The tensor decomposition addressed in this paper may be seen as a generalisation of Singular Value Decomposition of matrices. We consider general multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated moment matrix
problem and give a new criterion for flat extension of Quasi-Hankel matrices. We connect this criterion to the commutation characterisation of border bases. A new algorithm is described. It applies for general multihomogeneous tensors, extending the approach of J.J. Sylvester to binary forms. An example illustrates the algebraic operations involved in this approach and how the decomposition can be recovered from eigenvector computation.
The Hilbert scheme $mathbf{Hilb}_{p(t)}^{n}$ parametrizes closed subschemes and families of closed subschemes in the projective space $mathbb{P}^n$ with a fixed Hilbert polynomial $p(t)$. It is classically realized as a closed subscheme of a Grassman
nian or a product of Grassmannians. In this paper we consider schemes over a field $k$ of characteristic zero and we present a new proof of the existence of the Hilbert scheme as a subscheme of the Grassmannian $mathbf{Gr}_{p(r)}^{N(r)}$, where $N(r)= h^0 (mathcal{O}_{mathbb{P}^n}(r))$. Moreover, we exhibit explicit equations defining it in the Plucker coordinates of the Plucker embedding of $mathbf{Gr}_{p(r)}^{N(r)}$. Our proof of existence does not need some of the classical tools used in previous proofs, as flattening stratifications and Gotzmanns Persistence Theorem. The degree of our equations is $text{deg} p(t)+2$, lower than the degree of the equations given by Iarrobino and Kleiman in 1999 and also lower (except for the case of hypersurfaces) than the degree of those proved by Haiman and Sturmfels in 2004 after Bayers conjecture in 1982. The novelty of our approach mainly relies on the deeper attention to the intrinsic symmetries of the Hilbert scheme and on some results about Grassmannian based on the notion of extensors.
In this paper, we give new explicit representations of the Hilbert scheme of $mu$ points in $PP^{r}$ as a projective subvariety of a Grassmanniann variety. This new explicit description of the Hilbert scheme is simpler than the previous ones and glob
al. It involves equations of degree $2$. We show how these equations are deduced from the commutation relations characterizing border bases. Next, we consider infinitesimal perturbations of an input system of equations on this Hilbert scheme and describe its tangent space. We propose an effective criterion to test if it is a flat deformation, that is if the perturbed system remains on the Hilbert scheme of the initial equations. This criterion involves in particular formal reduction with respect to border bases.
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.
