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

Abstract

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 where are two complex, finite-dimensional vector spaces) if its rank with respect to the Segre–Veronese variety is at most . Such a polynomial may not have a unique minimal decomposition as with and coefficients, but we can show that there exist unique , , two unique linear forms , , and two unique bivariate polynomials and such that either or , ( 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 Segre–Veronese varieties.