## Integer Points in Polyhedra |

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### Contents

Introduction | 1 |

The algebra of polyhedra | 9 |

Linear transformations and polyhedra | 19 |

The structure of polyhedra | 27 |

Polarity | 41 |

Tangent cones Decompositions modulo polyhedra with lines | 49 |

Open polyhedra | 57 |

The exponential valuation | 63 |

The Minkowski Convex Body Theorem | 95 |

Reduced basis | 99 |

Exponential sums and generating functions | 107 |

Totally unimodular polytopes | 121 |

Decomposing a 2dimensional cone into unimodular cones via continued fractions | 129 |

Decomposing a rational cone of an arbitrary dimension into unimodular cones | 137 |

viii | 149 |

A valuation on rational cones | 167 |

Computing volumes | 77 |

Lattices bases and parallelepipeds | 81 |

A local formula for the number of integer points | 183 |

### Common terms and phrases

A C V affine hyperplane algorithm Cb(V Chapter closed convex set compute cone of feasible cone spanned contains a line converges absolutely convex hull convex set coprime d-dimensional decomposition Euclidean space Euler characteristic face F fcone(P feasible directions Figure finite formula G int halfspace Hence indicator functions integer points integer vertices intersection K C V L C V lattice points Lemma Let P C V Let us choose Let us consider let us define Let us fix linear inequalities linear transformation meromorphic function Minkowski sum modulo polyhedra non-empty polyhedron number of integer parallelepiped polyhedra with lines polyhedron without lines polynomial polytope positive integer proof of Theorem Prove rational cone rational function recession cone reduced basis relative interior spanned by 1,0 Suppose tangent cone Theorem 3.1 uniformly on compact unimodular cones valuation vector space vertex