## Grobner Bases and Convex PolytopesThis book is about the interplay of computational commutative algebra and the theory of convex polytopes. It centers around a special class of ideals in a polynomial ring: the class of toric ideals. They are characterized as those prime ideals that are generated by monomial differences or as the defining ideals of toric varieties (not necessarily normal). The interdisciplinary nature of the study of Grobner bases is reflected by the specific applications appearing in this book. These applications lie in the domains of integer programming and computational statistics. The mathematical tools presented in the volume are drawn from commutative algebra, combinatorics, and polyhedral geometry. |

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### Contents

Chapter 1 Gröbner Basics | 1 |

Chapter 2 The State Poly tope | 9 |

Chapter 3 Variation of Term Orders | 19 |

Chapter 4 Toric Ideals | 31 |

Chapter 5 Enumeration Sampling and Integer Programming | 39 |

Chapter 6 Primitive Partition Identities | 47 |

Chapter 7 Universal Grobner Bases | 55 |

Chapter 8 Regular Triangulations | 63 |

Chapter 10 Agraded Algebras | 85 |

Chapter 11 Canonical Subalgebra Bases | 99 |

Chapter 12 Generators Betti Numbers and Localizations | 113 |

Chapter 13 Toric Varieties in Algebraic Geometry | 127 |

Chapter 14 Some Specific Grobner Bases | 141 |

155 | |

161 | |

Back Cover | 163 |

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### Common terms and phrases

2-minors A-graded algebra A-graded ideal bases basis g basis of IA binomial Buchberger algorithm canonical basis Chapter circuits Compute the reduced cone configuration Consider conv(A convex hull coordinates Corollary corresponding defined denote edges Ehrhart polynomial element equals Example exists face finite geometry graded graph Graver basis Gröbner fan hence Hilbert Hilbert polynomial homogeneous ideal hypersimplex implies initial ideal initial monomial initial term integer programming isomorphic ker(A ker(T Lemma lexicographic term order lies linear matrix minimal Minkowski sum modulo monomial ideal non-zero order ideal Output p-matrix polyhedral polynomial ring polytope Q pos(A primitive partition identities projective toric variety proof of Theorem Proposition reduced Gröbner basis regular triangulation relatively prime reverse lexicographic right hand side semigroup simplicial complex square-free standard monomials Sturmfels subalgebra Subroutine subset supp(u Suppose syzygy toric ideal IA toric variety triangulation unimodular universal Gröbner basis variables vertex vertices weight vector zero