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 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 GEOMETRIC BIJECTIONS FOR REGULAR MATROIDS, ZONOTOPES, AND EHRHART THEORY

2019 ◽  
Vol 7 ◽  
Author(s):  
SPENCER BACKMAN ◽  
MATTHEW BAKER ◽  
CHI HO YUEN
Keyword(s):  
Spanning Trees ◽  
Finite Abelian Group ◽  
Tutte Polynomial ◽  
Geometric Interpretation ◽  
Lattice Points ◽  
Combinatorial Interpretation ◽  
Ehrhart Polynomial ◽  
Ehrhart Theory ◽  
Regular Matroid ◽  
Definition Of

Let $M$ be a regular matroid. The Jacobian group $\text{Jac}(M)$ of $M$ is a finite abelian group whose cardinality is equal to the number of bases of $M$ . This group generalizes the definition of the Jacobian group (also known as the critical group or sandpile group) $\operatorname{Jac}(G)$ of a graph $G$ (in which case bases of the corresponding regular matroid are spanning trees of $G$ ). There are many explicit combinatorial bijections in the literature between the Jacobian group of a graph $\text{Jac}(G)$ and spanning trees. However, most of the known bijections use vertices of $G$ in some essential way and are inherently ‘nonmatroidal’. In this paper, we construct a family of explicit and easy-to-describe bijections between the Jacobian group of a regular matroid $M$ and bases of $M$ , many instances of which are new even in the case of graphs. We first describe our family of bijections in a purely combinatorial way in terms of orientations; more specifically, we prove that the Jacobian group of $M$ admits a canonical simply transitive action on the set ${\mathcal{G}}(M)$ of circuit–cocircuit reversal classes of $M$ , and then define a family of combinatorial bijections $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}^{\ast }}$ between ${\mathcal{G}}(M)$ and bases of $M$ . (Here $\unicode[STIX]{x1D70E}$ (respectively $\unicode[STIX]{x1D70E}^{\ast }$ ) is an acyclic signature of the set of circuits (respectively cocircuits) of $M$ .) We then give a geometric interpretation of each such map $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}^{\ast }}$ in terms of zonotopal subdivisions which is used to verify that $\unicode[STIX]{x1D6FD}$ is indeed a bijection. Finally, we give a combinatorial interpretation of lattice points in the zonotope $Z$ ; by passing to dilations we obtain a new derivation of Stanley’s formula linking the Ehrhart polynomial of $Z$ to the Tutte polynomial of $M$ .

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 Commutative and non-commutative bialgebras of quasi-posets and applications to Ehrhart polynomials

2019 ◽  
Vol 10 (1) ◽  
pp. 27-63 ◽  
Author(s):  
Loïc Foissy
Keyword(s):  
Hopf Algebra ◽  
Point Of View ◽  
Duality Principle ◽  
Algebraic Proof ◽  
Ehrhart Polynomial ◽  
Algebraic Point ◽  
Ehrhart Polynomials ◽  
Lattice Polytope ◽  
Algebra Morphism

Abstract To any poset or quasi-poset is attached a lattice polytope, whose Ehrhart polynomial we study from a Hopf-algebraic point of view. We use for this two interacting bialgebras on quasi-posets. The Ehrhart polynomial defines a Hopf algebra morphism with values in \mathbb{Q}[X] . We deduce from the interacting bialgebras an algebraic proof of the duality principle, a generalization and a new proof of a result on B-series due to Whright and Zhao, using a monoid of characters on quasi-posets, and a generalization of Faulhaber’s formula. We also give non-commutative versions of these results, where polynomials are replaced by packed words. We obtain, in particular, a non-commutative duality principle.

scholarly journals Proof of Chapoton's Conjecture on Newton Polygons of $q$-Ehrhart Polynomials

10.37236/7322 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Jang Soo Kim ◽  
U-Keun Song
Keyword(s):  
Newton Polygon ◽  
Rational Functions ◽  
Newton Polygons ◽  
Ehrhart Polynomial ◽  
Ehrhart Polynomials ◽  
Order Polytope

Recently, Chapoton found a $q$-analog of Ehrhart polynomials, which are polynomials in $x$ whose coefficients are rational functions in $q$. Chapoton conjectured the shape of the Newton polygon of the numerator of the $q$-Ehrhart polynomial of an order polytope. In this paper, we prove Chapoton's conjecture.

scholarly journals Smooth Fano polytopes whose Ehrhart polynomial has a root with large real part

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Hidefumi Ohsugi ◽  
Kazuki Shibata
Keyword(s):  
Nous Avons ◽  
Ehrhart Polynomial ◽  
Ehrhart Polynomials ◽  
Nous Montrons ◽  
International Audience ◽  
Del Pezzo ◽  
Odd Cycles ◽  
Fano Polytope

International audience The symmetric edge polytopes of odd cycles (del Pezzo polytopes) are known as smooth Fano polytopes. In this extended abstract, we show that if the length of the cycle is 127, then the Ehrhart polynomial has a root whose real part is greater than the dimension. As a result, we have a smooth Fano polytope that is a counterexample to the two conjectures on the roots of Ehrhart polynomials. Les polytopes d'arêtes symètriques de cycles impaires (del Pezzo polytopes) sont connus sous le nom de polytopes de Fano lisses. Dans ce rèsumè ètendu, nous montrons que si la longueur du cycle est 127, alors le polynôme d'Ehrhart a une racine dont la partie rèele est plus grande que la dimension. En consèquence, nous avons un polytope de Fano lisse qui est un contre exemple à deux conjectures sur les racines de polynômes d'Ehrhart.

scholarly journals Mixed Ehrhart polynomials

10.37236/5815 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Christian Haase ◽  
Martina Juhnke-Kubitzke ◽  
Raman Sanyal ◽  
Thorsten Theobald
Keyword(s):  
Mixed Volume ◽  
Ehrhart Polynomial ◽  
Ehrhart Polynomials ◽  
Lattice Polytopes ◽  
Real Roots

For lattice polytopes $P_1,\ldots, P_k \subseteq \mathbb{R}^d$, Bihan (2016) introduced the discrete mixed volume $DMV(P_1,\dots,P_k)$ in analogy to the classical mixed volume.  In this note we study the associated mixed Ehrhart polynomial $ME_{P_1, \dots,P_k}(n) = DMV(nP_1, \dots, nP_k)$.  We provide a characterization of all mixed Ehrhart coefficients in terms of the classical multivariate Ehrhart polynomial. Bihan (2016) showed that the discrete mixed volume is always non-negative. Our investigations yield simpler proofs for certain special cases.We also introduce and study the associated mixed $h^*$-vector. We show that for large enough dilates $r  P_1, \ldots, rP_k$ the corresponding mixed $h^*$-polynomial has only real roots and as a consequence  the mixed $h^*$-vector becomes non-negative. 

scholarly journals Pick’s Theorem in Two-Dimensional Subspace ofR3

2015 ◽  
Vol 2015 ◽  
pp. 1-3
Author(s):  
Lin Si
Keyword(s):  
Euclidean Space ◽  
Dimensional Subspace ◽  
Lattice Points ◽  
Two Dimensional ◽  
Lattice Polygon ◽  
Simple Lattice ◽  
Pick's Theorem

In the Euclidean spaceR3, denote the set of all points with integer coordinate byZ3. For any two-dimensional simple lattice polygonP, we establish the following analogy version of Pick’s Theorem,kIP+1/2BP-1, whereBPis the number of lattice points on the boundary ofPinZ3,IPis the number of lattice points in the interior ofPinZ3, andkis a constant only related to the two-dimensional subspace includingP.

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