Waring identifiability for powers of forms via degenerations

TL;DR

This paper introduces a novel approach combining Terracini's lemma and degeneration techniques, including toric geometry, to analyze the identifiability and secant defectivity of varieties parametrizing powers of forms, with implications for Waring decompositions.

Contribution

It presents a new method for proving secant non-defectivity and identifiability of forms' powers, extending previous bounds on Waring rank using degenerations and toric geometry.

Findings

Established conditions for secant non-defectivity of certain varieties.

Proved identifiability of Waring decompositions for general forms of degree kd.

Provided bounds on Waring rank based on degeneration techniques.

Abstract

We discuss an approach to the secant non-defectivity of the varieties parametrizing $k$-th powers of forms of degree $d$. It employs a Terracini type argument along with certain degeneration arguments, some of which are based on toric geometry. This implies a result on the identifiability of the Waring decompositions of general forms of degree kd as a sum of $k$-th powers of degree $d$ forms, for which an upper bound on the Waring rank was proposed by Fr\"oberg, Ottaviani and Shapiro.