Waring decompositions and identifiability via Bertini and Macaulay2 software

Starting from our previous papers [AGMO] and [ABC], we prove the existence of a non-empty Euclidean open subset whose elements are polynomial vectors with 4 components, in 3 variables, degrees, respectively, 2,3,3,3 and rank 6, which are not identifiable over $\mathbb{C}$ but are identifiable over $\mathbb{R}$. This result has been obtained via computer-aided procedures suitably adapted to investigate the number of Waring decompositions for general polynomial vectors over the fields of complex and real numbers. Furthermore, by means of the Hessian criterion ([COV]), we prove identifiability over $\mathbb{C}$ for polynomial vectors in many cases of sub-generic rank.