Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra

✍ Scribed by Matthias Beck, Sinai Robins

• Publisher: Springer
• Year: 2015
• Tongue: English
• Leaves: 295
• Series: Undergraduate Texts in Mathematics
• Edition: 2nd
• Category: Library

✦ Synopsis

This richly illustrated textbook explores the amazing interaction between combinatorics, geometry, number theory, and analysis which arises in the interplay between polyhedra and lattices. Highly accessible to advanced undergraduates, as well as beginning graduate students, this second edition is perfect for a capstone course, and adds two new chapters, many new exercises, and updated open problems. For scientists, this text can be utilized as a self-contained tooling device.

The topics include a friendly invitation to Ehrhart’s theory of counting lattice points in polytopes, finite Fourier analysis, the Frobenius coin-exchange problem, Dedekind sums, solid angles, Euler–Maclaurin summation for polytopes, computational geometry, magic squares, zonotopes, and more.

With more than 300 exercises and open research problems, the reader is an active participant, carried through diverse but tightly woven mathematical fields that are inspired by an innocently elementary question: What are the relationships between the continuous volume of a polytope and its discrete volume?

Reviews of the first edition:

“You owe it to yourself to pick up a copy of Computing the Continuous Discretely to read about a number of interesting problems in geometry, number theory, and combinatorics.”

― MAA Reviews

“The book is written as an accessible and engaging textbook, with many examples, historical notes, pithy quotes, commentary integrating the mate

rial, exercises, open problems and an extensive bibliography.”

― Zentralblatt MATH

“This beautiful book presents, at a level suitable for advanced undergraduates, a fairly complete introduction to the problem of counting lattice points inside a convex polyhedron.”

― Mathematical Reviews

“Many departments recognize the need for capstone courses in which graduating students can see the tools they have acquired come together in some satisfying

way. Beck and Robins have written the perfect text for such a course.”

― CHOICE

✦ Table of Contents

Front Matter....Pages i-xx
Front Matter....Pages 1-1
The Coin-Exchange Problem of Frobenius....Pages 3-26
A Gallery of Discrete Volumes....Pages 27-58
Counting Lattice Points in Polytopes: The Ehrhart Theory....Pages 59-88
Reciprocity....Pages 89-100
Face Numbers and the Dehn–Sommerville Relations in Ehrhartian Terms....Pages 101-111
Magic Squares....Pages 113-129
Front Matter....Pages 131-131
Finite Fourier Analysis....Pages 133-148
Dedekind Sums, the Building Blocks of Lattice-Point Enumeration....Pages 149-165
Zonotopes....Pages 167-182
h-Polynomials and h ∗-Polynomials....Pages 183-197
The Decomposition of a Polytope into Its Cones....Pages 199-211
Euler–Maclaurin Summation in ℝd ....Pages 213-225
Solid Angles....Pages 227-239
A Discrete Version of Green’s Theorem Using Elliptic Functions....Pages 241-248
Back Matter....Pages 249-285

✦ Subjects

Combinatorics; Number Theory; Convex and Discrete Geometry; Computational Science and Engineering