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A TUTTE POLYNOMIAL FOR TORIC ARRANGEMENTS
"... Abstract. We introduce a multiplicity Tutte polynomial M(x, y), with applications to zonotopes and toric arrangements. We prove that M(x, y) satisfies a deletion-restriction recursion and has positive coefficients. The characteristic polynomial and the Poincaré polynomial of a toric arrangement are ..."
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Abstract. We introduce a multiplicity Tutte polynomial M(x, y), with applications to zonotopes and toric arrangements. We prove that M(x, y) satisfies a deletion-restriction recursion and has positive coefficients. The characteristic polynomial and the Poincaré polynomial of a toric arrangement are shown to be specializations of the associated polynomial M(x, y), likewise the corresponding polynomials for a hyperplane arrangement are specializations of the ordinary Tutte polynomial. Furthermore, M(1,y) is the Hilbert series of the related discrete Dahmen-Micchelli space, while M(x, 1) computes the volume and the number of integer points of the associated zonotope. Ad Alessandro Pucci, che ha ripreso in mano il timone della propria vita. 1.
Products of Linear Forms AND TUTTE POLYNOMIALS
, 2009
"... Let ∆ be a finite sequence of n vectors from a vector space over any field. We consider the subspace of Sym(V) spanned by Q v∈S v, where S is a subsequence of ∆. A result of Orlik and Terao provides a doubly indexed direct sum of this space. The main theorem is that the resulting Hilbert series is ..."
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Cited by 13 (1 self)
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Let ∆ be a finite sequence of n vectors from a vector space over any field. We consider the subspace of Sym(V) spanned by Q v∈S v, where S is a subsequence of ∆. A result of Orlik and Terao provides a doubly indexed direct sum of this space. The main theorem is that the resulting Hilbert series is the Tutte polynomial evaluation T(∆; 1+x, y). Results of Ardila and Postnikov, Orlik and Terao, Terao, and Wagner are obtained as corollaries.
Zonotopal algebra
, 2007
"... A wealth of geometric and combinatorial properties of a given linear endomorphism X of IR N is captured in the study of its associated zonotope Z(X), and, by duality, its associated hyperplane arrangement H(X). This well-known line of study is particularly interesting in case n:=rankX ≪ N. We enhanc ..."
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Cited by 12 (1 self)
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A wealth of geometric and combinatorial properties of a given linear endomorphism X of IR N is captured in the study of its associated zonotope Z(X), and, by duality, its associated hyperplane arrangement H(X). This well-known line of study is particularly interesting in case n:=rankX ≪ N. We enhance this study to an algebraic level, and associate X with three algebraic structures, referred herein as external, central, and internal. Each algebraic structure is given in terms of a pair of homogeneous polynomial ideals in n variables that are dual to each other: one encodes properties of the arrangement H(X), while the other encodes by duality properties of the zonotope Z(X). The algebraic structures are defined purely in terms of the combinatorial structure of X, but are subsequently proved to be equally obtainable by applying suitable algebro-analytic operations to either of Z(X) or H(X). The theory is universal in the sense that it requires no assumptions on the map X (the only exception being that the algebro-analytic operations on Z(X) yield sought-for results only in case X is unimodular), and provides new tools that can be used in enumerative combinatorics, graph theory, representation theory, polytope geometry, and approximation theory.
Combinatorial bounds on Hilbert functions of fat points in projective space
- J. Pure Appl. Algebra
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Sagbi bases of Cox-Nagata rings
, 2008
"... We degenerate Cox-Nagata rings to toric algebras by means of sagbi bases induced by configurations over the rational function field. For del Pezzo surfaces, this degeneration implies the Batyrev-Popov conjecture that these rings are presented by ideals of quadrics. For the blow-up of projective n-sp ..."
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Cited by 12 (2 self)
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We degenerate Cox-Nagata rings to toric algebras by means of sagbi bases induced by configurations over the rational function field. For del Pezzo surfaces, this degeneration implies the Batyrev-Popov conjecture that these rings are presented by ideals of quadrics. For the blow-up of projective n-space at n + 3 points, sagbi bases of Cox-Nagata rings establish a link between the Verlinde formula and phylogenetic algebraic geometry, and we use this to answer questions due to D’Cruz-Iarobbino and Buczyńska-Wi´sniewski. Inspired by the zonotopal algebras of Holtz and Ron, our study emphasizes explicit computations, and offers a new approach to Hilbert functions of fat points. 1 Powers of linear forms We fix n vector fields on a d-dimensional space with coordinates (z1,...,zd): ℓj = d∑
Hierarchical zonotopal power ideals
, 2010
"... Abstract. Zonotopal algebra deals with ideals and vector spaces of polynomials that are related to several combinatorial and geometric structures defined by a finite sequence of vectors. Given such a sequence X, an integer k ≥ −1 and an upper set in the lattice of flats of the matroid defined by X, ..."
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Cited by 10 (6 self)
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Abstract. Zonotopal algebra deals with ideals and vector spaces of polynomials that are related to several combinatorial and geometric structures defined by a finite sequence of vectors. Given such a sequence X, an integer k ≥ −1 and an upper set in the lattice of flats of the matroid defined by X, we define and study the associated hierarchical zonotopal power ideal. This ideal is generated by powers of linear forms. Its Hilbert series depends only on the matroid structure of X. It is related to various other matroid invariants, e. g. the shelling polynomial and the characteristic polynomial. This work unifies and generalizes results by Ardila-Postnikov on power ideals and by Holtz-Ron and Holtz-Ron-Xu on (hierarchical) zonotopal algebra. We also generalize a result on zonotopal Cox modules due to Sturmfels-Xu. Résumé. La théorie de l’algèbre “zonotopique ” s’occupe d’idéaux et d’espaces vectoriels de polynômes qui ont un rapport avec plusieurs structures combinatoires et géométriques définies par des suites finies de vecteurs. Étant donné une telle suite X, un nombre entier k ≥ −1 et un ensemble supérieur dans le treillis des plans du matroïde défini par X, nous définissons et étudions l’idéal hiérarchique zonotopique, engendré par des puissances de formes linéaires. Sa série de Hilbert dépend seulement de la structure matroïdale de X. Il existe des relations avec d’autres invariants de matroïdes, tels que le polynôme d’épluchage et le polynôme caractéristique. Ce travail unifie et généralise des résultats d’Ardila-Postnikov sur les idéaux de puissances et de Holtz-Ron et Holtz-Ron-Xu sur l’algèbre zonotopique (hiérarchique). Nous généralisons aussi un résultat sur les modules de Cox zonotopiques, dû à Sturmfels-Xu.
The f-vector of a realizable matroid complex is strictly log-concave
, 2011
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On the weak Lefschetz property for powers of linear forms
- Algebra & Number Theory
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ON THE WARING PROBLEM FOR POLYNOMIAL RINGS
"... Abstract. In this note we discuss an analog of the classical Waring problem for C[x0, x1,..., xn]. Namely, we show that a general homogeneous polynomial p ∈ C[x0, x1,..., xn] of degree divisible by k ≥ 2 can be represented as a sum of at most k n k-th powers of homogeneous polynomials in C[x0, x1,.. ..."
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Abstract. In this note we discuss an analog of the classical Waring problem for C[x0, x1,..., xn]. Namely, we show that a general homogeneous polynomial p ∈ C[x0, x1,..., xn] of degree divisible by k ≥ 2 can be represented as a sum of at most k n k-th powers of homogeneous polynomials in C[x0, x1,..., xn]. Noticeably, k n coincides with the number obtained by naive dimension count. 1.
