Journal ArticleJournal of Pure and Applied Algebra · March 1, 2025
A canonical minimal free resolution of an arbitrary co-artinian lattice ideal over the polynomial ring is constructed over any field whose characteristic is 0 or any but finitely many positive primes. The differential has a closed-form combinatorial descri ...
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Journal ArticleFoundations of Computational Mathematics · December 1, 2019
Can an ideal I in a polynomial ring k[x] over a field be moved by a change of coordinates into a position where it is generated by binomials xA- λxb with λ∈ k, or by unital binomials (i.e., with λ= 0 or 1)? Can a variety be moved into a position where it i ...
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Journal ArticleAnnals of Applied Statistics · 2016
New representations of tree-structured data objects, using ideas from
topological data analysis, enable improved statistical analyses of a population
of brain artery trees. A number of representations of each data tree arise from
persistence diagrams that ...
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Journal ArticleAdvances in Applied Mathematics · July 1, 2015
This paper investigates the computational geometry relevant to calculations of the Fréchet mean and variance for probability distributions on the phylogenetic tree space of Billera, Holmes and Vogtmann, using the theory of probability measures on spaces of ...
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Journal ArticlePacific Journal of Mathematics · January 1, 2015
We give an elementary proof of holonomicity for A-hypergeometric systems, with no requirements on the behavior of their singularities, a result originally due to Adolphson (1994) after the regular singular case by Gelfand and Gelfand (1986). Our method yie ...
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Journal ArticleSIAM Journal on Applied Dynamical Systems · January 1, 2014
This paper introduces the class of strongly endotactic networks, a subclass of the endotactic networks introduced by Craciun, Nazarov, and Pantea. The main result states that the global attractor conjecture holds for complex-balanced systems that are stron ...
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Journal ArticleJournal of Mathematical Imaging and Vision · January 1, 2014
Statistical analysis of magnetic resonance angiography (MRA) brain artery trees is performed using two methods for mapping brain artery trees to points in phylogenetic treespace: cortical landmark correspondence and descendant correspondence. The differenc ...
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Journal ArticleJournal of Algebraic Combinatorics · February 1, 2013
Given a morphism from an affine semigroup to an arbitrary commutative monoid, it is shown that every fiber possesses an affine stratification: a partition into a finite disjoint union of translates of normal affine semigroups. The proof rests on mesoprimar ...
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Journal ArticleAlgebra & Number Theory · 2012
We demonstrate how primary decomposition of
commutative monoid congruences fails to
capture the essence of primary
decomposition in commutative rings by
exhibiting a more sensitive theory of
mesoprimary decomposition of
congruences, complete with witnesses ...
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Journal ArticleSIGMA (Symmetry, Integrability, and Geometry: Methods and Applications) · 2012
Motivated by questions in mass-action
kinetics, we introduce the notion of
vertexical family of differential
inclusions. Defined on open hypercubes,
these families are characterized by
particular good behavior under projection
maps. The motivating exampl ...
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Journal ArticleMathematische Annalen · December 1, 2011
The face ring of a simplicial complex modulo m generic linear forms is shown to have finite local cohomology if and only if the link of every face of dimension m or more is nonsingular, i.e., has the homology of a wedge of spheres of the expected dimension ...
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ConferenceCombinatorial Aspects of Commutative Algebra and Algebraic Geometry: The Abel Symposium 2009 · December 1, 2011
This survey of methods surrounding lattice point methods for binomial ideals begins with a leisurely treatment of the geometric combinatorics of binomial primary decomposition. It then proceeds to three independent applications whose motivations come from ...
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Journal ArticleAdvances in Applied Mathematics · January 1, 2011
We encode arbitrary finite impartial combinatorial games in terms of lattice points in rational convex polyhedra. Encodings provided by these lattice games can be made particularly efficient for octal games, which we generalize to squarefree games. These e ...
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Journal ArticleJournal of the European Mathematical Society · January 1, 2011
We prove the conjectures of Graham-Kumar [GrKu08] and Griffeth-Ram [GrRa04] concerning the alternation of signs in the structure constants for torus-equivariant K-theory of generalized flag varieties G/P. These results are immediate consequences of an equi ...
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Journal ArticleMathematische Zeitschrift · April 1, 2010
An explicit lattice point realization is provided for the primary components of an arbitrary binomial ideal in characteristic zero. This decomposition is derived from a characteristic-free combinatorial description of certain primary components of binomial ...
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Journal ArticleDuke Mathematical Journal · January 1, 2010
We study quotients of the Weyl algebra by left ideals whose generators consist of an arbitrary Zd-graded binomial ideal I in C[∂1∂n] along with Euler operators defined by the grading and a parameter βεCd 2 Cd. We determine the parameters β for which these ...
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Journal ArticleJournal fur die Reine und Angewandte Mathematik · May 1, 2009
We relate a classic algebro-geometric degeneration technique, dating at least to Hodge 1941 (J. London Math. Soc. 16: 245-255), to the notion of vertex decompositions of simplicial complexes. The good case is when the degeneration is reduced, and we call t ...
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Journal ArticleMathematical Research Letters · January 1, 2008
Fix nonzero ideal sheaves a1, . . . ., ar and b on a normal ℚ-Gorenstein complex variety X. For any positive real numbers α and β, we construct a resolution of the multiplier ideal script T((a1 + . . . + ar)αbβ) by sheaves that are direct sums of multiplie ...
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Journal ArticleIsrael Journal of Mathematics · January 1, 2008
Let X, Y be finite sets and T a set of functions X → Y which we will call " tableaux". We define a simplicial complex whose facets, all of the same dimension, correspond to these tableaux. Such tableau complexes have many nice properties, and are frequentl ...
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Journal ArticleDiscrete and Computational Geometry · January 1, 2008
Let S be the boundary of a convex polytope of dimension d+1, or more generally let S be a convex polyhedral pseudomanifold. We prove that S has a polyhedral nonoverlapping unfolding into ℝd, so the metric space S is obtained from a closed (usually nonconve ...
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Journal ArticleJournal of Algebraic Geometry · January 1, 2008
Fix a variety X with a transitive (left) action by an algebraic group G. Let ε and ℱ be coherent sheaves on X. We prove that for elements g in a dense open subset of G, the sheaf Tor¡X- (ε, gℱ) vanishes for all i > 0. When ε and ℱ are structure sheaves of ...
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Journal ArticleMathematical Research Letters · 2008
Fix nonzero ideal sheaves a1, . . . ., ar and b on a normal ℚ-Gorenstein complex variety X. For any positive real numbers α and β, we construct a resolution of the multiplier ideal script T((a1 + . . . + ar)αbβ) by sheaves that are direct sums of multiplie ...
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Journal ArticleInventiones Mathematicae · November 1, 2006
We give four positive formulae for the (equioriented type A) quiver polynomials of Buch and Fulton [BF99 ]. All four formulae are combinatorial, in the sense that they are expressed in terms of combinatorial objects of certain types: Zelevinsky permutation ...
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Journal ArticleProceedings of the American Mathematical Society · May 1, 2006
Let A be an integer d × n matrix, and assume that the convex hull conv(A) of its columns is a simplex of dimension d - 1 not containing the origin. It is known that the semigroup ring ℂ[Ndbl;A] is Cohen-Macaulay if and only if the rank of the GKZ hypergeom ...
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Journal ArticleCommunications in Algebra · February 1, 2006
M. Masuda recently provided the missing piece proving a conjecture of R.P. Stanley on the characterization of f-vectors for Gorenstein *simplicial posets. We propose a slight simplification of Masuda's proof. ...
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Journal ArticleDuke Mathematical Journal · May 15, 2005
The main theorem here is the K-theoretic analogue of the cohomological "stable double component formula" for quiver polynomials in [KMS]. This K-theoretic version is still in terms of lacing diagrams, but nonminimal diagrams contribute terms of higher degr ...
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Journal ArticleAdvances in Mathematics · May 1, 2005
This note constructs the flat toric degeneration of the manifold ℱℓn of flags in ℂn due to Gonciulea and Lakshmibai (Transform. Groups 1(3) (1996) 215) as an explicit GIT quotient of the Gröbner degeneration due to Knutson and Miller (Gröbner geometry of S ...
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Journal ArticleJournal of Symbolic Computation · March 1, 2005
Let Q be an affine semigroup generating ℤd, and fix a finitely generated ℤd-graded module M over the semigroup algebra k[Q] for a field k. We provide an algorithm to compute a minimal ℤd-graded injective resolution of M up to any desired cohomological degr ...
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Journal ArticleElectronic Journal of Combinatorics · January 7, 2005
We extend a reciprocity theorem of Stanley about enumeration of integer points in polyhedral cones when one exchanges strict and weak inequalities. The proof highlights the roles played by Cohen-Macaulayness and canonical modules. The extension raises the ...
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Journal ArticleAnnals of Mathematics · January 1, 2005
Given a permutation w ∈ Sn, we consider a determinantal ideal Iw whose generators are certain minors in the generic n × n matrix (filled with independent variables). Using 'multidegrees' as simple algebraic substitutes for torus-equivariant cohomology clas ...
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Journal ArticleAdvances in Mathematics · May 1, 2004
Let (Π, Σ) be a Coxeter system. An ordered list of elements in Σ and an element in Π determine a subword complex, as introduced in Knutson and Miller (Ann. of Math. (2) (2003), to appear). Subword complexes are demonstrated here to be homeomorphic to balls ...
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Journal ArticleJournal of Combinatorial Theory. Series A · January 1, 2003
Mitosis is a rule introduced by Knutson and Miller for manipulating subsets of the n × n grid. It provides an algorithm that lists the reduced pipe dreams (also known as rc-graphs) of Fomin and Kirillov for a permutation w ∈ Sn by downward induction on wea ...
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Journal ArticlePacific Journal of Mathematics · January 1, 2003
Given a module M over a ring R that has a grading by a semigroup Q, we present a spectral sequence that computes the local cohomology HIi(M) at any graded ideal I in terms of Ext modules. We use this method to obtain flniteness results for the local cohomo ...
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Journal ArticleMathematical Research Letters · January 1, 2002
For a radical monomial ideal I in a normal semigroup ring κ[Q], there is a unique minimal irreducible resolution 0 → κ[Q]/I → W̄0 → W̄1 ... by modules W̄i of the form ⊕jk[Fij], where the Fij are (not necessarily distinct) faces of Q. That is, W̄i is a direct s ...
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Journal ArticleDocumenta Mathematica · January 1, 2002
We introduce the notion of rigid embedding in a grid surface, a new kind of plane drawing for simple triconnected planar graphs. Rigid embeddings provide methods to (1) find well-structured (cellular, here) minimal free resolutions for arbitrary monomial i ...
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Chapter · January 1, 2001
The duality theorem of Greenlees and May relating local cohomology with support on an ideal I and the left derived functors of J-adic completion [GM92) holds for rather general ideals in commutative rings. Here, simple formulas are provided for both local ...
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Journal ArticleJournal of Algebra · September 1, 2000
Alexander duality is made into a functor which extends the notion for monomial ideals to any finitely generated Nn-graded module. The functors associated with Alexander duality provide a duality on the level of free and injective resolutions, and numerous ...
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Journal ArticleJournal of Symbolic Computation · January 1, 2000
Monomial ideals which are generic with respect to either their generators or irreducible components have minimal free resolutions encoded by simplicial complexes. There are numerous equivalent ways to say that a monomial ideal is generic or cogeneric. For ...
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Journal ArticleDiscrete and Computational Geometry · January 1, 2000
It is possible to construct a figure in three dimensions which is combinatorially equivalent to a regular icosahedron, and whose faces are all congruent but not equilateral. Such icosamonohedra can be convex or nonconvex, and can be deformed continuously. ...
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Journal ArticleCompositio Mathematica
Building on coprincipal mesoprimary decomposition [Kahle and Miller,
2014], we combinatorially construct an irreducible decomposition of
any given binomial ideal. In a parallel manner, for congruences in
commutative monoids we construct decompositions tha ...
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