n]$ vanishing at the origin $0 \in \mathbb{R}^n$, we study the numerical range of the limit $$\lim_{x \to 0} \frac{f(x)}{g(x)},$$ denoted by $\mathcal{R}_{f,g}$, which is defined as the set of all possible limits of the rational function $\displaystyle\frac{f}{g}$ as $x$ tends to $0$. When $n = 2$ and $g$ has a non-isolated zero at $0$, we describe all possible forms of $\mathcal{R}_{f,g}$ 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 $g$.