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

We consider the inverse problem for the polynomial map that sends an m-tuple of quadratic forms in n variables to the sum of their dth powers. This map captures the moment problem for mixtures of m centered n-variate Gaussians. In the first nontrivial case d =3, we show that for any n is an element of N, this map is generically one-to-one (up to permutations of q1, ... ,q(m) and third roots of unity) in two ranges: m <= [(GRAPHICS)] + 1 for n < 16 and <= [(GRAPHICS) / [(GRAPHICS)] -[(GRAPHICS)] - 1 for n >= 16, thus proving generic identifiability for mixtures of centered Gaussians from their (exact) moments of degree at most 6. The first result is obtained by the explicit geometry of the tangential contact locus of the variety of sums of cubes of quadratic forms, as described by Chiantini and Ottaviani [SIAM J. Matrix Anal. Appl. 33 (2012), no. 3, 1018-1037], while the second result is accomplished using the link between secant nondefectivity with identifiability, proved by Casarotti and Mella [J. Eur. Math. Soc. (JEMS) (2022)]. The latter approach also generalizes to sums of dth powers of k-forms for d >= 3 and k >= 2.