The Number of Ramified Coverings of the Sphere by the Torus and Surfaces of Higher Genera
The Number of Ramified Coverings of the Sphere by the Torus and Surfaces of Higher Genera
复制标题
环面和高等属表面对球体的分支覆盖数
DOI：
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发表时间：
1999
期刊：
影响因子：
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通讯作者：
A. Vainshtein
中科院分区：
文献类型：
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作者：
I. Goulden;D. Jackson;A. Vainshtein
抽象的。我们得到了一个明确的表达式的数量的分歧覆盖的球面环面与给定的分歧类型的分歧点的少数，并猜测这是真实的分歧点的任意数量。此外，还证明了环面对球面的简单覆盖的猜想。我们得到相应的表达式为少数分歧点的曲面的高属，并猜测这个数字的一般形式的对称多项式，似乎是新的。该方法涉及到分析的行动的换位，以获得一个系统的线性偏微分方程，给出所需的号码的生成系列。
Abstract. We obtain an explicit expression for the number of ramified coverings of the sphere by the torus with given ramification type for a small number of ramification points, and conjecture this to be true for an arbitrary number of ramification points. In addition, the conjecture is proved for simple coverings of the sphere by the torus. We obtain corresponding expressions for surfaces of higher genera for small number of ramification points, and conjecture the general form for this number in terms of a symmetric polynomial that appears to be new. The approach involves the analysis of the action of a transposition to derive a system of linear partial differential equations that give the generating series for the desired numbers.