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 $\mathrm{O}(r)$-orbits, the problem becomes counting how many different $\mathrm{O}(r)$-orbits of decomposition exist. We say that $f$ is $\mathrm{O}(r)$-identifiable if there is a unique $\mathrm{O}(r)$-orbit. We give sufficient conditions for generic and specific $\mathrm{O}(r)$-identifiability. Moreover, we show the generic $\mathrm{O}(r)$-identifiability of ternary forms.

arXiv comment14 pages, to appear in Journal of Algebra

References in corpus (7)

• An algorithm for generic and low-rank specific identifiability of complex tensors
• On generic identifiability of symmetric tensors of subgeneric rank
• On the Waring problem for polynomial rings
• On generic and maximal k-ranks of binary forms
• On a notion of speciality of linear systems in P^n
• Identifiability for mixtures of centered Gaussians and sums of powers of quadratics
• On the degree of varieties of sum of squares

Cited by in corpus (1)

• Moment varieties of the inverse Gaussian and gamma distributions are nondefective