On the number of Waring decompositions for a generic polynomial vector

TL;DR

This paper establishes the uniqueness of a specific Waring decomposition for a generic polynomial vector of degrees (3,3,4) in three variables, introducing new cases and bounds in tensor decomposition theory.

Contribution

It proves a unique Waring decomposition for a specific polynomial vector and provides new insights into the identifiability and number of decompositions for related cases.

Findings

Unique Waring decomposition of rank 7 for (3,3,4) polynomial vector

No identifiable cases among pairs (f1, f2) with degrees (a, a+1) unless a=2

Lower bounds on the number of decompositions for certain polynomial pairs

Abstract

We prove that a general polynomial vector $(f_1, f_2, f_3)$ in three homogeneous variables of degrees $(3,3,4)$ has a unique Waring decomposition of rank 7. This is the first new case we are aware, and likely the last one, after five examples known since 19th century and the binary case. We prove that there are no identifiable cases among pairs $(f_1, f_2)$ in three homogeneous variables of degree $(a, a+1)$, unless $a=2$, and we give a lower bound on the number of decompositions. The new example was discovered with Numerical Algebraic Geometry, while its proof needs Nonabelian Apolarity.