Generalized identifiability of sums of squares

Abstract

Let f be a homogeneous polynomial of even degree d. We study the decompositions $f={\sum }_{i=1}^r{f}_i^2$ where $\mathrm{deg}{f}_i=d/2$. The minimal number of summands r is called the 2-rank of f, so that the polynomials having 2-rank equal to 1 are exactly the squares. Such decompositions are never unique and they are divided into $O\left(r\right)$-orbits, the problem becomes counting how many different $O\left(r\right)$-orbits of decomposition exist. We say that f is $O\left(r\right)$-identifiable if there is a unique $O\left(r\right)$-orbit. We give sufficient conditions for generic and specific $O\left(r\right)$-identifiability. Moreover, we show the generic $O\left(r\right)$-identifiability of ternary forms.