AI 中文总结
研究\(\mathbb{K}\)-域中理想分次族与相关凸区域（牛顿 - 奥昆科夫区域）的相互作用，结合阿图什 - 韦茨拓扑技术和凸几何性质，通过区域包含关系刻画一对理想分次族的渐近复苏数。
AI 中文摘要
我们研究了\(\mathbb{K}\)-域中理想的分次族与其相关凸区域之间的相互作用。这些区域称为牛顿 - 奥昆科夫区域，自然地源于与具有一维叶的赋值相关的理想分次族。我们主要关注计算一对理想分次族的渐近复苏数。通过结合阿图什 - 韦茨拓扑技术和牛顿 - 奥昆科夫区域的凸几何性质，我们通过相应的牛顿 - 奥昆科夫区域对之间的包含关系来刻画渐近复苏数。
英文摘要
We study the interplay between graded families of ideals in $\mathbb{K}$-domains and their associated convex regions. These regions, called Newton-Okounkov regions, arise naturally from graded families of ideals associated to a valuation with one-dimensional leaves. Our main focus is to compute asymptotic resurgence number of a pair of graded families of ideals. By combining techniques from Attouch--Wets topology and convex-geometric properties of Newton-Okounkov regions, we characterize the asymptotic resurgence number through containment relations between the pair of corresponding Newton-Okounkov regions.
