Lattice Points, Dedekind Sums, and Ehrhart Polynomials of Lattice Polyhedra

Chen, Guihai

doi:10.1007/s00454-002-2759-7

Cited by 32 publications

(18 citation statements)

References 16 publications

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“…The formulae of the above example can be used for obtaining a closed expression for the Ehrhart polynomial of a lattice d-simplex in R d , similar to those proved in [21,25] by very different methods. Indeed, the toric variety associated to such a simplex is a fake weighted projective space.…”

Section: Let Chmentioning

confidence: 99%

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Characteristic Classes of Singular Toric Varieties

Maxim

Schürmann

2014

Comm Pure Appl Math

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Abstract. In this paper we compute the motivic Chern classes and homology Hirzebruch characteristic classes of (possibly singular) toric varieties, which in the context of complete toric varieties fit nicely with a generalized Hirzebruch-Riemann-Roch theorem. As important special cases, we obtain new (or recover well-known) formulae for the Baum-Fulton-MacPherson Todd (or MacPherson-Chern) classes of toric varieties, as well as for the Thom-Milnor L-classes of simplicial projective toric varieties. We present two different perspectives for the computation of these characteristic classes of toric varieties. First, we take advantage of the torus-orbit decomposition and the motivic properties of the motivic Chern and resp. homology Hirzebruch classes to express the latter in terms of dualizing sheaves and resp. the (dual) Todd classes of closures of orbits. This method even applies to torus-invariant subspaces of a given toric variety. The obtained formula is then applied to weighted lattice point counting in lattice polytopes and their subcomplexes, yielding generalized Pick-type formulae. Secondly, in the case of simplicial toric varieties, we compute our characteristic classes by using the Lefschetz-Riemann-Roch theorem of Edidin-Graham in the context of the geometric quotient description of such varieties. In this setting, we define mock Hirzebruch classes of simplicial toric varieties (which specialize to the mock Chern, mock Todd and mock L-classes of such varieties) and investigate the difference between the (actual) homology Hirzebruch class and the mock Hirzebruch class. We show that this difference is localized on the singular locus, and we obtain a formula for it in which the contribution of each singular cone is identified explicitly. Finally, the two methods of computing characteristic classes are combined for proving several characteristic class formulae originally obtained by Cappell and Shaneson in the early 1990s.

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Section: Let Chmentioning

confidence: 99%

“…So, by evaluating the right-hand side of the above weighted lattice point counting formula (1.17), we obtain the following parametrized version of the classical Pick's formula (1.16): 21) where F runs over the facets (i.e., boundary segments) of P , v denotes the number of vertices of P , and…”

Section: Motivic Chern and Hirzebruch Classes Via Orbit Decompositionmentioning

confidence: 99%

Characteristic Classes of Singular Toric Varieties

Maxim

Schürmann

2014

Comm Pure Appl Math

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“…Their paper [108] also draws parallels between toric varieties and lattice polytopes. Subsequently, many attempts to provide formulas for Ehrhart quasipolynomialssome based on Theorem 10.7-have provided fertile ground for deeper connections and future work; a long but by no means complete list of references is [9,33,46,55,60,61,76,91,106,107,119,125,137,146,178]. …”

Section: The Todd Operator Was Introduced By Friedrich Hirzebruch In mentioning

confidence: 99%

Computing the continuous discretely: integer-point enumeration in polyhedra

Beck¹,

Robins²

2007

Choice Reviews Online

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We seek to bridge some critical gaps between various fields of mathematics by studying the interplay between the continuous volume and the discrete volume of polytopes. Examples of polytopes in three dimensions include crystals, boxes, tetrahedra, and any convex object whose faces are all flat. It is amusing to see how many problems in combinatorics, number theory, and many other mathematical areas can be recast in the language of polytopes that exist in some Euclidean space. Conversely, the versatile structure of polytopes gives us number-theoretic and combinatorial information that flows naturally from their geometry. The discrete volume of a body P can be described intuitively as the number of grid points that lie inside P, given a fixed grid in Euclidean space. The continuous volume of P has the usual intuitive meaning of volume that we attach to everyday objects we see in the real world. VIII PrefaceIndeed, the difference between the two realizations of volume can be thought of in physical terms as follows. On the one hand, the quantumlevel grid imposed by the molecular structure of reality gives us a discrete notion of space and hence discrete volume. On the other hand, the Newtonian notion of continuous space gives us the continuous volume. We see things continuously at the Newtonian level, but in practice we often compute things discretely at the quantum level. Mathematically, the grid we impose in space-corresponding to the grid formed by the atoms that make up an object-helps us compute the usual continuous volume in very surprising and charming ways, as we shall discover.In order to see the continuous/discrete interplay come to life among the three fields of combinatorics, number theory, and geometry, we begin our focus with the simple-to-state coin-exchange problem of Frobenius. The beauty of this concrete problem is that it is easy to grasp, it provides a useful computational tool, and yet it has most of the ingredients of the deeper theories that are developed here.In the first chapter, we give detailed formulas that arise naturally from the Frobenius coin-exchange problem in order to demonstrate the interconnections between the three fields mentioned above. The coin-exchange problem provides a scaffold for identifying the connections between these fields. In the ensuing chapters we shed this scaffolding and focus on the interconnections themselves:(1) Enumeration of integer points in polyhedra-combinatorics, (2) Dedekind sums and finite Fourier series-number theory, (3) Polygons and polytopes-geometry.We place a strong emphasis on computational techniques, and on computing volumes by counting integer points using various old and new ideas. Thus, the formulas we get should not only be pretty (which they are!) but should also allow us to efficiently compute volumes by using some nice functions. In the very rare instances of mathematical exposition when we have a formulation that is both "easy to write" and "quickly computable," we have found a mathematical nugget. We have endeavored to fill this boo...

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“…The above direct proof for the reciprocity law of Dedekind sums is a special case of [3] for the computation of the co-dimension two coefficient of a special lattice simplex. In higher dimensions, the coefficient formulas similar to (17) have been given in [3,4]; and one can apply those coefficient formulas to an n-dimensional lattice simplex to obtain Zagier's reciprocity law of higher dimensional Dedekind sums; see [10,20].…”

mentioning

confidence: 99%

“…In higher dimensions, the coefficient formulas similar to (17) have been given in [3,4]; and one can apply those coefficient formulas to an n-dimensional lattice simplex to obtain Zagier's reciprocity law of higher dimensional Dedekind sums; see [10,20]. However, I would like to mention another proof given by Beck [1] for the reciprocity law of Dedekind sums, using the generating functions of Ehrhart polynomials of [5].…”

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confidence: 99%

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

Chen

2003

Annals of Combinatorics

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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 using the idea of weight function of [2] to obtain a weighted version; other Pick type theorems known to the author for integral lattice Z 2 are reduced to some special cases of the generalization. Finally, using the 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. Point-Set Version of the Pick TheoremLet P be a lattice polygon of R 2 , i.e., the vertices of P are points of the integral lattice Z 2 . Let i(P ) be the number of lattice points of P and i(∂P ) the number of lattice points of its boundary ∂σ. The Pick Theorem says thatGiven a bounded lattice polyhedron X of R 2 ; we denote byX the closure of X and by int X the interior of X; the frontier of X is the setX − intX. The link of X near a point x ∈X is the intersection of X and a circle S 1 (x, r) centered at x with small enough radius r; the Euler characteristic χ(lk (x, X)) is a local topological invariant, which plays an important role in our Pick type theorems. For an interior point x ∈ int X, the link lk (x, X) is a circle and has Euler characteristic zero. If X is closed, then for any x ∈ fr X, the link lk (x, X) is a collection of finite number of arcs and points, so χ(lk (x, X)) = the number of branches near x.

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Lattice Points, Dedekind Sums, and Ehrhart Polynomials of Lattice Polyhedra

Cited by 32 publications

References 16 publications

Characteristic Classes of Singular Toric Varieties

Characteristic Classes of Singular Toric Varieties

Computing the continuous discretely: integer-point enumeration in polyhedra

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

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