Equal sums of two cubes of quadratic forms

Abstract

We give a complete description of all solutions to the equation [Formula: see text] for quadratic forms [Formula: see text] and show how Ramanujan’s example can be extended to three equal sums of pairs of cubes. We also give a complete census in counting the number of ways a sextic [Formula: see text] can be written as a sum of two cubes. The extreme example is [Formula: see text], which has six such representations.

Identifiability for mixtures of centered Gaussians and sums of powers of quadratics

“…As for general identifiability, that is, uniqueness of the decomposition for a general point of a secant variety, a complete answer is given in the case of sums of powers of linear forms ($$k=1$$): In [11], it was shown that in all but a few exceptional cases, identifiability holds for all subgeneric ranks, while Galuppi [14] completed the classification of cases in which identifiability also holds for generic ranks. For $$k \geqslant 2$$, as far as we know, before the present work, only identifiability for sextics as sums of cubes was recently addressed for rank $$\hskip.001pt 2$$ [28].…”