Dedekind sums: a combinatorial-geometric viewpoint
Dedekind sums: a combinatorial-geometric viewpoint
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戴德金求和：组合几何观点
DOI：
10.1090/dimacs/064/04
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发表时间：
2001
期刊：
影响因子：
--
通讯作者：
S. Robins
中科院分区：
文献类型：
--
作者：
M. Beck;S. Robins
关于DedekindSums的文献很多。在这篇说明性论文中，我们证明了Dedekind和的许多推广都有一条共同的线索，即通过研究有理多面体的格点计数。具体地说，有一些自然的有限傅立叶级数，我们称之为傅里叶-德德金德和，它们构成了正整数的有限集的整数分拆个数的基础。这个问题也被称为“硬币兑换问题”。Dedekind和最近在拓扑学、数论和组合几何等不同领域重新引起了人们的兴趣。作为特例，我们在这里研究的傅立叶-德德金德和包括Berndt，Carlitz，Grosswald，Knuth，Rademacher和Zagier所研究的广义Dedekind和。我们对这些和的兴趣源于Dedekin和Zagier的和在多面体的格点计数公式中的出现。利用一些简单的母函数，我们证明了广义Dedekind和是这类公式的自然成分。作为我们公式的直接“几何”推论，我们得到并推广了Dedekin、Zagier和Gessel的互易定律。最后，我们证明了Zagier的高维Dedekind和的一个多项式时间复杂性结果。
The literature on Dedekind sums is vast. In this expository paper we show that there is a common thread to many generalizations of Dedekind sums, namely through the study of lattice point enumeration of rational polytopes. In particular, there are some natural finite Fourier series which we call Fourier-Dedekind sums, and which form the building blocks of the number of partitions of an integer from a finite set of positive integers. This problem also goes by the name of the `coin exchange problem'. Dedekind sums have enjoyed a resurgence of interest recently, from such diverse fields as topology, number theory, and combinatorial geometry. The Fourier-Dedekind sums we study here include as special cases generalized Dedekind sums studied by Berndt, Carlitz, Grosswald, Knuth, Rademacher, and Zagier. Our interest in these sums stems from the appearance of Dedekind's and Zagier's sums in lattice point count formulas for polytopes. Using some simple generating functions, we show that generalized Dedekind sums are natural ingredients for such formulas. As immediate `geometric' corollaries to our formulas, we obtain and generalize reciprocity laws of Dedekind, Zagier, and Gessel. Finally, we prove a polynomial-time complexity result for Zagier's higher-dimensional Dedekind sums.