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

2002 ◽  
Vol 28 (2) ◽  
pp. 175-199 ◽  
Author(s):  
Chen
Keyword(s):  
Lattice Points ◽  
Dedekind Sums ◽  
Ehrhart Polynomials ◽  
Lattice Polyhedra

scholarly journals On formulas for Dedekind sums and the number of lattice points in tetrahedra

2009 ◽  
Vol 129 (8) ◽  
pp. 1931-1955
Author(s):  
Stephen T. Yau ◽  
Letian Zhang
Keyword(s):  
Lattice Points ◽  
Dedekind Sums

scholarly journals Generalized Ehrhart polynomials

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Sheng Chen ◽  
Nan Li ◽  
Steven V Sam
Keyword(s):  
Diophantine Equations ◽  
Rational Functions ◽  
Lattice Points ◽  
Classical Theorem ◽  
Number Of Solutions ◽  
Ehrhart Polynomials ◽  
Linear Diophantine Equations ◽  
International Audience

International audience Let $P$ be a polytope with rational vertices. A classical theorem of Ehrhart states that the number of lattice points in the dilations $P(n) = nP$ is a quasi-polynomial in $n$. We generalize this theorem by allowing the vertices of $P(n)$ to be arbitrary rational functions in $n$. In this case we prove that the number of lattice points in $P(n)$ is a quasi-polynomial for $n$ sufficiently large. Our work was motivated by a conjecture of Ehrhart on the number of solutions to parametrized linear Diophantine equations whose coefficients are polynomials in $n$, and we explain how these two problems are related. Soit $P$ un polytope avec sommets rationelles. Un théorème classique des Ehrhart déclare que le nombre de points du réseau dans les dilatations $P(n) = nP$ est un quasi-polynôme en $n$. Nous généralisons ce théorème en permettant à des sommets de $P(n)$ comme arbitraire fonctions rationnelles en $n$. Dans ce cas, nous prouvons que le nombre de points du réseau en $P(n)$ est une quasi-polynôme pour $n$ assez grand. Notre travail a été motivée par une conjecture d'Ehrhart sur le nombre de solutions à linéaire paramétrée Diophantine équations dont les coefficients sont des polyômes en $n$, et nous expliquer comment ces deux problèmes sont liés.

scholarly journals Ehrhart $f^*$-Coefficients of Polytopal Complexes are Non-negative Integers

10.37236/2106 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Felix Breuer
Keyword(s):  
Complete Characterization ◽  
Lattice Points ◽  
Ehrhart Polynomial ◽  
Simplicial Cone ◽  
Ehrhart Theory ◽  
Ehrhart Polynomials ◽  
Integer Points ◽  
Polytopal Complex ◽  
Main Technical Tool

The Ehrhart polynomial $L_P$ of an integral polytope $P$ counts the number of integer points in integral dilates of $P$. Ehrhart polynomials of polytopes are often described in terms of their Ehrhart $h^*$-vector (aka Ehrhart $\delta$-vector), which is the vector of coefficients of $L_P$ with respect to a certain binomial basis and which coincides with the $h$-vector of a regular unimodular triangulation of $P$ (if one exists). One important result by Stanley about $h^*$-vectors of polytopes is that their entries are always non-negative. However, recent combinatorial applications of Ehrhart theory give rise to polytopal complexes with $h^*$-vectors that have negative entries.In this article we introduce the Ehrhart $f^*$-vector of polytopes or, more generally, of polytopal complexes $K$. These are again coefficient vectors of $L_K$ with respect to a certain binomial basis of the space of polynomials and they have the property that the $f^*$-vector of a unimodular simplicial complex coincides with its $f$-vector. The main result of this article is a counting interpretation for the $f^*$-coefficients which implies that $f^*$-coefficients of integral polytopal complexes are always non-negative integers. This holds even if the polytopal complex does not have a unimodular triangulation and if its $h^*$-vector does have negative entries. Our main technical tool is a new partition of the set of lattice points in a simplicial cone into discrete cones. Further results include a complete characterization of Ehrhart polynomials of integral partial polytopal complexes and a non-negativity theorem for the $f^*$-vectors of rational polytopal complexes.

scholarly journals On the volume of lattice manifolds

2000 ◽  
Vol 61 (2) ◽  
pp. 313-318 ◽  
Author(s):  
Krzysztof Kołodziejczyk
Keyword(s):  
Lattice Points ◽  
General Lattice ◽  
Volume Formula ◽  
Lattice Polyhedra ◽  
Lattice Polyhedron

The volume of a general lattice polyhedron P in ℝN can be determined in terms of numbers of lattice points from N − 1 different lattices in P Ehrhart gave a formula for the volume of “polyèdre entier” in even-dimensional spaces involving only N/2 lattices. The aim of this note is to comment on Ehrhart's formula and provide a similar volume formula applicable to lattice polyhedra that are N-dimensional manifolds in ℝN.

scholarly journals On Ehrhart Polynomials of Lattice Triangles

10.37236/6624 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Johannes Hofscheier ◽  
Benjamin Nill ◽  
Dennis Öberg
Keyword(s):  
Lattice Points ◽  
Ehrhart Polynomial ◽  
Ehrhart Polynomials ◽  
Lattice Polygon

The Ehrhart polynomial of a lattice polygon $P$ is completely determined by the pair $(b(P),i(P))$ where $b(P)$ equals the number of lattice points on the boundary and $i(P)$ equals the number of interior lattice points. All possible pairs $(b(P),i(P))$ are completely described by a theorem due to Scott. In this note, we describe the shape of the set of pairs $(b(T),i(T))$ for lattice triangles $T$ by finding infinitely many new Scott-type inequalities.

scholarly journals Toric varieties, lattice points and Dedekind sums

1993 ◽  
Vol 295 (1) ◽  
pp. 1-24 ◽  
Author(s):  
James E. Pommersheim
Keyword(s):  
Toric Varieties ◽  
Lattice Points ◽  
Dedekind Sums

Circles on lattices and Hadamard matrices

2019 ◽  
pp. 2-9 ◽  
Author(s):  
N. A. Balonin ◽  
M. B. Sergeev ◽  
J. Seberry ◽  
O. I. Sinitsyna
Keyword(s):  
Hadamard Matrices ◽  
Lattice Points ◽  
Practical Significance ◽  
Upper And Lower Bounds ◽  
Problem Size ◽  
Orthogonal Matrices ◽  
Video Information ◽  
Scientific Schools ◽  
Gauss Theorem ◽  
Triangular Numbers

Introduction: The Hadamard conjecture about the existence of Hadamard matrices in all orders multiple of 4, and the Gauss problem about the number of points in a circle are among the most important turning points in the development of mathematics. They both stimulated the development of scientific schools around the world with an immense amount of works. There are substantiations that these scientific problems are deeply connected. The number of Gaussian points (Z3 lattice points) on a spheroid, cone, paraboloid or parabola, along with their location, determines the number and types of Hadamard matrices.Purpose: Specification of the upper and lower bounds for the number of Gaussian points (with odd coordinates) on a spheroid depending on the problem size, in order to specify the Gauss theorem (about the solvability of quadratic problems in triangular numbers by projections onto the Liouville plane) with estimates for the case of Hadamard matrices. Methods: The authors, in addition to their previous ideas about proving the Hadamard conjecture on the base of a one-to-one correspondence between orthogonal matrices and Gaussian points, propose one more way, using the properties of generalized circles on Z3 .Results: It is proved that for a spheroid, the lower bound of all Gaussian points with odd coordinates is equal to the equator radius R, the upper limit of the points located above the equator is equal to the length of this equator L=2πR, and the total number of points is limited to 2L. Due to the spheroid symmetry in the sector with positive coordinates (octant), this gives the values of R/8 and L/4. Thus, the number of Gaussian points with odd coordinates does not exceed the border perimeter and is no less than the relative share of the sector in the total volume of the figure.Practical significance: Hadamard matrices associated with lattice points have a direct practical significance for noise-resistant coding, compression and masking of video information.

ON FINDING THE COEFFICIENTS d 3 c  OF THE EHRHART POLYNOMIALS OF A POLYHEDRON IN d 

2008 ◽  
Vol 11 (1) ◽  
pp. 105-119
Author(s):  
Shatha Assaad Al-Najjar ◽  
◽  
Manal N. Al-Harere ◽  
Vian A. Al. Al-Salehy ◽  
◽  
...  
Keyword(s):  
Ehrhart Polynomials
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