论文标题
庞加莱系列的循环图和链条上
Poincaré Series of Divisors on Graphs and Chains of Loops
论文作者
论文摘要
我们研究了与i上有限的除数集有关的庞加莱系列。有限图和II。某种称为循环链的公制图。我们的主要结果是在这两种情况下，Poincaré系列的合理性和计算算法的合理性证明。理性证明中使用的主要工具如下。对于图，我们研究了从自由的Abelian有限等级到图形和整数的直接总和的某种同态。对于环的链条，我们的主要工具是Lang在一系列环路上对Brill-Noether Loci的猜想的类似物，并适应了代数曲线上Poincaré系列除数的合理性证明（在特征性零的代数封闭场上）。我们的算法是基于对合理证明所涉及的对象的仔细研究，例如计算某些同态纤维和理性多面体中的晶格枚举。
We study Poincaré series associated to a finite collection of divisors on i. a finite graph and ii. a certain family of metric graphs called chain of loops. Our main results are proofs of rationality of the Poincaré series and algorithms for computing it in both these cases. The main tools used in the proof of rationality are the following. For graphs, we study a certain homomorphism from a free Abelian group of finite rank to the direct sum of the Jacobian of the graph and the integers. For chains of loops, our main tool is an analogue of Lang's conjecture for Brill-Noether loci on a chain of loops and adapts the proof of rationality of the Poincaré series of divisors on an algebraic curve (over an algebraically closed field of characteristic zero). Our algorithms are based on a closer study of the objects involved in the proof of rationality, for instance, computing the fibres of certain homomorphisms and lattice point enumeration in rational polyhedra.
