𝔖 Bobbio Scriptorium
✦   LIBER   ✦

Gröbner Bases and Polyhedral Geometry of Reducible and Cyclic Models

✍ Scribed by Serkan Hoşten; Seth Sullivant

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
234 KB
Volume
100
Category
Article
ISSN
0097-3165

No coin nor oath required. For personal study only.

✦ Synopsis

This article studies the polyhedral structure and combinatorics of polytopes that arise from hierarchical models in statistics, and shows how to construct Gro¨bner bases of toric ideals associated to a subset of such models. We study the polytopes for cyclic models, and we give a complete polyhedral description of these polytopes in the binary cyclic case. Further, we show how to build Gro¨bner bases of a reducible model from the Gro¨bner bases of its pieces. This result also gives a different proof that decomposable models have quadratic Gro¨bner bases. Finally, we present the solution of a problem posed by Vlach (Discrete Appl. Math. 13 (1986) 61-78) concerning the dimension of fibers coming from models corresponding to the boundary of a simplex.

📜 SIMILAR VOLUMES

Reduced Gröbner Bases, Free Difference–D
Reduced Gröbner Bases, Free Difference–Differential Modules and Difference–Differential Dimension Polynomials
✍ Alexander Levin 📂 Article 📅 2000 🏛 Elsevier Science 🌐 English ⚖ 442 KB

We define a special type of reduction in a free left module over a ring of differencedifferential operators and use the idea of the Gröbner basis method to develop a technique that allows us to determine the Hilbert function of a finitely generated differencedifferential module equipped with the nat

A Note on Gröbner Bases and Integration
A Note on Gröbner Bases and Integration of Rational Functions
✍ Günter Czichowski 📂 Article 📅 1995 🏛 Elsevier Science 🌐 English ⚖ 136 KB

It will be shown that for rational functions the logarithmic part of the integral can be computed in a very simple manner by Buchberger's algorithm.

Properties of Gröbner Bases and Applicat
Properties of Gröbner Bases and Applications to Doubly Periodic Arrays
✍ Mulan Liu; Lei Hu 📂 Article 📅 1998 🏛 Elsevier Science 🌐 English ⚖ 430 KB

In this paper the properties of Gröbner bases of zero-dimensional ideals are studied. A basis of the space of linear recurring arrays and the trace expression of linear recurring arrays are given.

An Optimal Algorithm for Constructing th
An Optimal Algorithm for Constructing the Reduced Gröbner Basis of Binomial Ideals, and Applications to Commutative Semigroups
✍ Ulla Koppenhagen; Ernst W. Mayr 📂 Article 📅 2001 🏛 Elsevier Science 🌐 English ⚖ 404 KB

It is known that the reduced Gröbner basis of general polynomial ideals can be computed in exponential space. The algorithm, obtained by Kühnle and Mayr, is, however, based on rather complex parallel computations, and, above that, makes extensive use of the parallel computation thesis. In this paper

Gröbner Bases for the Rings of Special O
Gröbner Bases for the Rings of Special Orthogonal and 2 × 2 Matrix Invariants
✍ M Domokos; V Drensky 📂 Article 📅 2001 🏛 Elsevier Science 🌐 English ⚖ 110 KB

We present a Gröbner basis for the ideal of relations among the standard generators of the algebra of invariants of the special orthogonal group acting on k-tuples of vectors. The cases of SO 3 and SO 4 are interpreted in terms of the algebras of invariants and semi-invariants of k-tuples of 2 × 2 m