# Integer Points in Polyhedra

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*Alexander Barvinok*

A publication of the European Mathematical Society

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.