Reconstruction of a homogeneous polynomial from its additive decompositions when identifiability fails

Abstract: Let $X\subset \mathbb {P}^r$ be an integral and non-degenerate variety. For any $q\in \mathbb {P}^r$ let $r_X(q)$ be its $X$-rank and $\mathcal {S} (X,q)$ the set of all finite subsets of $X$ such that $|S|=r_X(q)$ and $q\in \langle S\rangle$, where $\langle \ \ \rangle$ denotes the linear span. We consider the case $|\mathcal {S} (X,q)|>1$ (i.e. when $q$ is not $X$-identifiable) and study the set $W(X)_q:= \cap _{S\in\mathcal {S}}\langle S\rangle$, which we call the non-uniqueness set of $q$. We study the case $\dim X=1$ and the case $X$ a Veronese embedding of $\mathbb {P}^n$.