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Structured low-rank approximation is the problem of minimizing a weighted Frobenius distance to a given matrix among all matrices of fixed rank in a linear space of matrices. We study the critical points of this optimization problem using algebraic geometry. A particular focus lies on Hankel matrices, Sylvester matrices and generic linear spaces.

Solving multihomogeneous systems, as a wide range of structured algebraic systems occurring frequently in practical problems, is of first importance. Experimentally, solving these systems with Gröbner bases algorithms seems to be easier than solving homogeneous systems of the same degree. Nevertheless, the reasons of this behaviour are not clear. In this… (More)

Computing loci of rank defects of linear matrices (also called the MinRank problem) is a fundamental NP-hard problem of linear algebra which has applications in Cryptology, in Error Correcting Codes and in Geometry. Given a square linear matrix (i.e. a matrix whose entries are <i>k</i>-variate linear forms) of size <i>n</i> and an integer <i>r</i>, the… (More)

A fundamental problem in computer science is to find all the common zeroes of m quadratic polynomials in n unknowns over F 2. The cryptanalysis of several modern ciphers reduces to this problem. Up to now, the best complexity bound was reached by an exhaustive search in 4 log 2 n 2 n operations. We give an algorithm that reduces the problem to a combination… (More)

We consider the problem of computing critical points of the restriction of a polynomial map to an algebraic variety. This is of first importance since the global minimum of such a map is reached at a critical point. Thus, these points appear naturally in non-convex polynomial optimization which occurs in a wide range of scientific applications (control… (More)

Toric (or sparse) elimination theory is a framework developped during the last decades to exploit monomial structures in systems of Laurent polynomials. Roughly speaking, this amounts to computing in a <i>semigroup algebra</i>, <i>i.e</i>. an algebra generated by a subset of Laurent monomials. In order to solve symbolically sparse systems, we introduce… (More)

Computing the critical points of a polynomial function q ∈ is of first importance in several applications in optimization and in real algebraic geometry. These points are solutions of a highly structured system of multivariate polynomial equations involving maximal minors of a Jacobian matrix. We investigate the complexity of solving this problem by using… (More)

Structured Low-Rank Approximation is a problem arising in a wide range of applications in Numerical Analysis and Engineering Sciences. Given an input matrix M , the goal is to compute a matrix M ′ of given rank r in a linear or affine subspace E of matrices (usually encoding a specific structure) such that the Frobenius distance M − M ′ is small. We propose… (More)

In this paper, we fully break the Algebraic Surface Cryptosystem (ASC for short) proposed at PKC'2009 [3]. This system is based on an unusual problem in multivari-ate cryptography: the Section Finding Problem. Given an algebraic surface X(x, y,t) ∈ F p [x, y,t] such that deg xy X(x, y,t) = w, the question is to find a pair of polynomials of degree d, u x… (More)