We consider the inverse problem for the polynomial map that sends an -tuple of quadratic forms in variables to the sum of their th powers. This map cap- tures the moment problem for mixtures of centered -variate Gaussians. In the first nontrivial case = 3, we show that for any ∈ ℕ, this map is generically one- to-one (up to permutations of 1 , … , and third roots of unity) in two ranges: ⩽ ( 2 ) + 1 for < 16 and ⩽ (+5 6 )∕(+1 2 ) − (+1 2 ) − 1 for ⩾ 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 tan- gential 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 th powers of -forms for ⩾ 3 and ⩾ 2.

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Taveira Blomenhofer, F. A., Casarotti, A., Michalek, M., & Oneto, A. (2023). Identifiability for mixtures of centered Gaussians and sums of powers of quadratics. Bulletin of the London Mathematical Society, 55(5), 2087–2556. doi:10.1112/blms.12871