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Integer Points in Polyhedra
Alexander Barvinok, University of Michigan, Ann Arbor, MI
A publication of the European Mathematical Society.
cover
Zurich Lectures in Advanced Mathematics
2008; 200 pp; softcover
Volume: 9
ISBN-10: 3-03719-052-3
ISBN-13: 978-3-03719-052-4
List Price: US$44
Member Price: US$35.20
Order Code: EMSZLEC/9
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This is a self-contained exposition of several core aspects of the theory of rational polyhedra with a view towards algorithmic applications to efficient counting of integer points, a problem arising in many areas of pure and applied mathematics. The approach is based on the consistent development and application of the apparatus of generating functions and the algebra of polyhedra. Topics range from classical, such as the Euler characteristic, continued fractions, Ehrhart polynomial, Minkowski Convex Body Theorem, and the Lenstra-Lenstra-Lovász lattice reduction algorithm, to recent advances such as the Berline-Vergne local formula.

The text is intended for graduate students and researchers. Prerequisites are a modest background in linear algebra and analysis as well as some general mathematical maturity. Numerous figures, exercises of varying degree of difficulty as well as references to the literature and publicly available software make the text suitable for a graduate course.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

Readership

Graduate students and research mathematicians interested in geometry and topology.

Table of Contents

  • Introduction
  • The algebra of polyhedra
  • Linear transformations and polyhedra
  • The structure of polyhedra
  • Polarity
  • Tangent cones. Decompositions modulo polyhedra with lines
  • Open polyhedra
  • The exponential valuation
  • Computing volumes
  • Lattices, bases, and parallelepipeds
  • The Minkowski Convex Body Theorem
  • Reduced basis
  • Exponential sums and generating functions
  • Totally unimodular polytopes
  • Decomposing a 2-dimensional cone into unimodular cones via continued fractions
  • Decomposing a rational cone of an arbitrary dimension into unimodular cones
  • Efficient counting of integer points in rational polytopes
  • The polynomial behavior of the number of integer points in polytopes
  • A valuation on rational cones
  • A "local" formula for the number of integer points in a polytope
  • Bibliography
  • Index
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