关于本质不可约映射的多项式计数
On polynomials counting essentially irreducible maps
摘要：
我们考虑亏格- $g$曲面上的映射，其$n$（标记）面具有规定的偶数度。众所周知，由于诺伯里的工作，如果一个不允许顶点的程度，这种地图的枚举是有关的计数格点的模空间的属- $g$曲线与$n$标记的点，并给出了一个对称多项式$N_{g,n}(\ell_1,\ldots,\ell_n)$在面对度$2\ell_1, \ldots, 2\ell_n$。我们推广这一点，通过限制到亏格- $g$地图，基本上是$2b$ -不可约的$b\geq 0$，这松散地说，这意味着他们不允许拥有长度小于$2b$的可收缩的循环，每个这样的循环长度$2b$是需要绑定一个面的程度$2b$。这种地图的枚举示出再次由一个对称多项式$\hat{N}_{g,n}^{(b)}(\ell_1,\ldots,\ell_n)$在面对度与多项式依赖$b$。这些多项式满足（广义的）弦方程和狄龙方程，这对于$g\leq 1$是唯一确定的。证明在很大程度上依赖于替代的方法，由Bouttier和Guitter和枚举的平面映射的属- $g$表面。
英文摘要：
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.
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