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An invitation to Ehrhart theory: polyhedral geometry and its applications in enumerative combinatorics. (English) Zbl 1437.52010

Gutierrez, Jaime (ed.) et al., Computer algebra and polynomials. Applications of algebra and number theory. Berlin: Springer. Lect. Notes Comput. Sci. 8942, 1-29 (2015).
Summary: In this expository article we give an introduction to Ehrhart theory, i.e., the theory of integer points in polyhedra, and take a tour through its applications in enumerative combinatorics. Topics include geometric modeling in combinatorics, Ehrhart’s method for proving that a counting function is a polynomial, the connection between polyhedral cones, rational functions and quasisymmetric functions, methods for bounding coefficients, combinatorial reciprocity theorems, algorithms for counting integer points in polyhedra and computing rational function representations, as well as visualizations of the greatest common divisor and the Euclidean algorithm.
For the entire collection see [Zbl 1335.68004].

MSC:

52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
13M10 Polynomials and finite commutative rings
11C08 Polynomials in number theory
05A17 Combinatorial aspects of partitions of integers
90C05 Linear programming
90C10 Integer programming
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