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

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 $S^{d_1}V_1\otimes S^{d_2}V_2$ where $V_1,V_2$ are two complex, finite dimensional vector spaces) if its rank with respect to the Segre-Veronese variety $S_{d_1,d_2}(V_1,V_2)$ is at most $\min \{d_1,d_2\}$. Such a polynomial may not have a unique minimal decomposition as $p=\sum_{i=1}^r\lambda_i p_i$ with $p_i\in S_{d_1,d_2}(V_1,V_2)$ and $\lambda_i$ coefficients, but we can show that there exist unique $p_1, \ldots , p_{r'}$, $p_{1}', \ldots , p_{r''}'\in S_{d_1,d_2}(V_1,V_2) $, two unique linear forms $l\in V_1^*$, $l'\in V_2^*$, and two unique bivariate polynomials $q\in S^{d_2}V_2^*$ and $q'\in S^{d_1}V_1^*$ such that either $p=\sum_{i=1}^{r'} \lambda_i p_i+l^{d_1}q $ or $ p= \sum_{i=1}^{r''}\lambda'_i p_i'+l'^{d_2}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 Segre-Veronese varieties.