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 captures the moment problem for mixtures of centered ‐variate Gaussians. In the first nontrivial case , we show that for any , this map is generically one‐to‐one (up to permutations of and third roots of unity) in two ranges: for and for , thus proving generic identifiability for mixtures of centered Gaussians from their (exact) moments of degree at most . 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 th powers of ‐forms for and .
The variety Sing n,m consists of all tuples X = (X 1 , . . . , Xm) of n×n matrices such that every linear combination of X 1 , . . . , Xm is singular. Equivalently, X ∈ Sing n,m if and only if det(λ 1 X 1 +. . .+λmXm) = 0 for all λ 1 , . . . , λm ∈ Q. Makam and Wigderson [12] asked whether the ideal generated by these equations is always radical, that is, if any polynomial identity that is valid on Sing n,m lies in the ideal generated by the polynomials det(λ 1 X 1 + . . . + λmXm). We answer this question in the negative by determining the vanishing ideal of Sing 2,m for all m ∈ N. Our results exhibit that there are additional equations arising from the tensor structure of X. More generally, for any n and m ≥ n 2 − n + 1, we prove there are equations vanishing on Sing n,m that are not in the ideal generated by polynomials of type det(λ 1 X 1 + . . . + λmXm). Our methods are based on classical results about Fano schemes, representation theory and Gröbner bases.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.