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172
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 ..."
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Cited by 173 (16 self)
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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.
On the geometry and algebra of the point and line correspondences between N images
, 1995
"... We explore the geometric and algebraic relations that exist between correspondences of points and lines in an arbitrary number of images. We propose to use the formalism of the GrassmannCayley algebra as the simplest way to make both geometric and algebraic statements in a very synthetic and effect ..."
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Cited by 164 (7 self)
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We explore the geometric and algebraic relations that exist between correspondences of points and lines in an arbitrary number of images. We propose to use the formalism of the GrassmannCayley algebra as the simplest way to make both geometric and algebraic statements in a very synthetic and effective way (i.e. allowing actual computation if needed). We have a fairly complete picture of the situation in the case of points: there are only three types of algebraic relations which are satisfied by the coordinates of the images of a 3D point: bilinear relations arising when we consider pairs of images among the N and which are the wellknown epipolar constraints, trilinear relations arising when we consider triples of images among the N , and quadrilinear relations arising when we consider fourtuples of images among the N . In the case of lines, we show how the traditional perspective projection equation can be suitably generalized and that in the case of three images there exist two in...
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 ..."
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Cited by 144 (60 self)
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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.
Quantum Schubert Polynomials
 J. AMER. MATH. SOC
, 1997
"... We compute GromovWitten invariants of the flag manifold using a new combinatorial construction for its quantum cohomology ring. Our construction provides quantum analogues of the BernsteinGelfandGelfand results on the cohomology of the flag manifold, and the LascouxSchutzenberger theory of S ..."
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Cited by 86 (7 self)
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We compute GromovWitten invariants of the flag manifold using a new combinatorial construction for its quantum cohomology ring. Our construction provides quantum analogues of the BernsteinGelfandGelfand results on the cohomology of the flag manifold, and the LascouxSchutzenberger theory of Schubert polynomials. We also derive the quantum Monk's formula.
Affine Structure from Line Correspondences with Uncalibrated Affine Cameras
 IEEE Trans. Pattern Analysis and Machine Intelligence
, 1997
"... This paper presents a linear algorithm for recovering 3D affine shape and motion from line correspondences with uncalibrated affine cameras. The algorithm requires a minimum of seven line correspondences over three views. The key idea is the introduction of a onedimensional projective camera. This ..."
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Cited by 83 (9 self)
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This paper presents a linear algorithm for recovering 3D affine shape and motion from line correspondences with uncalibrated affine cameras. The algorithm requires a minimum of seven line correspondences over three views. The key idea is the introduction of a onedimensional projective camera. This converts 3D affine reconstruction of "line directions" into 2D projective reconstruction of "points". In addition, a linebased factorisation method is also proposed to handle redundant views. Experimental results both on simulated and real image sequences validate the robustness and the accuracy of the algorithm.
Cyclic SelfDual Codes
, 1983
"... It is shown that if the automorphism group of a binary selfdual code satisfies a certain condition then the code contains words of weight congruent to 2 modulo 4. In particular, no cyclic binary selfdual code can have all its weights divisible by 4. The number of cyclic binary selfdual codes of l ..."
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Cited by 71 (5 self)
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It is shown that if the automorphism group of a binary selfdual code satisfies a certain condition then the code contains words of weight congruent to 2 modulo 4. In particular, no cyclic binary selfdual code can have all its weights divisible by 4. The number of cyclic binary selfdual codes of length n is determined, and the shortest nontrivial code in this class is shown to have length 14.
Computational Invariant Theory
 Encyclopaedia of Mathematical Sciences, SpringerVerlag
, 1998
"... This article is an expanded version of the material presented there. The main topic is the calculation of the invariant ring of a finite group acting on a polynomial ring by linear transformations of the indeterminates. By "calculation" I mean finding a finite system of generators for the ..."
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Cited by 70 (3 self)
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This article is an expanded version of the material presented there. The main topic is the calculation of the invariant ring of a finite group acting on a polynomial ring by linear transformations of the indeterminates. By "calculation" I mean finding a finite system of generators for the invariant ring, and (optionally) determining structural properties of it. In this exposition particular emphasis is placed on the case that the ground field has positive characteristic dividing the group order. We call this the modular case, and it is important for several reasons. First, many theoretical questions about the structure of modular invariant rings are still open. I will address the problems which I consider the most important or fascinating in the course of the paper. Thus it is very helpful to be able to compute modular invariant rings in order to gain experience, formulate or check conjectures, and gather some insight which in fortunate cases leads to proofs. Furthermore, the computation of modular invariant ring can be very useful for the study of cohomology of finite groups (see Adem and Milgram [1]). This exposition also treats the nonmodular case (characteristic zero or coprime to the group order), where computations are much easier and the theory is for the most part settled. There are also various applications in this case, such as the solution of algebraic equations or the study of dynamical systems with symmetries (see, for example, Gatermann [11], Worfolk [26]).
A Survey on the Theorema Project
 IN INTERNATIONAL SYMPOSIUM ON SYMBOLIC AND ALGEBRAIC COMPUTATION
, 1997
"... The Theorema project aims at extending current computer algebra systems by facilities for supporting mathematical proving. The present earlyprototype version of the Theorema software system is implemented in Mathematica 3.0. The system consists of a general higherorder predicate logic prover and ..."
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Cited by 54 (13 self)
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The Theorema project aims at extending current computer algebra systems by facilities for supporting mathematical proving. The present earlyprototype version of the Theorema software system is implemented in Mathematica 3.0. The system consists of a general higherorder predicate logic prover and a collection of special provers that call each other depending on the particular proof situations. The individual provers imitate the proof style of human mathematicians and aim at producing humanreadable proofs in natural language presented in nested cells that facilitate studying the computergenerated proofs at various levels of detail. The special provers are intimately connected with the functors that build up the various mathematical domains.