Results 1  10
of
223
The tropical Grassmannian
, 2003
"... In tropical algebraic geometry, the solution sets of polynomial equations are piecewiselinear. We introduce the tropical variety of a polynomial ideal, and we identify it with a polyhedral subcomplex of the Gröbner fan. The tropical Grassmannian arises in this manner from the ideal of quadratic Plü ..."
Abstract

Cited by 168 (15 self)
 Add to MetaCart
In tropical algebraic geometry, the solution sets of polynomial equations are piecewiselinear. We introduce the tropical variety of a polynomial ideal, and we identify it with a polyhedral subcomplex of the Gröbner fan. The tropical Grassmannian arises in this manner from the ideal of quadratic Plücker relations. It is shown to parametrize 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 offer a natural generalization of the space of trees. Their facets correspond to binomial initial ideals of the Plücker ideal. The tropical Grassmannian G3,6 is a simplicial complex glued from 1035 tetrahedra.
Eigenvalues of a real supersymmetric tensor
 J. Symbolic Comput
"... In this paper, we define the symmetric hyperdeterminant, eigenvalues and Eeigenvalues of a real supersymmetric tensor. We show that eigenvalues are roots of a onedimensional polynomial, and when the order of the tensor is even, Eeigenvalues are roots of another onedimensional polynomial. These t ..."
Abstract

Cited by 145 (62 self)
 Add to MetaCart
(Show Context)
In this paper, we define the symmetric hyperdeterminant, eigenvalues and Eeigenvalues of a real supersymmetric tensor. We show that eigenvalues are roots of a onedimensional polynomial, and when the order of the tensor is even, Eeigenvalues are roots of another onedimensional polynomial. These two onedimensional polynomials are associated with the symmetric hyperdeterminant. We call them the characteristic polynomial and the Echaracteristic polynomial of that supersymmetric tensor. Real eigenvalues (Eeigenvalues) with real eigenvectors (Eeigenvectors) are called Heigenvalues (Zeigenvalues). When the order of the supersymmetric tensor is even, Heigenvalues (Zeigenvalues) exist and the supersymmetric tensor is positive definite if and only if all of its Heigenvalues (Zeigenvalues) are positive. An mthorder ndimensional supersymmetric tensor where m is even has exactly n(m − 1) n−1 eigenvalues, and the number of its Eeigenvalues is strictly less than n(m − 1) n−1 when m ≥ 4. We show that the product of all the eigenvalues is equal to the value of the symmetric hyperdeterminant, while the sum of all the eigenvalues is equal to the sum of the diagonal elements of that supersymmetric tensor, multiplied by (m − 1) n−1.The n(m −1) n−1 eigenvalues are distributed in n disks in C.Thecenters and radii of these n disks are the diagonal elements, and the sums of the absolute values of the corresponding offdiagonal elements, of that supersymmetric tensor. On the other hand, Eeigenvalues are invariant under orthogonal transformations.
Tropical geometry and its applications
 International Congress of Mathematicians vol. II, 827–852, Eur. Math. Soc
, 2006
"... Abstract. From a formal perspective tropical geometry can be viewed as a branch of geometry manipulating with certain piecewiselinear objects that take over the rôle of classical algebraic varieties. This talk outlines some basic notions of this area and surveys some of its applications for the pr ..."
Abstract

Cited by 141 (6 self)
 Add to MetaCart
(Show Context)
Abstract. From a formal perspective tropical geometry can be viewed as a branch of geometry manipulating with certain piecewiselinear objects that take over the rôle of classical algebraic varieties. This talk outlines some basic notions of this area and surveys some of its applications for the problems in classical (real and complex) geometry.
First Steps in Tropical Geometry
 CONTEMPORARY MATHEMATICS
"... Tropical algebraic geometry is the geometry of the tropical semiring (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 descr ..."
Abstract

Cited by 123 (10 self)
 Add to MetaCart
(Show Context)
Tropical algebraic geometry is the geometry of the tropical semiring (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.
Algebraic Geometry of Bayesian Networks
 Journal of Symbolic Computation
, 2005
"... We study the algebraic varieties defined by the conditional independence statements of Bayesian networks. A complete algebraic classification is given for Bayesian networks on at most five random variables. Hidden variables are related to the geometry of higher secant varieties. 1 ..."
Abstract

Cited by 84 (6 self)
 Add to MetaCart
(Show Context)
We study the algebraic varieties defined by the conditional independence statements of Bayesian networks. A complete algebraic classification is given for Bayesian networks on at most five random variables. Hidden variables are related to the geometry of higher secant varieties. 1
Detecting global optimality and extracting solutions in GloptiPoly
 Chapter in D. Henrion, A. Garulli (Editors). Positive polynomials in control. Lecture Notes in Control and Information Sciences
, 2005
"... GloptiPoly is a Matlab/SeDuMi addon to build and solve convex linear matrix inequality (LMI) relaxations of nonconvex optimization problems with multivariate polynomial objective function and constraints, based on the theory of moments. In contrast with the dual sumofsquares decompositions of po ..."
Abstract

Cited by 80 (12 self)
 Add to MetaCart
(Show Context)
GloptiPoly is a Matlab/SeDuMi addon to build and solve convex linear matrix inequality (LMI) relaxations of nonconvex optimization problems with multivariate polynomial objective function and constraints, based on the theory of moments. In contrast with the dual sumofsquares decompositions of positive polynomials, the theory of moments allows to detect global optimality of an LMI relaxation and extract globally optimal solutions. In this report, we describe and illustrate the numerical linear algebra algorithm implemented in GloptiPoly for detecting global optimality and extracting solutions. We also mention some related heuristics that could be useful to reduce the number of variables in the LMI relaxations. 1
Tropical discriminants
, 2005
"... Tropical geometry is used to develop a new approach to the theory of discriminants and resultants in the sense of Gel’fand, Kapranov and Zelevinsky. The tropical Adiscriminant, which is the tropicalization of the dual variety of the projective toric variety given by an integer matrix A, is shown t ..."
Abstract

Cited by 62 (6 self)
 Add to MetaCart
Tropical geometry is used to develop a new approach to the theory of discriminants and resultants in the sense of Gel’fand, Kapranov and Zelevinsky. The tropical Adiscriminant, which is the tropicalization of the dual variety of the projective toric variety given by an integer matrix A, is shown to coincide with the Minkowski sum of the row space of A and of the tropicalization of the kernel of A. This leads to an explicit positive formula for the extreme monomials of any Adiscriminant, and to a combinatorial rule for deciding when two regular triangulations of A correspond to the same monomial of the Adiscriminant.
Robust game theory
, 2006
"... We present a distributionfree model of incompleteinformation games, both with and without private information, in which the players use a robust optimization approach to contend with payoff uncertainty. Our “robust game” model relaxes the assumptions of Harsanyi’s Bayesian game model, and provides ..."
Abstract

Cited by 55 (0 self)
 Add to MetaCart
We present a distributionfree model of incompleteinformation games, both with and without private information, in which the players use a robust optimization approach to contend with payoff uncertainty. Our “robust game” model relaxes the assumptions of Harsanyi’s Bayesian game model, and provides an alternative distributionfree equilibrium concept, which we call “robustoptimization equilibrium, ” to that of the ex post equilibrium. We prove that the robustoptimization equilibria of an incompleteinformation game subsume the ex post equilibria of the game and are, unlike the latter, guaranteed to exist when the game is finite and has bounded payoff uncertainty set. For arbitrary robust finite games with bounded polyhedral payoff uncertainty sets, we show that we can compute a robustoptimization equilibrium by methods analogous to those for identifying a Nash equilibrium of a finite game with complete information. In addition, we present computational results.
MATROID POLYTOPES, NESTED SETS AND BERGMAN FANS
, 2004
"... The tropical variety defined by linear equations with constant coefficients is the Bergman fan of the corresponding matroid. Building on a selfcontained introduction to matroid polytopes, we present a geometric construction of the Bergman fan, and we discuss its relationship with the simplicial com ..."
Abstract

Cited by 55 (6 self)
 Add to MetaCart
The tropical variety defined by linear equations with constant coefficients is the Bergman fan of the corresponding matroid. Building on a selfcontained introduction to matroid polytopes, we present a geometric construction of the Bergman fan, and we discuss its relationship with the simplicial complex of nested sets in the lattice of flats. The Bergman complex is triangulated by the nested set complex, and the two complexes coincide if and only if every connected flat remains connected after contracting along any subflat. This sharpens a result of ArdilaKlivans who showed that the Bergman complex is triangulated by the order complex of the lattice of flats. The nested sets specify the De ConciniProcesi compactification of the complement of a hyperplane arrangement, while the Bergman fan specifies the tropical compactification. These two compactifications are almost equal, and we highlight the subtle differences.