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375
On the complexity of Gröbner basis computation of semiregular overdetermined . . .
, 2004
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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 76 (11 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.
Complete search in continuous global optimization and constraint satisfaction, Acta Numerica 13
, 2004
"... A chapter for ..."
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 67 (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 52 (9 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.
Asymptotic Behaviour of the Degree of Regularity of SemiRegular Polynomial Systems
 In MEGA’05, 2005. Eighth International Symposium on Effective Methods in Algebraic Geometry
"... We compute the asymptotic expansion of the degree of regularity for overdetermined semiregular sequences of algebraic equations. This degree implies bounds for the generic complexity of Gröbner bases algorithms, in particular the F5 [Fau02] algorithm. Bounds can also be derived for the XL [SPCK00] ..."
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Cited by 45 (25 self)
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We compute the asymptotic expansion of the degree of regularity for overdetermined semiregular sequences of algebraic equations. This degree implies bounds for the generic complexity of Gröbner bases algorithms, in particular the F5 [Fau02] algorithm. Bounds can also be derived for the XL [SPCK00] family of algorithms used by the cryptographic community. 1 Motivations and Results The worstcase complexity of Gröbner bases has been the object of extensive studies. In the most general case, it is well known after work by Mayr and Meyer that the complexity is doubly exponential in the number of variables. For subclasses of polynomial systems, the complexity may be much smaller. Of particular importance is the class of regular sequences of polynomials. There, it is known that after a generic linear change of variables the complexity of the computation for the degreereverselexicographic order is simply exponential in the number of variables. Moreover, in characteristic 0, these systems are generic. Our goal is to give similar complexity bounds for overdetermined systems, for a class of systems that we
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 41 (18 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
On the Validity of Implicitization by Moving Quadrics for Rational Surfaces with No Base Points
 J. Symbolic Computation
, 2000
"... Techniques from algebraic geometry and commutative algebra are adopted to establish sufficient polynomial conditions for the validity of implicitization by the method of moving quadrics both for rectangular tensor product surfaces of bidegree (m; n) and for triangular surfaces of total degree n in ..."
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Cited by 40 (3 self)
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Techniques from algebraic geometry and commutative algebra are adopted to establish sufficient polynomial conditions for the validity of implicitization by the method of moving quadrics both for rectangular tensor product surfaces of bidegree (m; n) and for triangular surfaces of total degree n in the absence of base points. 1 Introduction Several years ago, Tom Sederberg introduced a new technique for finding the implicit equation of a rational surface [Sederberg & Chen 1995]. The classical method for finding the implicit equation of a rational parametric surface x = x(s; t) w(s; t) ; y = y(s; t) w(s; t) ; z = z(s; t) w(s; t) is to compute the bivariate resultant of the three polynomials: x(s; t) \Gamma x \Delta w(s; t); y(s; t) \Gamma y \Delta w(s; t); z(s; t) \Gamma z \Delta w(s; t): Unfortunately for many applications, the resultant of these three polynomials vanishes identically when the surface has base points  that is, parameter values (s 0 ; t 0 ) for which x(s 0 ;...
LengthBased Attacks for Certain Group Based Encryption Rewriting Systems
, 2002
"... In this note, we describe a probabilistic attack on public key cryptosystems based on the word/conjugacy problems for finitely presented groups of the type proposed recently by Anshel, Anshel and Goldfeld. In such a scheme, one makes use of the property that in the given group the word problem has a ..."
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Cited by 40 (1 self)
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In this note, we describe a probabilistic attack on public key cryptosystems based on the word/conjugacy problems for finitely presented groups of the type proposed recently by Anshel, Anshel and Goldfeld. In such a scheme, one makes use of the property that in the given group the word problem has a polynomial time solution, while the conjugacy problem has no known polynomial solution. An example is the braid group from topology in which the word problem is solvable in polynomial time while the only known solutions to the conjugacy problem are exponential. The attack in this paper is based on having a canonical representative of each string relative to which a length function may be computed. Hence the term length attack. Such canonical representatives are known to exist for the braid group.