Computing the Continuous Discretely
Integerpoint Enumeration in Polyhedra
Authors: Beck, Matthias, Robins, Sinai
Free Preview The authors write with flair and have chosen a unique set of topics
 Places a strong emphasis on computational techniques
 Contains more than 200 exercises and has been heavily classtested
 Includes hints to selected exercises
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 About this Textbook

This muchanticipated textbook illuminates the field of discrete mathematics with examples, theory, and applications of the discrete volume of a polytope. The authors have weaved a unifying thread through basic yet deep ideas in discrete geometry, combinatorics, and number theory. Because there is no other book that puts together all of these ideas in one place, this text is truly a service to the mathematical community.
We encounter here a friendly invitation to the field of "counting integer points in polytopes," also known as Ehrhart theory, and its various connections to elementary finite Fourier analysis, generating functions, the Frobenius coinexchange problem, solid angles, magic squares, Dedekind sums, computational geometry, and more. With 250 exercises and open problems, the reader feels like an active participant, and the authors' engaging style encourages such participation. The many compelling pictures that accompany the proofs and examples add to the inviting style.
For teachers, this text is ideally suited as a capstone course for undergraduate students or as a compelling text in discrete mathematical topics for beginning graduate students. For scientists, this text can be utilized as a quick tooling device, especially for those who want a selfcontained, easytoread introduction to these topics.
 Reviews

From the reviews:
"You owe it to yourself to pick up a copy … to read about a number of interesting problems in geometry, number theory, and combinatorics … . Even people who are familiar with the material would almost certainly learn something from the clear and engaging exposition … . It contains a large number of exercises … . Each chapter also ends with a series of relevant open problems … . it is also full of mathematics that is selfcontained and worth reading on its own." (Darren Glass, MathDL, February, 2007)
"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. … Most importantly the book gives a complete presentation of the use of generating functions of various kinds to enumerate lattice points, as well as an introduction to the theory of Erhart quasipolynomials. … This book provides many wellcrafted exercises, and even a list of open problems in each chapter." (Jesús A. De Loera, Mathematical Reviews, Issue 2007 h)
"All mathematics majors study the topics they will need to know, should they want to go to graduate school. But most will not, and 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 (San Francisco State Univ.) and Robins (Temple Univ.) have written the perfect text for such a course. … Summing Up: Highly recommended. General readers; lowerdivision undergraduates through faculty." (D. V. Feldman, CHOICE, Vol. 45 (2), 2007)
"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 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." (Margaret M. Bayer, Zentralblatt MATH, Vol. 1114 (16), 2007)
"The main topic of the book is initiated by a theorem of Ehrhart … . This is a wonderful book for various readerships. Students, researchers, lecturers in enumeration, geometry and number theory all find it very pleasing and useful. The presentation is accessible for mature undergraduates. … it is a clear introduction to graduate students and researchers with many exercises and with a list of open problems at the end of each chapter." (Péter Hajnal, Acta Scientiarum Mathematicarum, Vol. 75, 2009)
“The theme of this textbook … is the relation between the continuous and the discrete, namely, the interplay between the volume of a polytope and the number of grid points contained in it. … The text contains many exercises, some of which present material needed later (for these exercises hints are provided), and lists also many open research problems. – The book can be recommended both for its style and for its interesting … content.” (P. Schmitt, Monatshefte für Mathematik, Vol. 155 (2), October, 2008)
 Table of contents (12 chapters)


The CoinExchange Problem of Frobenius
Pages 324

A Gallery of Discrete Volumes
Pages 2555

Counting Lattice Points in Polytopes:The Ehrhart Theory
Pages 5782

Reciprocity
Pages 8393

Face Numbers and the Dehn—Sommerville Relations in Ehrhartian Terms
Pages 95104

Table of contents (12 chapters)
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Bibliographic Information
 Bibliographic Information

 Book Title
 Computing the Continuous Discretely
 Book Subtitle
 Integerpoint Enumeration in Polyhedra
 Authors

 Matthias Beck
 Sinai Robins
 Series Title
 Undergraduate Texts in Mathematics
 Copyright
 2007
 Publisher
 SpringerVerlag New York
 Copyright Holder
 SpringerVerlag New York
 eBook ISBN
 9780387461120
 DOI
 10.1007/9780387461120
 Softcover ISBN
 9781441921192
 Series ISSN
 01726056
 Edition Number
 1
 Number of Pages
 XVIII, 227
 Topics
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