Rank 1 decompositions of symmetric tensors outside a fixed support
Summary: Let \(\nu_d:\mathbb P^m \to \mathbb P^n\), \( n:= \binom {n+d}{n} - 1\), denote the degree \(d\) Veronese embedding of \(\mathbb P^m\). For any \(P \in \mathbb P^n\), let \(\mathrm{sr}(P)\) be the minimal cardinality of \(S \subset \nu_d(\mathbb P^m)\) such that \(P \in \langle S \rangle\). Identifying \(P\) with a homogeneous polynomial \(q\) (or a symmetric tensor), \(S\) corresponds to writing \(q\) as a sum of \(\sharp(S)\) powers \(L^d\) with \(L\) a linear form (or as a sum of \(\sharp(S)\,d\)-powers of vectors). Here we fix an integral variety \(T \varsubsetneq \mathbb P^m\) and \(P \in \langle \nu_d(T) \rangle\) and study a similar decomposition with \(S \nsubseteq T\) and \(\sharp(S)\) minimal. For instance, if \(T\) is a linear subspace, then we prove that \(\sharp(S) \geq \sharp(S \cap T) + d+1\) and classify all \((S,P)\) such that \(\sharp(S) - \sharp(S \cap T) \leq 2d-1\).
scientific article; zbMATH DE number 6304704