Normal polytope (original) (raw)
In mathematics, specifically in combinatorial commutative algebra, a convex lattice polytope P is called normal if it has the following property: given any positive integer n, every lattice point of the dilation nP, obtained from P by scaling its vertices by the factor n and taking the convex hull of the resulting points, can be written as the sum of exactly n lattice points in P. This property plays an important role in the theory of toric varieties, where it corresponds to projective normality of the toric variety determined by P. Normal polytopes have popularity in algebraic combinatorics. These polytopes also represent the homogeneous case of the Hilbert bases of finite positive rational cones and the connection to algebraic geometry is that they define projectively normal embeddings o
| Property | Value |
|---|---|
| dbo:abstract | In mathematics, specifically in combinatorial commutative algebra, a convex lattice polytope P is called normal if it has the following property: given any positive integer n, every lattice point of the dilation nP, obtained from P by scaling its vertices by the factor n and taking the convex hull of the resulting points, can be written as the sum of exactly n lattice points in P. This property plays an important role in the theory of toric varieties, where it corresponds to projective normality of the toric variety determined by P. Normal polytopes have popularity in algebraic combinatorics. These polytopes also represent the homogeneous case of the Hilbert bases of finite positive rational cones and the connection to algebraic geometry is that they define projectively normal embeddings of toric varieties. (en) |
| dbo:wikiPageExternalLink | http://math.sfsu.edu/gubeladze/publications/kripo/kripo.pdf |
| dbo:wikiPageID | 11391827 (xsd:integer) |
| dbo:wikiPageLength | 8542 (xsd:nonNegativeInteger) |
| dbo:wikiPageRevisionID | 1099637176 (xsd:integer) |
| dbo:wikiPageWikiLink | dbr:Bernd_Sturmfels dbr:Cancellative dbr:Invariant_(mathematics) dbr:Commutative dbr:Mathematics dbr:Ehrhart_polynomial dbr:Monoid dbr:Convex_cone dbr:Convex_hull dbr:Simplex dbr:Combinatorial_commutative_algebra dbr:Hall's_marriage_theorem dbr:Algebraic_geometry dbr:Ambient_space dbr:Euclidean_space dbr:Number_theory dbr:Gordan's_lemma dbr:Hilbert_basis_(linear_programming) dbr:Isomorphism dbr:Ring_theory dbr:Hyperplane dbr:Abelian_group dbc:Polytopes dbr:Birkhoff_polytope dbr:Toric_variety dbr:Polygons dbr:Convex_lattice_polytope dbr:Grothendieck_group dbr:Unimodular_matrix dbr:Rational_cone dbr:Polytope dbr:Affine_subspace dbr:Edge_vector dbr:Projective_normality dbr:Rational_pointed_cone dbr:Unimodular_simplex |
| dbp:wikiPageUsesTemplate | dbt:Math dbt:Reflist dbt:Isbn |
| dct:subject | dbc:Polytopes |
| rdfs:comment | In mathematics, specifically in combinatorial commutative algebra, a convex lattice polytope P is called normal if it has the following property: given any positive integer n, every lattice point of the dilation nP, obtained from P by scaling its vertices by the factor n and taking the convex hull of the resulting points, can be written as the sum of exactly n lattice points in P. This property plays an important role in the theory of toric varieties, where it corresponds to projective normality of the toric variety determined by P. Normal polytopes have popularity in algebraic combinatorics. These polytopes also represent the homogeneous case of the Hilbert bases of finite positive rational cones and the connection to algebraic geometry is that they define projectively normal embeddings o (en) |
| rdfs:label | Normal polytope (en) |
| owl:sameAs | freebase:Normal polytope wikidata:Normal polytope https://global.dbpedia.org/id/4su6Q |
| prov:wasDerivedFrom | wikipedia-en:Normal_polytope?oldid=1099637176&ns=0 |
| foaf:isPrimaryTopicOf | wikipedia-en:Normal_polytope |
| is dbo:wikiPageDisambiguates of | dbr:Normal |
| is dbo:wikiPageWikiLink of | dbr:Normal |
| is foaf:primaryTopic of | wikipedia-en:Normal_polytope |