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Computer Algebra and Polynomials

Volume 8942 of the series Lecture Notes in Computer Science pp 1-29

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An Invitation to Ehrhart Theory: Polyhedral Geometry and its Applications in Enumerative Combinatorics

  • Felix BreuerAffiliated withResearch Institute for Symbolic Computation, Johannes Kepler University Email author 

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Abstract

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.

Keywords

Polynomial Quasipolynomial Rational function Quasisymmetric function Partial polytopal complex Simplicial cone Fundamental parallelepiped Combinatorial reciprocity theorem Barvinok’s algorithm Euclidean algorithm Greatest common divisor Generating function Formal power series Integer linear programming