Title: Refined algebraic domains with finite sets in the boundaries Show Chinese title

Title: 边界上具有有限集合的精化代数域

Abstract: Refined algebraic domains are regions in the plane surrounded by finitely many non-singular real algebraic curves which may intersect with normal crossing. We are interested in shapes of such regions with surrounding real algebraic curves. Poincar'e-Reeb Graphs of them are graphs the regions naturally collapse to respecting the projection to a straight line. Such graphs were first formulated by Sorea, for example, around 2020, and regions surrounded by mutually disjoint non-singular real algebraic curves were mainly considered. The author has generalized the studies to several general situations. We find classes of such objects defined inductively by adding curves. We respect characteristic finite sets in the curves. We consider regions surrounded by the curves and of a new type. We investigate geometric properties and combinatorial ones of them and discuss important examples. We also previously studied explicit classes defined inductively in this way and review them. Show Chinese abstract

Abstract: 精化的代数域是被有限多个非奇异实代数曲线包围的平面区域，这些曲线可能以正常交叉的方式相交。 我们关注由周围实代数曲线构成的区域的形状。 它们的Poincaré-Reeb图是区域在投影到一条直线时自然坍缩成的图。 这样的图最早由Sorea在2020年左右提出，主要考虑的是由互不相交的非奇异实代数曲线包围的区域。 作者已将这些研究推广到几种一般情况。 我们发现通过添加曲线递归定义的此类对象的类。 我们尊重曲线中的特征有限集。 我们考虑由这些曲线包围的新类型区域。 我们研究它们的几何性质和组合性质，并讨论重要的例子。 我们还之前研究了以这种方式递归定义的显式类并进行回顾。