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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 145 (62 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.
Algebraic cryptanalysis of hidden field equation (HFE) cryptosystems using Gröbner bases
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On the complexity of Gröbner basis computation of semiregular overdetermined . . .
, 2004
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Complete search in continuous global optimization and constraint satisfaction
 ACTA NUMERICA 13
, 2004
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Symmetric tensors and symmetric tensor rank
 Scientific Computing and Computational Mathematics (SCCM
, 2006
"... Abstract. A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank1 orderk tensor is the outer product of k nonzero vectors. An ..."
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Cited by 99 (20 self)
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Abstract. A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank1 orderk tensor is the outer product of k nonzero vectors. Any symmetric tensor can be decomposed into a linear combination of rank1 tensors, each of them being symmetric or not. The rank of a symmetric tensor is the minimal number of rank1 tensors that is necessary to reconstruct it. The symmetric rank is obtained when the constituting rank1 tensors are imposed to be themselves symmetric. It is shown that rank and symmetric rank are equal in a number of cases, and that they always exist in an algebraically closed field. We will discuss the notion of the generic symmetric rank, which, due to the work of Alexander and Hirschowitz, is now known for any values of dimension and order. We will also show that the set of symmetric tensors of symmetric rank at most r is not closed, unless r = 1. Key words. Tensors, multiway arrays, outer product decomposition, symmetric outer product decomposition, candecomp, parafac, tensor rank, symmetric rank, symmetric tensor rank, generic symmetric rank, maximal symmetric rank, quantics AMS subject classifications. 15A03, 15A21, 15A72, 15A69, 15A18 1. Introduction. We
Effective lattice point counting in rational convex polytopes
 JOURNAL OF SYMBOLIC COMPUTATION
, 2003
"... This paper discusses algorithms and software for the enumeration of all lattice points inside a rational convex polytope: we describe LattE, a computer package for lattice point enumeration which contains the first implementation of A. Barvinok's algorithm [8]. We report on computational experi ..."
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Cited by 95 (14 self)
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This paper discusses algorithms and software for the enumeration of all lattice points inside a rational convex polytope: we describe LattE, a computer package for lattice point enumeration which contains the first implementation of A. Barvinok's algorithm [8]. We report on computational experiments with multiway contingency tables, knapsack type problems, rational polygons, and flow polytopes. We prove that this kind of symbolicalgebraic ideas surpasses the traditional branchandbound enumeration and in some instances LattE is the only software capable of counting. Using LattE, we have also computed new formulas of Ehrhart (quasi)polynomials for interesting families of polytopes (hypersimplices, truncated cubes, etc). We end with a survey of other "algebraicanalytic" algorithms, including a "polar" variation of Barvinok's algorithm which is very fast when the number of facetdefining inequalities is much smaller compared to the number of vertices.
Recent Developments on Direct Relative Orientation
, 2006
"... This paper presents a novel version of the fivepoint relative orientation algorithm given in Nister (2004). The name of the algorithm arises from the fact that it can operate even on the minimal five point correspondences required for a finite number of solutions to relative orientation. For the mi ..."
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Cited by 91 (0 self)
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This paper presents a novel version of the fivepoint relative orientation algorithm given in Nister (2004). The name of the algorithm arises from the fact that it can operate even on the minimal five point correspondences required for a finite number of solutions to relative orientation. For the minimal five correspondences the algorithm returns up to ten real solutions. The algorithm can also operate on many points. Like the previous version of the fivepoint algorithm, our method can operate correctly even in the face of critical surfaces, including planar and ruled quadric scenes. The paper
Factoring Multivariate Polynomials via Partial Differential Equations
 Math. Comput
, 2000
"... A new method is presented for factorization of bivariate polynomials over any field of characteristic zero or of relatively large characteristic. It is based on a simple partial differential equation that gives a system of linear equations. Like Berlekamp's and Niederreiter's algorithms fo ..."
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Cited by 56 (10 self)
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A new method is presented for factorization of bivariate polynomials over any field of characteristic zero or of relatively large characteristic. It is based on a simple partial differential equation that gives a system of linear equations. Like Berlekamp's and Niederreiter's algorithms for factoring univariate polynomials, the dimension of the solution space of the linear system is equal to the number of absolutely irreducible factors of the polynomial to be factored and any basis for the solution space gives a complete factorization by computing gcd's and by factoring univariate polynomials over the ground field. The new method finds absolute and rational factorizations simultaneously and is easy to implement for finite fields, local fields, number fields, and the complex number field. The theory of the new method allows an effective Hilbert irreducibility theorem, thus an efficient reduction of polynomials from multivariate to bivariate.
Eigenvalues and invariants of tensors
, 2007
"... A tensor is represented by a supermatrix under a coordinate system. In this paper, we define Eeigenvalues and Eeigenvectors for tensors and supermatrices. By the resultant theory, we define the Echaracteristic polynomial of a tensor. An Eeigenvalue of a tensor is a root of the Echaracteristic p ..."
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Cited by 54 (23 self)
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A tensor is represented by a supermatrix under a coordinate system. In this paper, we define Eeigenvalues and Eeigenvectors for tensors and supermatrices. By the resultant theory, we define the Echaracteristic polynomial of a tensor. An Eeigenvalue of a tensor is a root of the Echaracteristic polynomial. In the regular case, a complex number is an Eeigenvalue if and only if it is a root of the Echaracteristic polynomial. We convert the Echaracteristic polynomial of a tensor to a monic polynomial and show that the coefficients of that monic polynomial are invariants of that tensor, i.e., they are invariant under coordinate system changes. We call them principal invariants of that tensor. The maximum number of principal invariants of mth order ndimensional tensors is a function of m and n. We denote it by d(m,n) and show that d(1, n) = 1, d(2, n) = n, d(m,2) = m for m 3 and d(m,n) mn−1 + · · · + m for m,n 3. We also define the rank of a tensor. All real eigenvectors associated with nonzero Eeigenvalues are in a subspace with dimension equal to its rank.