Identifiability of homogeneous polynomials and Cremona Transformations

Abstract

A homogeneous polynomial of degree $d$ in $n+1$ variables is identifiable if it admits a unique additive decomposition in powers of linear forms. Identifiability is expected to be very rare. In this paper we conclude a work started more than a century ago and we describe all values of $d$ and $n$ for which a general polynomial of degree $d$ in $n+1$ variables is identifiable. This is done by classifying a special class of Cremona transformations of projective spaces.

arXiv comment25 pages. Proof of Proposition 17 and Lemma 18 fixed, some typos corrected

References in corpus (3)

• An algorithm for generic and low-rank specific identifiability of complex tensors
• On generic identifiability of symmetric tensors of subgeneric rank
• Birational aspects of the Geometry of Varieties of Sum of Powers

Cited by in corpus (10)

• The Hitchhiker guide to: Secant Varieties and Tensor Decomposition
• On generic identifiability of symmetric tensors of subgeneric rank
• Secant non-defectivity via collisions of fat points
• Waring, tangential and cactus decompositions
• Algebraic stories from one and from the other pockets
• Identifiability for mixtures of centered Gaussians and sums of powers of quadratics
• Decompositions and Terracini loci of cubic forms of low rank
• Minimality and uniqueness for decompositions of specific ternary forms
• Generic identifiability of pairs of ternary forms
• Cremona equivalence and log Kodaira dimension