A uniqueness result on the decompositions of a bi-homogeneous polynomial

作者：Ballico Edoardo; Bernardi Alessandra^*

来源：Linear and Multilinear Algebra, 2017, 65(4): 677-698.

DOI：10.1080/03081087.2016.1202182

• Bi-homogeneous polynomials
• tensor decomposition
• identifiability
• Segre-Veronese varieties
• tangential variety

In the first part of this paper we give a precise description of all the minimal decompositions of any bi-homogeneous polynomial p (i. e. a partially symmetric tensor of (SV1)-V-d1 circle times (SV2)-V-d2 where V-1, V-2 are two complex, finite-dimensional vector spaces) if its rank with respect to the Segre-Veronese variety S-d1,S- (d2) (V-1, V-2) is at most min{d(1), d(2)}. Such a polynomial may not have a unique minimal decomposition as p = Sigma(r)(i) = 1 lambda(ipi) with p(i) is an element of S-d1,S- (d2) (V-1, V-2) and lambda(i) coefficients, but we can show that there exist unique p(1),..., p(r') , p'(1),..., p'(r ''). S-d1,S- (d2) (V-1, V-2), two unique linear forms l is an element of V*(1), l' is an element of V*(2), and two unique bivariate polynomials q is an element of (SV)-V-d2*(2) and q'. (SV)-V-d1*(1) such that either p = Sigma(i=1) r' lambda(ipi) + l(d1)q or p =Sigma(i=1) r' lambda(ipi) + l' (d2)q', (lambda(i),lambda'(i) being appropriate coefficients). In the second part of the paper we focus on the tangential variety of the Segre-Veronese varieties. We compute the rank of their tensors (that is valid also in the case of Segre-Veronese of more factors) and we describe the structure of the decompositions of the elements in the tangential variety of the two factors SegreVeronese varieties.

• 出版日期2017