论文标题
普遍的Riemann功能，其权重和完整的图
Generalized Riemann Functions, Their Weights, and the Complete Graph
论文作者
论文摘要
通过a {\ em riemann函数}，我们的意思是一个函数$ f \ colon {\ mathbb z}^n \ to {\ mathbb z} $，这样$ f（{\ bf d}）$等于$ 0 $ 0 $ 0 $ for $ d_1+d_1+\ cd+d_n $ gud_n $ gud_n $ affors $ d_n $ a $ d_1+d_1+d_1+d_1+d_1+d_1+d_1+d_1+d_1 $ c $，对于$ d_1+\ cdots+d_n $足够大。通过向图形的Baker-Norine等级函数添加$ 1 $，可以获得等效的Riemann函数，并且对于相关的等级函数类似。 对于每个riemann函数，我们通过möbiusiNversion将相关功能$ w \ colon {\ mathbb z}^n \与{\ mathbb z} $相关联，我们称之为riemann函数的{\ em weight}。我们提供的证据表明，重量似乎以更简单的方式组织了Riemann功能的结构：首先，Riemann功能$ f $满足Riemann-Roch公式，如果其重量可以满足更简单的对称条件。其次，我们将计算某些图表的贝克 - 北林等级的重量，并表明重量函数非常简单。我们为在两个顶点和完整的图表上的图形进行此操作。 对于完整的图，我们建立在Cori和Le Borgne的工作基础上，他们给出了一种线性时间方法来计算完整图的贝克 - 诺林等级。相关的重量函数具有简单的公式，并且非常稀疏（即主要为零）。我们对体重函数的计算导致了另一种线性时间算法来计算Baker -Norine等级，该公式可能与Cori和Le Borgne相关的公式，但似乎更简单，但$$ R _ {{\ rm bn}，k_n}，k_n}，k_n}，k_n}（ \ biggl \ {i = 0，\ ldots，{\ rm deg}（{\ bf d}）\ \ bigm | \ \ sum_ {j = 1}^{n-2} \ bigl（（d_j-d_ {n-1}+i）\ bmod n \ bigr）\ le {\ rm deg}（{\ bf d}） - i \ biggr \} \ biggr |。 $$ 我们对体重功能的研究导致了Riemann功能的自然概括，Riemann函数表现出许多相同的特性。
By a {\em Riemann function} we mean a function $f\colon{\mathbb Z}^n\to{\mathbb Z}$ such that $f({\bf d})$ is equals $0$ for $d_1+\cdots+d_n$ sufficiently small, and equals $d_1+\cdots+d_n+C$ for a constant, $C$, for $d_1+\cdots+d_n$ sufficiently large. By adding $1$ to the Baker-Norine rank function of a graph, one gets an equivalent Riemann function, and similarly for related rank functions. To each Riemann function we associate a related function $W\colon{\mathbb Z}^n\to{\mathbb Z}$ via Möbius inversion that we call the {\em weight} of the Riemann function. We give evidence that the weight seems to organize the structure of a Riemann function in a simpler way: first, a Riemann function $f$ satisfies a Riemann-Roch formula iff its weight satisfies a simpler symmetry condition. Second, we will calculate the weight of the Baker-Norine rank for certain graphs and show that the weight function is quite simple to describe; we do this for graphs on two vertices and for the complete graph. For the complete graph, we build on the work of Cori and Le Borgne who gave a linear time method to compute the Baker-Norine rank of the complete graph. The associated weight function has a simple formula and is extremely sparse (i.e., mostly zero). Our computation of the weight function leads to another linear time algorithm to compute the Baker-Norine rank, via a formula likely related to one of Cori and Le Borgne, but seemingly simpler, namely $$ r_{{\rm BN},K_n}({\bf d}) = -1+\biggl| \biggl\{ i=0,\ldots,{\rm deg}({\bf d}) \ \Bigm| \ \sum_{j=1}^{n-2} \bigl( (d_j-d_{n-1}+i) \bmod n \bigr) \le {\rm deg}({\bf d})-i \biggr\} \biggr|. $$ Our study of weight functions leads to a natural generalization of Riemann functions, with many of the same properties exhibited by Riemann functions.
