标题： 分支覆盖的组合表示 显示英文标题

标题： A Combinatorial Presentation for Branched Coverings of the 2-Sphere

摘要： 威廉·瑟斯顿（1946-2012）通过为两球面的泛化分支自映射关联一个平面图，给出了一个组合特征 10.48550/arXiv.1502.04760。 通过推广局部平衡的概念，作者将瑟斯顿的结果扩展到涵盖任何两球面的分支覆盖。 作为应用，我们为每个给定的分歧类型提供了实有理函数等价类的数量下界。 此外，作为结果，我们得到了一个新证明的定理（ 10.2307/3062151 ， 10.4007/annals.2009.170.863 ， 10.1090/S0894-0347-09-00640-7 ），该定理对应于枚举几何中一个已知的现实问题的特殊情况，即B。 & M. Shapiro 猜想，现在已成为定理\cite{MR2552110}。 我们证明的定理版本涉及泛化的有理函数，确保如果该函数的所有临界点都是实数，那么可以通过与$\mathbb{C}\mathbb{P}^{1}$的自同构后复合将其转换为具有实系数的有理映射。 我们提供的证明是构造性的，并基于基本的论证。 显示英文摘要

摘要： William Thurston (1946-2012) gave a combinatorial characterization for generic branched self-coverings of the two-sphere by associating a planar graph to them 10.48550/arXiv.1502.04760. By generalizing the notion of local balancing, the author extends the Thurston result to encompass any branched covering of the two-sphere. As an application, we supply a lower bound for the number of equivalence classes of real rational functions for each given ramification profile. Furthermore, as a consequence, we obtain a new proof for a theorem ( 10.2307/3062151 , 10.4007/annals.2009.170.863 , 10.1090/S0894-0347-09-00640-7 ) that corresponds to a special case of a reality problem in enumerative geometry which was known as the B. \& M. Shapiro Conjecture, now it is a theorem \cite{MR2552110}. The theorem version that we prove concerns generic rational functions, assuring that if all critical points of that function are real, then we can transform it into a rational map with real coefficients by post-composition with an automorphism of $\mathbb{C}\mathbb{P}^{1}$. The proof we present is constructive and founded on elementary arguments.}