论文标题
关于多项式计数基本上不可还原图
On polynomials counting essentially irreducible maps
论文作者
论文摘要
我们考虑属于$ g $ $ g $表面的地图，其面孔的面孔均匀。自从诺伯里（Norbury）的工作以来，人们已经知道，如果一个人不允许第一学位的顶点，则此类地图的列举与属于属于$ n $标记点的模量曲线中的晶格点的计数有关$ 2 \ ell_1，\ ldots，2 \ ell_n $。我们通过将其限制到本质上是$ 2B $ $ b \ geq 0 $的$ g $地图来概括这一点，这宽松地说，它们不允许其长度低于$ 2B $的合同周期，每个这样的长度$ 2B $都需要$ 2B $的长度限制$ 2B $ $ 2B $。此类地图的枚举被证明是由对称的多项式$ \ hat {n} _ {g，n}^{（b）}（\ ell_1，\ ldots，\ ell_n）$在面部依赖于$ b $的面部程度的。这些多项式满足（广义）字符串和dilaton方程，对于$ g \ \ leq 1 $唯一地确定它们。这些证据在很大程度上依赖于Bouttier和Guitter的替代方法，以及在属属表面上列举了平面地图。
We consider maps on genus-$g$ surfaces with $n$ (labeled) faces of prescribed even degrees. It is known since work of Norbury that, if one disallows vertices of degree one, the enumeration of such maps is related to the counting of lattice point in the moduli space of genus-$g$ curves with $n$ labeled points and is given by a symmetric polynomial $N_{g,n}(\ell_1,\ldots,\ell_n)$ in the face degrees $2\ell_1, \ldots, 2\ell_n$. We generalize this by restricting to genus-$g$ maps that are essentially $2b$-irreducible for $b\geq 0$, which loosely speaking means that they are not allowed to possess contractible cycles of length less than $2b$ and each such cycle of length $2b$ is required to bound a face of degree $2b$. The enumeration of such maps is shown to be again given by a symmetric polynomial $\hat{N}_{g,n}^{(b)}(\ell_1,\ldots,\ell_n)$ in the face degrees with a polynomial dependence on $b$. These polynomials satisfy (generalized) string and dilaton equations, which for $g\leq 1$ uniquely determine them. The proofs rely heavily on a substitution approach by Bouttier and Guitter and the enumeration of planar maps on genus-$g$ surfaces.
