Commuting matrices: Difference between revisions

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[[File:IP_polytope_with_LP_relaxation.png|right|thumb|300px|The red dots are the integer lattice points within the blue polygon, the latter representing a two-dimensional [[linear program]] ]]
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Study of '''integer points in convex polyhedra'''<ref>In some contexts convex polyhedra are called simply "polyhedra".</ref> is motivated by the questions, such as "how many nonnegative [[integer]]-valued solutions does a [[system of linear equations]] with nonnegative coefficients have" or "how many solutions does an [[integer linear program]] have". Counting integer points in [[polyhedra]] or other questions about them arise in [[representation theory]], [[commutative algebra]], [[algebraic geometry]],  [[statistics]], and [[computer science]].<ref>[http://math.sfsu.edu/beck/src06.html Integer points in polyhedra. Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics], ACM--SIAM Joint Summer Research Conference, 2006</ref>
 
The set of integer points, or, more generally, the set of points of an [[affine lattice]],  in a polyhedron is called '''Z-polyhedron''',<ref>The term "Z-polyhedron" is also used as a synonym to [[convex lattice polytope]], the [[convex hull]] of finitely many points in an affine lattice.</ref> from the mathematical notation <math>\mathbb{Z}</math> or '''Z''' for the set of integer numbers.<ref>"Computations on Iterated Spaces" in: The Compiler Design Handbook: Optimizations and Machine Code Generation, [[CRC Press]] 2007, 2nd edition, ISBN 1-4200-4382-X, [http://books.google.com/books?id=1kqAv-uDEPEC&pg=PT465&dq=%22Z-polyhedron%22#v=onepage&q=%22Z-polyhedron%22&f=false p.15-7] </ref>
 
==Properties==
For a lattice &Lambda;, [[Minkowski's theorem]] relates the number d(&Lambda;) and the volume of a symmetric [[convex set]] ''S'' to the number of lattice points contained in ''S''.  
 
The number of lattice points contained in a [[polytope]] all of whose vertices are elements of the lattice is described by the polytope's [[Ehrhart polynomial]]. Formulas for some of the coefficients of this polynomial involve d(&Lambda;) as well.
 
==Applications==
===Loop optimization===
In certain approaches to [[loop optimization]], the set of the executions of the loop body is viewed as the set of integer points in a polyhedron defined by loop constraints.
 
==See also==
*[[Convex lattice polytope]]
*[[Pick's theorem]]
 
==References and notes==
{{reflist}}
 
==Further reading==
*"Integer Points In Polyhedra: Geometry, Number Theory, Algebra, Optimization: Proceedings of an AMS-IMS-SIAM Joint Summer Research Conference on Integer Points in Polyhedra, 2003 (''[[Contemporary Mathematics]]'' series, v. 374), 2005, ISBN 0-8218-3459-2 
*[[Alexander Barvinok]], ''Integer Points in Polyhedra'', 2008, ISBN 3-03719-052-3
 
[[Category:Lattice points]]
[[Category:Linear algebra]]
[[Category:Linear programming]]
[[Category:Polyhedra| ]]
[[Category:Polytopes]]

Latest revision as of 20:12, 12 December 2014

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