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**Computing the continuous discretely. Integer-point enumeration in polyhedra. With illustrations by David Austin.
2nd edition.**
*(English)*
Zbl 1339.52002

Undergraduate Texts in Mathematics. New York, NY: Springer (ISBN 978-1-4939-2968-9/hbk; 978-1-4939-2969-6/ebook). xx, 285 p. (2015).

This book is an outstanding book on counting integer points of polytopes (usually with integer vertices). The book is divided into two parts. In part I the authors discuss the basics of theory of Ehrhart polynomials and various preliminary subjects related to polytopes, e.g., face structure of the polytopes, Dedekind summation, magic squares, etc. Second part of the book contains recent results and approaches to integer-point enumeration in polyhedra. The book contains main results of Ehrhart theory related to Dedekind summation, finite Fourier analysis, various questions regarding triangulation of polytopes and decompositions into its cones. Numerous important examples are exhaustively studied. The reader will find here both basic examples of certain polyhedra (simplices, cubes, cross-polytopes and pyramids) and more advance series of examples (zonotopes, and in particular, permutahedra, etc.).

The book contains lots of exercises with very helpful hints. Another essential feature of the book is a vast collection of open problems on different aspects of integer point counting and related areas. To my opinion it is very important to have such a list of open problems for further references. The book is reader-friendly written, self-contained and contains numerous beautiful illustrations. The reader is always accompanied with deep research jokes by famous researchers and valuable historical notes.

The book contains lots of exercises with very helpful hints. Another essential feature of the book is a vast collection of open problems on different aspects of integer point counting and related areas. To my opinion it is very important to have such a list of open problems for further references. The book is reader-friendly written, self-contained and contains numerous beautiful illustrations. The reader is always accompanied with deep research jokes by famous researchers and valuable historical notes.

Reviewer: Oleg Karpenkov (Liverpool)

### MSC:

52-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to convex and discrete geometry |

52B05 | Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.) |

05-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to combinatorics |

05A15 | Exact enumeration problems, generating functions |

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |