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

TL;DR

This paper characterizes all minimal decompositions of bi-homogeneous polynomials with low rank, showing their uniqueness up to certain components, and analyzes tensor ranks and decompositions within the tangential variety of Segre-Veronese varieties.

Contribution

It provides a complete description of minimal decompositions for bi-homogeneous polynomials with low rank, including their uniqueness and structure, and studies tensor ranks in the tangential variety.

Findings

Unique decomposition forms involving linear forms and bivariate polynomials.

Explicit rank calculations for tensors in the tangential variety.

Structural description of tensor decompositions in the tangential variety.

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 $S^{d_1}V_1\otimes S^{d_2}V_2$ 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.