Partially complex ranks for real projective varieties. (English) Zbl 1460.14117

Ballico, Edoardo

Riv. Mat. Univ. Parma (N.S.) 11, No. 2, 207-216 (2020).

Summary: Let \(X(C)\subset \mathbb{P}^r(\mathbb{C})\) 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.

MSC:

14N07	Secant varieties, tensor rank, varieties of sums of powers
15A69	Multilinear algebra, tensor calculus
14P05	Real algebraic sets

Keywords:

tensor rank; real tensor rank; real symmetric tensor rank; additive decomposition of polynomials; typical rank