Computing the continuous discretely. Integer-point enumeration in polyhedra. (English) Zbl 1114.52013

Beck, Matthias; Robins, Sinai

Undergraduate Texts in Mathematics. New York, NY: Springer (ISBN 978-0-387-29139-0/hbk; 978-0-387-46112-0/ebook). xviii, 226 p. (2007).

Pick’s Theorem gives the area of an integral convex polygon in terms of the number of integer points in the interior and the number of integer points on the boundary of the polygon. Intuitively, the number of integer points in a convex \(d\)-dimensional polytope is a good estimate (the “discrete volume”) for the \(d\)-dimensional volume of the polytope. This book is concerned with the mathematics of that connection between the discrete and the continuous, with significance for geometry, number theory and combinatorics.
The authors give a coherent and tightly developed picture that encompasses the Frobenius coin-exchange problem, Bernoulli polynomials, Ehrhart polynomials for lattice point enumeration, Dehn-Sommerville relations on numbers of faces of polytopes, magic squares, the Birkhoff-von Neumann polytope, finite Fourier series, Dedekind sums, Brion’s Theorem on decomposition of polytopes, Euler-Maclaurin summation, volume via solid angles, and Green’s Theorem in the plane.
The book is written as an accessible and engaging textbook, with many examples, historical notes, pithy quotes, commentary integrating the material, exercises, open problems and an extensive bibliography.

Reviewer: Margaret M. Bayer (Lawrence)

MSC:

52B20	Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
11P21	Lattice points in specified regions
05A15	Exact enumeration problems, generating functions
11F20	Dedekind eta function, Dedekind sums
11H06	Lattices and convex bodies (number-theoretic aspects)
52-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to convex and discrete geometry
11-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
05-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to combinatorics
52B05	Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)

Keywords:

convex polytope; generating function; Ehrhart polynomial; volume; Frobenius number; magic square; finite Frobenius series; Dedekind sum; solid angle

Software:

barvinok; LattE; OEIS