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SYMMETRIC TENSOR DECOMPOSITION
, 2009
"... We present an algorithm for decomposing a symmetric tensor of dimension n and order d as a sum of of rank1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for symmetric tensors of dimension 2. We exploit the known fact that every symmetric tensor is equivalently represented ..."
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Cited by 30 (5 self)
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We present an algorithm for decomposing a symmetric tensor of dimension n and order d as a sum of of rank1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for symmetric tensors of dimension 2. We exploit the known fact that every symmetric tensor is equivalently represented by a homogeneous polynomial in n variables of total degree d. Thus the decomposition corresponds to a sum of powers of linear forms. The impact of this contribution is twofold. First it permits an efficient computation of the decomposition of any tensor of subgeneric rank, as opposed to widely used iterative algorithms with unproved convergence (e.g. Alternate Least Squares or gradient descents). Second, it gives tools for understanding uniqueness conditions, and for detecting the tensor rank.
Characterizations of border bases
 Journal of Pure and Applied Algebra
"... This paper presents characterizations of border bases of zerodimensional polynomial ideals that are analogous to the known characterizations of Gröbner bases. Based on a Border Division Algorithm, a variant of the usual Division Algorithm, we characterize border bases as border prebases with one of ..."
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Cited by 30 (4 self)
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This paper presents characterizations of border bases of zerodimensional polynomial ideals that are analogous to the known characterizations of Gröbner bases. Based on a Border Division Algorithm, a variant of the usual Division Algorithm, we characterize border bases as border prebases with one of the following equivalent properties: special generation, generation of the border form ideal, confluence of the corresponding rewrite relation, reduction of Spolynomials to zero, and lifting of syzygies. The last characterization relies on a detailed study of the relative position of the border terms and their syzygy module. In particular, a border prebasis is a border basis if and only if all fundamental syzygies of neighboring border terms lift; these liftings are easy to compute. Key words: border basis, division algorithm, syzygy module 1
Semidefinite characterization and computation of zerodimensional real radical ideals
, 2007
"... real radical ideals ..."
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Computing border bases
 JOURNAL OF PURE AND APPLIED ALGEBRA
, 2005
"... This paper presents several algorithms that compute border bases of a zerodimensional ideal. The first relates to the FGLM algorithm as it uses a linear basis transformation. In particular, it is able to compute border bases that do not contain a reduced Gröbner basis. The second algorithm is based ..."
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Cited by 19 (2 self)
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This paper presents several algorithms that compute border bases of a zerodimensional ideal. The first relates to the FGLM algorithm as it uses a linear basis transformation. In particular, it is able to compute border bases that do not contain a reduced Gröbner basis. The second algorithm is based on a generic algorithm by Bernard Mourrain originally designed for computing an ideal basis that need not be a border basis. Our fully detailed algorithm computes a border basis of a zerodimensional ideal from a given set of generators. To obtain concrete instructions we appeal to a degreecompatible term ordering σ and hence compute a border basis that contains the reduced σGröbner basis. We show an example in which this computation actually has advantages over Buchberger’s algorithm. Moreover, we formulate and prove two optimizations of the Border Basis Algorithm which reduce the dimensions of the linear algebra subproblems.
Solving Projective Complete Intersection Faster
 Proc. Intern. Symp. on Symbolic and Algebraic Computation
, 2000
"... In this paper, we present a new method for solving square polynomial systems with no zero at infinity. We analyze its complexity, which indicates substantial improvements, compared with the previously known methods for solving such systems. We describe a framework for symbolic and numeric computatio ..."
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Cited by 19 (6 self)
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In this paper, we present a new method for solving square polynomial systems with no zero at infinity. We analyze its complexity, which indicates substantial improvements, compared with the previously known methods for solving such systems. We describe a framework for symbolic and numeric computations, developed in C++, in which we have implemented this algorithm. We mention the techniques that are involved in order to build efficient codes and compare with existing softwares. We end by some applications of this method, considering in particular an autocalibration problem in Computer Vision and an identification problem in Signal Processing, and report on the results of our first implementation.
Solving polynomial systems via symbolicnumeric reduction to geometric involutive form
 J. Symbolic Comput
"... We briefly survey several existing methods for solving polynomial system with inexact coefficients, then introduce our new symbolicnumeric method which is based on the geometric (Jet) theory of partial differential equations. The method is stable and robust. Numerical experiments illustrate the pe ..."
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Cited by 17 (5 self)
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We briefly survey several existing methods for solving polynomial system with inexact coefficients, then introduce our new symbolicnumeric method which is based on the geometric (Jet) theory of partial differential equations. The method is stable and robust. Numerical experiments illustrate the performance of the new method.
Stable normal forms for polynomial system solving
 HTTP://HAL.INRIA.FR/INRIA00343103/EN/. INTERNATIONAL PEERREVIEWED CONFERENCE/PROCEEDINGS
, 2008
"... This paper describes and analyzes a method for computing border bases of a zerodimensional ideal I. The criterion used in the computation involves specific commutation polynomials and leads to an algorithm and an implementation extending the one provided in [29]. This general border basis algorithm ..."
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Cited by 15 (1 self)
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This paper describes and analyzes a method for computing border bases of a zerodimensional ideal I. The criterion used in the computation involves specific commutation polynomials and leads to an algorithm and an implementation extending the one provided in [29]. This general border basis algorithm weakens the monomial ordering requirement for Gröbner bases computations. It is up to date the most general setting for representing quotient algebras, embedding into a single formalism Gröbner bases, Macaulay bases and new representation that do not fit into the previous categories. With this formalism we show how the syzygies of the border basis are generated by commutation relations. We also show that our construction of normal form is stable under small perturbations of the ideal, if the number of solutions remains constant. This new feature for a symbolic algorithm has a huge impact on the practical efficiency as it is illustrated by the experiments on classical benchmark polynomial systems, at the end of the paper.
Computing the multiplicity structure from geometric involutive form
 Proc. ISSAC’ 08, ACM Press
, 2008
"... Abstract. We present a method based on symbolicnumeric reduction to geometric involutive form to compute the primary component and the differential operators for an isolated singular solution of a polynomial ideal. The singular solution can be exact or approximate. If the singular solution is known ..."
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Cited by 14 (3 self)
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Abstract. We present a method based on symbolicnumeric reduction to geometric involutive form to compute the primary component and the differential operators for an isolated singular solution of a polynomial ideal. The singular solution can be exact or approximate. If the singular solution is known with limited accuracy, then we propose a new method to refine it to high accuracy. 1.
Semidefinite characterization and computation of real radical ideals
 Foundations of Computational Mathematics
"... For an ideal I ⊆ R[x] given by a set of generators, a new semidefinite characterization of its real radical I(VR(I)) is presented, provided it is zerodimensional (even if I is not). Moreover we propose an algorithm using numerical linear algebra and semidefinite optimization techniques, to compute ..."
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Cited by 13 (8 self)
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For an ideal I ⊆ R[x] given by a set of generators, a new semidefinite characterization of its real radical I(VR(I)) is presented, provided it is zerodimensional (even if I is not). Moreover we propose an algorithm using numerical linear algebra and semidefinite optimization techniques, to compute all (finitely many) points of the real variety VR(I) as well as a set of generators of the real radical ideal. The latter is obtained in the form of a border or Gröbner basis. The algorithm is based on moment relaxations and, in contrast to other existing methods, it exploits the real algebraic nature of the problem right from the beginning and avoids the computation of complex components. AMS: 14P05 13P10 12E12 12D10 90C22 1