论文标题
实际双变量有理功能的限制
Limits of real bivariate rational functions
论文作者
论文摘要
给定两个非零多项式$ f，g \ in \ mathbb r [x，y] $和a point $（a，b）\ in \ mathbb {r}^2，$，我们给出了一些限制$ \ \ \ \ \ \ \ \ display \ lim _ lim _ {（x，y）{（x，a，a，b）}的限制$ \ \ displiststyle \ lim _ { y）} {g（x，y）}。$我们还表明，如果分母$ g $在给定点$（a，b）的孤立零是零的，则$，则可能的限制$ \ displayStyle \ lim _ {（x，x，y）\ to（a，b）}}} \ frac {y y y y y y y y y y y y lim \ lim _ {（x，y） $ \ OVILLINE {\ MATHBB {R}} $，可以明确确定。作为应用程序，我们提出了一种有效的算法来验证限制的存在并计算限制（如果存在）。我们的方法是几何形状，是基于Puiseux扩展的。
Given two nonzero polynomials $f, g \in\mathbb R[x,y]$ and a point $(a, b) \in \mathbb{R}^2,$ we give some necessary and sufficient conditions for the existence of the limit $\displaystyle \lim_{(x, y) \to (a, b)} \frac{f(x, y)}{g(x, y)}.$ We also show that, if the denominator $g$ has an isolated zero at the given point $(a, b),$ then the set of possible limits of $\displaystyle \lim_{(x, y) \to (a, b)} \frac{f(x, y)}{g(x, y)}$ is a closed interval in $\overline{\mathbb{R}}$ and can be explicitly determined. As an application, we propose an effective algorithm to verify the existence of the limit and compute the limit (if it exists). Our approach is geometric and is based on Puiseux expansions.
