In this paper we derive aggregate separation bounds, named after Davenport-Mahler-Mignotte (dmm), on the isolated roots of polynomial systems, specifically on the minimum distance between any two such roots. The bounds exploit the structure of the sy
stem and the height of the sparse (or toric) resultant by means of mixed volume, as well as recent advances on aggregate root bounds for univariate polynomials, and are applicable to arbitrary positive dimensional systems. We improve upon Cannys gap theorem cite{c-crmp-87} by a factor of $OO(d^{n-1})$, where $d$ bounds the degree of the polynomials, and $n$ is the number of variables. One application is to the bitsize of the eigenvalues and eigenvectors of an integer matrix, which also yields a new proof that the problem is polynomial. We also compare against recent lower bounds on the absolute value of the root coordinates by Brownawell and Yap cite{by-issac-2009}, obtained under the hypothesis there is a 0-dimensional projection. Our bounds are in general comparable, but exploit sparseness; they are also tighter when bounding the value of a positive polynomial over the simplex. For this problem, we also improve upon the bounds in cite{bsr-arxix-2009,jp-arxiv-2009}. Our analysis provides a precise asymptotic upper bound on the number of steps that subdivision-based algorithms perform in order to isolate all real roots of a polynomial system. This leads to the first complexity bound of Milnes algorithm cite{Miln92} in 2D.
We present an algorithm for decomposing a symmetric tensor, of dimension n and order d as a sum of rank-1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for binary forms. We recall the correspondence between the decomposition
of a homogeneous polynomial in n variables of total degree d as a sum of powers of linear forms (Warings problem), incidence properties on secant varieties of the Veronese Variety and the representation of linear forms as a linear combination of evaluations at distinct points. Then we reformulate Sylvesters approach from the dual point of view. Exploiting this duality, we propose necessary and sufficient conditions for the existence of such a decomposition of a given rank, using the properties of Hankel (and quasi-Hankel) matrices, derived from multivariate polynomials and normal form computations. This leads to the resolution of polynomial equations of small degree in non-generic cases. We propose a new algorithm for symmetric tensor decomposition, based on this characterization and on linear algebra computations with these Hankel matrices. The impact of this contribution is two-fold. First it permits an efficient computation of the decomposition of any tensor of sub-generic rank, as opposed to widely used iterative algorithms with unproved global convergence (e.g. Alternate Least Squares or gradient descents). Second, it gives tools for understanding uniqueness conditions, and for detecting the rank.
