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 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 Lattice polytopes, Hecke operators, and the Ehrhart polynomial

2007 ◽  
Vol 13 (2) ◽  
pp. 253-276 ◽  
Author(s):  
Paul E. Gunnells ◽  
Fernando Rodriguez Villegas
Keyword(s):  
Hecke Operators ◽  
Ehrhart Polynomial ◽  
Lattice Polytopes

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 An Ehrhart Series Formula For Reflexive Polytopes

10.37236/1153 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Benjamin Braun
Keyword(s):  
Ehrhart Polynomial ◽  
Lattice Polytopes ◽  
Reflexive Polytopes ◽  
Ehrhart Series ◽  
Multiplicative Relationship ◽  
Free Sum

It is well known that for $P$ and $Q$ lattice polytopes, the Ehrhart polynomial of $P\times Q$ satisfies $L_{P\times Q}(t)=L_P(t)L_Q(t)$. We show that there is a similar multiplicative relationship between the Ehrhart series for $P$, for $Q$, and for the free sum $P\oplus Q$ that holds when $P$ is reflexive and $Q$ contains $0$ in its interior.

scholarly journals Nuij Type Pencils of Hyperbolic Polynomials

2017 ◽  
Vol 60 (3) ◽  
pp. 561-570
Author(s):  
Krzysztof Kurdyka ◽  
Laurentiu Paunescu
Keyword(s):  
Toeplitz Matrices ◽  
Hyperbolic Polynomials ◽  
Real Roots ◽  
Full Characterization

AbstractNuij’s theorem states that if a polynomial p ∈ ℝ[z] is hyperbolic (i.e., has only real roots), then p+sp'' is also hyperbolic for any s ∈ ℝ. We study other perturbations of hyperbolic polynomials of the form pa(z, s) := . We give a full characterization of those a = (a1 , . . . , ad ) ∈ ℝd for which pa(z, s) is a pencil of hyperbolic polynomials. We also give a full characterization of those a = (a1 , . . . , ad ) ∈ ℝd for which the associated families pa(z, s) admit universal determinantal representations. In fact, we show that all these sequences come fromspecial symmetric Toeplitz matrices.

Solvability by Real Radicals and Fermat Primes

2004 ◽  
Vol 47 (2) ◽  
pp. 229-236
Author(s):  
C. U. Jensen
Keyword(s):  
Real Number ◽  
Number Field ◽  
The Real ◽  
Real Roots ◽  
Fermat Primes ◽  
Real Number Field

AbstractWe give a survey of old and new results concerning the expressibility of the real roots of a solvable polynomial over a real number field by real radicals. A characterization of Fermat primes is obtained in terms of solvability by real radicals for certain ploynomials.

scholarly journals Classification of Triples of Lattice Polytopes with a Given Mixed Volume

2020 ◽  
Author(s):  
Gennadiy Averkov ◽  
Christopher Borger ◽  
Ivan Soprunov
Keyword(s):  
Recent Result ◽  
Mixed Volume ◽  
Polynomial Systems ◽  
Lattice Polytopes ◽  
Complex Torus ◽  
Sparse Polynomial ◽  
Sparse Polynomial Systems

Abstract We present an algorithm for the classification of triples of lattice polytopes with a given mixed volume m in dimension 3. It is known that the classification can be reduced to the enumeration of so-called irreducible triples, the number of which is finite for fixed m. Following this algorithm, we enumerate all irreducible triples of normalized mixed volume up to 4 that are inclusion-maximal. This produces a classification of generic trivariate sparse polynomial systems with up to 4 solutions in the complex torus, up to monomial changes of variables. By a recent result of Esterov, this leads to a description of all generic trivariate sparse polynomial systems that are solvable by radicals.

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.

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