Partially complex ranks for real projective varieties

Abstract

Let $X(\mathbb{C})\subset {\mathbb{P}}^r(\mathbb{C})$ be an integral non-degenerate variety defined over $\mathbb{R}$. For any $q\in {\mathbb{P}}^r(\mathbb{R})$ we study the existence of $S\subset X(\mathbb{C})$ with small cardinality, invariant for the complex conjugation and with $q$ contained in the real linear space spanned by $S$. We discuss the advantages of these additive decompositions with respect to the $X(\mathbb{R})$-rank, i.e. the rank of $q$ with respect to $X(\mathbb{R})$. We describe the case of hypersurfaces and Veronese varieties.

References in corpus (2)

• Identifiability beyond Kruskal's bound for symmetric tensors of degree 4
• On complex and real identifiability of tensors