Waring decompositions and identifiability via Bertini and Macaulay2 software

TL;DR

This paper investigates the identifiability of certain polynomial vectors using Waring decompositions, employing computer algebra tools like Bertini and Macaulay2 to analyze complex and real cases, and establishing new results on real identifiability.

Contribution

It demonstrates the existence of polynomial vectors with specific properties that are not identifiable over complex numbers but are over real numbers, and applies the Hessian criterion for identifiability in sub-generic cases.

Findings

Existence of polynomial vectors with non-unique complex decompositions but unique real decompositions.

Application of computer-aided methods to study Waring decompositions.

Proved identifiability over complex numbers for many sub-generic rank cases.

Abstract

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.