Book•
Gröbner bases and convex polytopes
14 Dec 1995-
TL;DR: Grobner basics The state polytope Variation of term orders Toric ideals Enumeration, sampling and integer programming Primitive partition identities Universal Grobner bases Regular triangulations The second hypersimplex $\mathcal A$-graded algebras Canonical subalgebra bases Generators, Betti numbers and localizations Toric varieties in algebraic geometry as mentioned in this paper.
Abstract: Grobner basics The state polytope Variation of term orders Toric ideals Enumeration, sampling and integer programming Primitive partition identities Universal Grobner bases Regular triangulations The second hypersimplex $\mathcal A$-graded algebras Canonical subalgebra bases Generators, Betti numbers and localizations Toric varieties in algebraic geometry Some specific Grobner bases Bibliography Index.
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Book•
12 Aug 2008
TL;DR: A singular introduction to commutative algebra as mentioned in this paper is one of the most widely used works in algebraic geometry, with a broad coverage of theoretical topics in the portions of the algebra closest to algebraic geometrical geometry.
Abstract: From the reviews of the first edition: "It is certainly no exaggeration to say that A Singular Introduction to Commutative Algebra aims to lead a further stage in the computational revolution in commutative algebra . Among the great strengths and most distinctive features is a new, completely unified treatment of the global and local theories. making it one of the most flexible and most efficient systems of its type....another strength of Greuel and Pfister's book is its breadth of coverage of theoretical topics in the portions of commutative algebra closest to algebraic geometry, with algorithmic treatments of almost every topic....Greuel and Pfister have written a distinctive and highly useful book that should be in the library of every commutative algebraist and algebraic geometer, expert and novice alike." J.B. Little, MAA, March 2004 The second edition is substantially enlarged by a chapter on Groebner bases in non-commtative rings, a chapter on characteristic and triangular sets with applications to primary decomposition and polynomial solving and an appendix on polynomial factorization including factorization over algebraic field extensions and absolute factorization, in the uni- and multivariate case.
869 citations
12 Sep 2002
TL;DR: Polynomials in one variable Grobner bases of zero-dimensional ideals Bernstein's theorem and fewnomials as mentioned in this paper are the primary decomposition of polynomial systems in economics and statistics.
Abstract: Polynomials in one variable Grobner bases of zero-dimensional ideals Bernstein's theorem and fewnomials Resultants Primary decomposition Polynomial systems in economics Sums of squares Polynomial systems in statistics Tropical algebraic geometry Linear partial differential equations with constant coefficients Bibliography Index.
860 citations
Cites background from "Gröbner bases and convex polytopes..."
...See Chapter 2 in [Stu95]....
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...In the setting of [Stu95], this polytope is the convex hull of the columns of an integer matrix A....
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...9 in [Stu95]....
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TL;DR: The tropical Grassmannian G2; n as mentioned in this paper is a simplicial complex glued from 1035 tetrahedra, and it is a polyhedral subcomplex of the Grobner fan.
Abstract: In tropical algebraic geometry, the solution sets of polynomial equations are piecewise-linear. We introduce the tropical variety of a polynomial ideal, and we identify it with a polyhedral subcomplex of the Grobner fan. The tropical Grassmannian arises in this manner from the ideal of quadratic Plucker relations. It parametrizes all tropical linear spaces. Lines in tropical projective space are trees, and their tropical Grassmannian G2; n equals the space of phylogenetic trees studied by Billera, Holmes and Vogtmann. Higher Grassmannians oer a natural generalization of the space of trees. Their faces correspond to monomial-free initial ideals of the Plucker ideal. The tropical Grassmannian G3; 6 is a simplicial complex glued from 1035 tetrahedra.
562 citations
Posted Content•
TL;DR: Tropical algebraic geometry is the geometry of the tropical semiring as mentioned in this paper, where the objects are polyhedral cell complexes which behave like complex algebraic varieties and have a complete description of the families of quadrics through four points in the tropical projective plane and a counterexample to the incidence version of Pappus' Theorem.
Abstract: Tropical algebraic geometry is the geometry of the tropical semiring $(\mathbb{R},\min,+)$. Its objects are polyhedral cell complexes which behave like complex algebraic varieties. We give an introduction to this theory, with an emphasis on plane curves and linear spaces. New results include a complete description of the families of quadrics through four points in the tropical projective plane and a counterexample to the incidence version of Pappus' Theorem.
437 citations
Cites background from "Gröbner bases and convex polytopes..."
...Further let v(1), . . . , v(k) and w(1), . . . , w(l) be the weighted direction vectors of the outgoing edges of p into the two open half planes defined by ℓ....
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Book•
01 Jan 2010
TL;DR: In this paper, the Hungarian method for the assignment problem was used to solve the traveling salesman problem and a group-theoretic approach in mixed integer linear programming was proposed for solving the problem.
Abstract: I The Early Years.- Solution of a Large-Scale Traveling-Salesman Problem.- The Hungarian Method for the Assignment Problem.- Integral Boundary Points of Convex Polyhedra.- Outline of an Algorithm for Integer Solutions to Linear Programs An Algorithm for the Mixed Integer Problem.- An Automatic Method for Solving Discrete Programming Problems.- Integer Programming: Methods, Uses, Computation.- Matroid Partition.- Reducibility Among Combinatorial Problems.- Lagrangian Relaxation for Integer Programming.- Disjunctive Programming.- II From the Beginnings to the State-of-the-Art.- Polyhedral Approaches to Mixed Integer Linear Programming.- Fifty-Plus Years of Combinatorial Integer Programming.- Reformulation and Decomposition of Integer Programs.- III Current Topics.- Integer Programming and Algorithmic Geometry of Numbers.- Nonlinear Integer Programming.- Mixed Integer Programming Computation.- Symmetry in Integer Linear Programming.- Semidefinite Relaxations for Integer Programming.- The Group-Theoretic Approach in Mixed Integer Programming.
412 citations