59 research outputs found
Multihomogeneous Polynomial Decomposition using Moment Matrices
International audienceIn the paper, we address the important problem of tensor decomposition which can be seen as a generalisation of Sin- gular Value Decomposition for matrices. We consider gen- eral multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated moment matrix problem and we give a new criterion for flat extension of Quasi-Hankel matrices. We connect this criterion to the commutation characterisation of border bases. A new algo- rithm is described: it applies for general multihomogeneous tensors, extending the approach of J.J. Sylvester on binary forms. An example illustrates the algebraic operations in- volved in this approach and how the decomposition can be recovered from eigenvector computation
Geometric lower bounds for generalized ranks
We revisit a geometric lower bound for Waring rank of polynomials (symmetric
rank of symmetric tensors) of Landsberg and Teitler and generalize it to a
lower bound for rank with respect to arbitrary varieties, improving the bound
given by the "non-Abelian" catalecticants recently introduced by Landsberg and
Ottaviani. This is applied to give lower bounds for ranks of multihomogeneous
polynomials (partially symmetric tensors); a special case is the simultaneous
Waring decomposition problem for a linear system of polynomials. We generalize
the classical Apolarity Lemma to multihomogeneous polynomials and give some
more general statements. Finally we revisit the lower bound of Ranestad and
Schreyer, and again generalize it to multihomogeneous polynomials and some more
general settings.Comment: 43 pages. v2: minor change
Gr\"obner bases of syzygies and Stanley depth
Let F. be a any free resolution of a Z^n-graded submodule of a free module
over the polynomial ring K[x_1, ..., x_n]. We show that for a suitable term
order on F., the initial module of the p'th syzygy module Z_p is generated by
terms m_ie_i where the m_i are monomials in K[x_{p+1}, ..., x_n]. Also for a
large class of free resolutions F., encompassing Eliahou-Kervaire resolutions,
we show that a Gr\"obner basis for Z_p is given by the boundaries of generators
of F_p. We apply the above to give lower bounds for the Stanley depth of the
syzygy modules Z_p, in particular showing it is at least p+1. We also show that
if I is any squarefree ideal in K[x_1, ..., x_n], the Stanley depth of I is at
least of order the square root of 2n.Comment: 13 page
General Tensor Decomposition, Moment Matrices and Applications
SubmittedInternational audienceThe 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
Partially Symmetric Variants of Comon's Problem Via Simultaneous Rank
A symmetric tensor may be regarded as a partially symmetric tensor in several
different ways. These produce different notions of rank for the symmetric
tensor which are related by chains of inequalities. By exploiting algebraic
tools such as apolarity theory, we show how the study of the simultaneous
symmetric rank of partial derivatives of the homogeneous polynomial associated
to the symmetric tensor can be used to prove equalities among different
partially symmetric ranks. This approach aims to understand to what extent the
symmetries of a tensor affect its rank. We apply this to the special cases of
binary forms, ternary and quaternary cubics, monomials, and elementary
symmetric polynomials.Comment: 28 p
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