Normal polytopes, triangulations, and Koszul algebras.

Bruns, Winfried; Gubeladze, Joseph; NguyenKhac, V.

doi:10.1515/crll.1997.485.123

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

1997

DOI: 10.1515/crll.1997.485.123

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Normal polytopes, triangulations, and Koszul algebras.

Winfried Bruns¹,

Joseph Gubeladze²,

V. NguyenKhac³

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Cited by 34 publications

References 17 publications

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Hilbert Bases, Unimodular Triangulations, and Binary Covers of Rational Polyhedral Cones

Firla

Ziegler

1999

Discrete Comput Geom

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We present a hierarchy of covering properties of rational convex cones with respect to the unimodular subcones spanned by the Hilbert basis.For two of the concepts from the hierarchy we derive characterizations: a description of partitions that leads to a natural integer programming formulation for the HILBERT PAR-TITION problem, and a characterization of "binary covers" that admits a linear algebra test over GF(2) for the existence of BINARY HILBERT COVERS.Implementation of our test leads to interesting new examples, among them: cones that have a HILBERT PARTITION but no REGULAR one; a four-dimensional cone with unimodular facets that has no HILBERT PARTITION; and two five-dimensional cones that do not have any BINARY HILBERT COVER.

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Hilbert Bases, Unimodular Triangulations, and Binary Covers of Rational Polyhedral Cones

Firla

Ziegler

1999

Discrete Comput Geom

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Problems and algorithms for affine semigroups

Bruns

Gubeladze²,

Trung

2001

Semigroup Forum

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Lattice polytopes cut out by root systems and the Koszul property

Payne

2009

Advances in Mathematics

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We show that lattice polytopes cut out by root systems of classical type are normal and Koszul, generalizing a well-known result of Bruns, Gubeladze, and Trung in type A. We prove similar results for Cayley sums of collections of polytopes whose Minkowski sums are cut out by root systems. The proofs are based on a combinatorial characterization of diagonally split toric varieties. arXiv:0805.1252v2 [math.CO]

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Normal polytopes, triangulations, and Koszul algebras.

Cited by 34 publications

References 17 publications

Hilbert Bases, Unimodular Triangulations, and Binary Covers of Rational Polyhedral Cones

Hilbert Bases, Unimodular Triangulations, and Binary Covers of Rational Polyhedral Cones

Problems and algorithms for affine semigroups

Lattice polytopes cut out by root systems and the Koszul property

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