scholarly journals Viewing counting polynomials as Hilbert functions via Ehrhart theory

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Felix Breuer ◽  
Aaron Dall
Keyword(s):  
Hilbert Function ◽  
Chromatic Polynomial ◽  
Point Of View ◽  
Hilbert Functions ◽  
Sufficient Criterion ◽  
Ehrhart Polynomial ◽  
Ehrhart Theory ◽  
International Audience ◽  
Polytopal Complex ◽  
Nous Utilisons

International audience Steingrímsson (2001) showed that the chromatic polynomial of a graph is the Hilbert function of a relative Stanley-Reisner ideal. We approach this result from the point of view of Ehrhart theory and give a sufficient criterion for when the Ehrhart polynomial of a given relative polytopal complex is a Hilbert function in Steingrímsson's sense. We use this result to establish that the modular and integral flow and tension polynomials of a graph are Hilbert functions. Steingrímsson (2001) a montré que le polynôme chromatique d'un graphe est la fonction de Hilbert d'un idéal relatif de Stanley-Reisner. Nous abordons ce résultat du point de vue de la théorie d'Ehrhart et donnons un critère suffisant pour que le polynôme d'Ehrhart d'un complexe polytopal relatif donné soit une fonction de Hilbert au sens de Steingrímsson. Nous utilisons ce résultat pour établir que les polynômes de flux et de tension modulaires et intégraux d'un graphe sont des fonctions de Hilbert.

scholarly journals Quasisymmetric functions from combinatorial Hopf monoids and Ehrhart Theory

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Jacob White
Keyword(s):  
Commutative Algebra ◽  
Hopf Algebras ◽  
Simplicial Complexes ◽  
Hilbert Functions ◽  
Quasisymmetric Functions ◽  
Ehrhart Theory ◽  
International Audience ◽  
Hopf Monoid

International audience We investigate quasisymmetric functions coming from combinatorial Hopf monoids. We show that these invariants arise naturally in Ehrhart theory, and that some of their specializations are Hilbert functions for relative simplicial complexes. This class of complexes, called forbidden composition complexes, also forms a Hopf monoid, thus demonstrating a link between Hopf algebras, Ehrhart theory, and commutative algebra. We also study various specializations of quasisymmetric functions.

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 Bundles on with vanishing lower cohomologies

2020 ◽  
pp. 1-20
Author(s):  
Mengyuan Zhang
Keyword(s):  
Hilbert Function ◽  
Betti Numbers ◽  
Projective Spaces ◽  
Hilbert Functions ◽  
Free Resolutions ◽  
Rational Varieties ◽  
Minimal Free Resolutions ◽  
Graded Lattice

Abstract We study bundles on projective spaces that have vanishing lower cohomologies using their short minimal free resolutions. We partition the moduli $\mathcal{M}$ according to the Hilbert function H and classify all possible Hilbert functions H of such bundles. For each H, we describe a stratification of $\mathcal{M}_H$ by quotients of rational varieties. We show that the closed strata form a graded lattice given by the Betti numbers.

Fat Points in ℙ1× ℙ1and Their Hilbert Functions

2004 ◽  
Vol 56 (4) ◽  
pp. 716-741 ◽  
Author(s):  
Elena Guardo ◽  
Adam Van Tuyl
Keyword(s):  
Finite Number ◽  
Hilbert Function ◽  
Free Resolution ◽  
Hilbert Functions ◽  
Minimal Free Resolution ◽  
Fat Points ◽  
Fat Point ◽  
Point Scheme

AbstractWe study the Hilbert functions of fat points in ℙ1× ℙ1. IfZ⊆ ℙ1× ℙ1is an arbitrary fat point scheme, then it can be shown that for everyiandjthe values of the Hilbert functionHZ(l,j) andHZ(i,l) eventually become constant forl≫ 0. We show how to determine these eventual values by using only the multiplicities of the points, and the relative positions of the points in ℙ1× ℙ1. This enables us to compute all but a finite number values ofHZwithout using the coordinates of points. We also characterize the ACM fat point schemes using our description of the eventual behaviour. In fact, in the case thatZ⊆ ℙ1× ℙ1is ACM, then the entire Hilbert function and its minimal free resolution depend solely on knowing the eventual values of the Hilbert function.

scholarly journals The Pantelides algorithm for delay differential-algebraic equations

2020 ◽  
Vol 4 (1) ◽  
Author(s):  
Ines Ahrens ◽  
Benjamin Unger
Keyword(s):  
Differential Algebraic Equations ◽  
Equivalence Classes ◽  
Point Of View ◽  
Algebraic Equations ◽  
Sufficient Criterion ◽  
Differential Algebraic Equation ◽  
Graph Theoretical Approach ◽  
Delay Differential ◽  
Necessary And Sufficient ◽  
Structural Point

Abstract We present a graph-theoretical approach that can detect which equations of a delay differential-algebraic equation (DDAE) need to be differentiated or shifted to construct a solution of the DDAE. Our approach exploits the observation that differentiation and shifting are very similar from a structural point of view, which allows us to generalize the Pantelides algorithm for differential-algebraic equations to the DDAE setting. The primary tool for the extension is the introduction of equivalence classes in the graph of the DDAE, which also allows us to derive a necessary and sufficient criterion for the termination of the new algorithm.

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 A preorder-free construction of the Kazhdan-Lusztig representations of $S_n$, with connections to the Clausen representations

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Charles Buehrle ◽  
Mark Skandera
Keyword(s):  
Polynomial Ring ◽  
Transition Matrices ◽  
Free Construction ◽  
Nous Montrons ◽  
International Audience ◽  
Original Construction ◽  
Nous Utilisons ◽  
Invent Math

International audience We use the polynomial ring $\mathbb{C}[x_{1,1},\ldots,x_{n,n}]$ to modify the Kazhdan-Lusztig construction of irreducible $S_n$-modules. This modified construction produces exactly the same matrices as the original construction in [$\textit{Invent. Math}$ $\mathbf{53}$ (1979)], but does not employ the Kazhdan-Lusztig preorders. We also show that our modules are related by unitriangular transition matrices to those constructed by Clausen in [$\textit{J. Symbolic Comput.}$ $\textbf{11}$ (1991)]. This provides a $\mathbb{C}[x_{1,1},\ldots,x_{n,n}]$-analog of results of Garsia-McLarnan in [$\textit{Adv. Math.}$ $\textbf{69}$ (1988)]. Nous utilisons l'anneau $\mathbb{C}[x_{1,1},\ldots,x_{n,n}]$ pour modifier la construction Kazhdan-Lusztig des modules-$S_n$ irréductibles dans $\mathbb{C}[S_n]$. Cette construction modifiée produit exactement les mêmes matrices que la construction originale dans [$\textit{Invent. Math}$ $\mathbf{53}$ (1979)], mais sans employer les préordres de Kazhdan-Lusztig. Nous montrons aussi que nos modules sont reliés par des matrices unitriangulaires aux modules construits par Clausen dans [$\textit{J. Symbolic Comput.}$ $\textbf{11}$ (1991)]. Ce résultat donne un $\mathbb{C}[x_{1,1},\ldots,x_{n,n}]$-analogue des résultats de Garsia-McLarnan dans [$\textit{Adv. Math.}$ $\textbf{69}$ (1988)].

scholarly journals Counting Quiver Representations over Finite Fields Via Graph Enumeration

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Geir Helleloid ◽  
Fernando Rodriguez-Villegas
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Quiver Representations ◽  
Dimension Vector ◽  
Compte Rendu ◽  
Isomorphism Classes ◽  
International Audience ◽  
Derivatives Of ◽  
Nous Utilisons ◽  
Complete Account

International audience Let $\Gamma$ be a quiver on $n$ vertices $v_1, v_2, \ldots , v_n$ with $g_{ij}$ edges between $v_i$ and $v_j$, and let $\boldsymbol{\alpha} \in \mathbb{N}^n$. Hua gave a formula for $A_{\Gamma}(\boldsymbol{\alpha}, q)$, the number of isomorphism classes of absolutely indecomposable representations of $\Gamma$ over the finite field $\mathbb{F}_q$ with dimension vector $\boldsymbol{\alpha}$. We use Hua's formula to show that the derivatives of $A_{\Gamma}(\boldsymbol{\alpha}, q)$ with respect to $q$, when evaluated at $q = 1$, are polynomials in the variables $g_{ij}$, and we can compute the highest degree terms in these polynomials. The formulas for these coefficients depend on the enumeration of certain families of connected graphs. This note simply gives an overview of these results; a complete account of this research is available on the arXiv and has been submitted for publication. Soit $\Gamma$ un carquois sur $n$ sommets $ v_1, v_2, \ldots , v_n$ avec $g_{ij}$ arêtes entre $v_i$ et $v_j$, et soit $\boldsymbol{\alpha} \in \mathbb{N}^n$. Hua a donné une formule pour $A_{\Gamma}(\boldsymbol{\alpha}, q)$, le nombre de classes d'isomorphisme absolument indécomposables de représentations de $\Gamma$ sur le corps fini $\mathbb{F}_q$ avec vecteur de dimension $\boldsymbol{\alpha}$. Nous utilisons la formule de Hua pour montrer que les dérivées de $A_{\Gamma}(\boldsymbol{\alpha}, q)$ par rapport à $q$, alors évaluée à $q=1$, sont des polynômes dans les variables $g_{ij}$, et on peut calculer les termes de plus haut degré de ces polynômes. Les formules pour ces coefficients dépendent de l'énumération de certaines familles de graphes connectés. Cette note donne simplement un aperçu de ces résultats, un compte rendu complet de cette recherche est disponible sur arXiv et a été soumis pour publication.

scholarly journals The short toric polynomial

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Gábor Hetyei
Keyword(s):  
Lower Bound ◽  
Reflection Principle ◽  
Lattice Path ◽  
Simple Polytope ◽  
Independent Form ◽  
International Audience ◽  
Bound Theorem ◽  
Lattice Path Enumeration ◽  
Nous Utilisons ◽  
Eulerian Poset

International audience We introduce the short toric polynomial associated to a graded Eulerian poset. This polynomial contains the same information as Stanley's pair of toric polynomials, but allows different algebraic manipulations. Stanley's intertwined recurrence may be replaced by a single recurrence, in which the degree of the discarded terms is independent of the rank. A short toric variant of the formula by Bayer and Ehrenborg, expressing the toric h-vector in terms of the cd-index, may be stated in a rank-independent form, and it may be shown using weighted lattice path enumeration and the reflection principle. We use our techniques to derive a formula expressing the toric h-vector of a dual simplicial Eulerian poset in terms of its f-vector. This formula implies Gessel's formula for the toric h-vector of a cube, and may be used to prove that the nonnegativity of the toric h-vector of a simple polytope is a consequence of the Generalized Lower Bound Theorem holding for simplicial polytopes. Nous introduisons le polynôme torique court associé à un ensemble ordonné Eulérien. Ce polynôme contient la même information que le couple de polynômes toriques de Stanley, mais il permet des manipulations algébriques différentes. La récurrence entrecroisée de Stanley peut être remplacée par une seule récurrence dans laquelle le degré des termes écartés est indépendant du rang. La variante torique courte de la formule de Bayer et Ehrenborg, qui exprime le vecteur torique d'un ensemble ordonné Eulérien en termes de son cd-index, est énoncée sous une forme qui ne dépend pas du rang et qui peut être démontrée en utilisant une énumération des chemins pondérés et le principe de réflexion. Nous utilisons nos techniques pour dériver une formule exprimant le vecteur h-torique d'un ensemble ordonné Eulérien dont le dual est simplicial, en termes de son f-vecteur. Cette formule implique la formule de Gessel pour le vecteur h-torique d'un cube, et elle peut être utilisée pour démontrer que la positivité du vecteur h-torique d'un polytope simple est une conséquence du Théorème de la Borne Inférieure Généralisé appliqué aux polytopes simpliciaux.

scholarly journals A stochastic modelling of phytoplankton aggregation

2006 ◽  
Vol Volume 5, Special Issue TAM... ◽  
Author(s):  
Nadjia El Saadi ◽  
Ovide Arino
Keyword(s):  
Differential Equation ◽  
Stochastic Partial Differential Equation ◽  
Stochastic Modelling ◽  
Point Of View ◽  
Modèle Mathématique ◽  
Initial Number ◽  
Microscopic Description ◽  
Random Dispersal ◽  
International Audience ◽  
Number Of Cells

International audience The aim of this work is to provide a stochastic mathematical model of aggregation in phytoplankton, from the point of view of modelling a system of a large but finite number of phytoplankton cells that are subject to random dispersal, mutual interactions allowing the cell motions some dependence and branching (cell division or death). We present the passage from the ''microscopic'' description to the ''macroscopic'' one, when the initial number of cells tends to infinity (large phytoplankton populations). The limit of the system is an extension of the Dawson-Watanabe superprocess: it is a superprocess with spatial interactions which can be described by a nonlinear stochastic partial differential equation. L'objectif de ce travail est de fournir un modèle mathématique stochastique qui décrit l'aggrégation du hytoplancton,à partir de la modélisation d'un système de grande taille, mais finie, de cellules de phytoplancton sujettes à une dispersion aléatoire, des interactions spatiales qui donnent aux mouvements des cellules une certaine dépendance et un branchement (division cellulaire ou mort). Nous présentons le passage de la description microscopique à une description macroscopique, lorsque le nombre de cellules devient très grand (grandes populations de phytoplancton). La limite du système est une extension du superprocessus de Dawson-Watanabe: c'est un superprocessus avec interactions qui peut être décrit par une équation aux dérivées partielles stochastique non linéaire.

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