The Holonomic Ansatz I. Foundations and Applications to Lattice Path Counting

Many combinatorial quantities belong to the holonomic ansatz. For example, sequences enumerating lattice paths. Once this fact is known, many times empirically obtained “conjectures” can be easily proved automatically. Accompanied by Maple packages GuessHolo2 and GuessHolo3 available from Zeilberger’s website. Supported in part by the NSF. . Received October 19, 2006     

Lattice Points Inside Rational Simplices and the Casson Invariant of Brieskorn Spheres

We express the number of lattice points inside certain simplices with vertices in Q³ or Q⁴ in terms of Dedekind–Rademacher sums. This leads to an elementary proof of a formula relating the Euler characteristic of the Seiberg–Witten-Floer homology of a Brieskorn Z-homology sphere to the Casson invariant.     

The Pick Theorem and the Proof of the Reciprocity Law for Dedekind Sums

This paper is to provide some new generalizations of the Pick Theorem. We first derive a point-set version of the Pick Theorem for an arbitrary bounded lattice polyhedron. Then, we use the idea of a weight function of [2] to obtain a weighted version. Other Pick type theorems known to the author for the integral lattice Z² are reduced to some special cases of this generalization. Finally, using an idea of Ehrhart [6] and the Pick Theorem, we give a direct proof of the reciprocity law for Dedekind sums. The ideas and methods presented here may be pushed to higher dimensions.AMS Subject Classification: 52C05, 11H06, 57N05, 57N15, 57N35.     

Lattice points in bodies of revolution II

In [3] it was shown that when a three-dimensional smooth convex body has rotational symmetry around a coordinate axis one can find better bounds for the lattice point discrepancy than what is known for more general convex bodies. To accomplish this, however, it was necessary to assume a non-vanishing condition on the third derivative of the generatrix. In this article we drop this condition, showing that the aforementioned bound holds for a wider family of revolution bodies, which includes those with analytic boundary. A novelty in our approach is that, besides the usual analytic methods, it requires studying some Diophantine properties of the Taylor coefficients of the phase on the Fourier transform side.     

On the enumeration of certain weighted graphs

We enumerate weighted simple graphs with a natural upper bound condition on the sum of the weight of adjacent vertices. We also compute the generating function of the numbers of these graphs, and prove that it is a rational function. In particular, we show that the generating function for connected bipartite simple graphs is of the form p₁(x)/(1-x)^m+1. For nonbipartite simple graphs, we get a generating function of the form p₂(x)/(1-x)^m+1(1+x)^l. Here m is the number of vertices of the graph, p₁(x) is a symmetric polynomial of degree at most m, p₂(x) is a polynomial of degree at most m+l, and l is a nonnegative integer. In addition, we give computational results for various graphs.     

Asymptotic of Eigenvalues and Lattice Points

dimension. elementary domains of In this work we study the spectral counting function for the p-Laplace operator in one We show the existence of a two-term Weyl-type asymptote. The method of proof is rather based on the Dirichlet lattice points problem, which enables us to obtain similar results for infinite measure.     

Higher Power Residue Codes and the Leech Lattice

We shall consider higher power residue codes over the ring Z₄. We will briefly introduce these codes over Z₄ and then we will find a new construction for the Leech lattice. A similar construction is used to construct some of the other lattices of rank 24.     

一种判断多项式函数极值点和拐点个数的简单方法

给出了判断一类多项式函数极值点和拐点个数的一种快捷方法,得到了几个相关结论,并通．过几个典型例题验证了该方法解题的有效性和快捷性．     

探讨Abel积分零点的一个定理

本文研究了具有中心环域的可积多项式系统,探讨此系统在用多项式进行微扰的情况下,其所对应的Abel积分的零点个数的计算方法,给出了一个较为实用的定理,并举出了若干个应用的例子．     

Covering Convex Bodies by Cylinders and Lattice Points by Flats

In connection with an unsolved problem of Bang (1951) we give a lower bound for the sum of the base volumes of cylinders covering a d-dimensional convex body in terms of the relevant basic measures of the given convex body. As an application we establish lower bounds on the number of k-dimensional flats (i.e. translates of k-dimensional linear subspaces) needed to cover all the integer points of a given convex body in d-dimensional Euclidean space for 1≤k≤d−1. K. Bezdek and A.E. Litvak are partially supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant.     

Periodicity of hyperplane arrangements with integral coefficients modulo positive integers

We study central hyperplane arrangements with integral coefficients modulo positive integers q. We prove that the cardinality of the complement of the hyperplanes is a quasi-polynomial in two ways, first via the theory of elementary divisors and then via the theory of the Ehrhart quasi-polynomials. This result is useful for determining the characteristic polynomial of the corresponding real arrangement. With the former approach, we also prove that intersection lattices modulo q are periodic except for a finite number of q’s. This work was supported by the MEXT and the JSPS.     

de Bruijn定理的推广

de Bruijn定理是一种重要的组合计数方法，本文以非常自然的方式推广了这种方法．p图是图G在其顶点上的置换群P作用下形成的轨道．文中引进了P-图，P-图的色容指标，P-图关于色置换群H的色权多项式以及色对称与全色对称图等概念，建立了色权多项式的计算公式和一系列的组合公式及性质．     

Lattice polytopes cut out by root systems and the Koszul property

We show that lattice polytopes cut out by root systems of classical type are normal and Koszul, generalizing a well-known result of Bruns, Gubeladze, and Trung in type A. We prove similar results for Cayley sums of collections of polytopes whose Minkowski sums are cut out by root systems. The proofs are based on a combinatorial characterization of diagonally split toric varieties.     

分配格中理想诱导的幂格

本文在文 [1 ]的基础上 ,给出由分配格中理想诱导的两种幂格 .     

关于广义Dedekind和的2k次均值

本文利用 Ｄｉｒｉｃｈｌｅｔ Ｌ－函数的均值定理,研究广义 Ｄｅｄｅｋｉｎｄ和的渐近性质,并给出一个有趣的渐近公式．     

On the Mean Value of the Square of a Generalized Dedekind Sum

The main purpose of this paper is to use the mean value theorem of the Dirichlet L-functions to study the distribution property of a generalized Dedekind sum, and give a sharper mean square value formula.     

Products of foldable triangulations

Regular triangulations of products of lattice polytopes are constructed with the additional property that the dual graphs of the triangulations are bipartite. The (weighted) size difference of this bipartition is a lower bound for the number of real roots of certain sparse polynomial systems by recent results of Soprunova and Sottile [E. Soprunova, F. Sottile, Lower bounds for real solutions to sparse polynomial systems, Adv. Math. 204 (1) (2006) 116–151]. Special attention is paid to the cube case.