Riv. Mat. Univ. Parma, Vol. 11, No. 2, 2020

Edoardo Ballico [a]

Partially complex ranks for real projective varieties

Pages: 207-216
Received: 4 January 2019
Accepted in revised form: 17 July 2019
Mathematics Subject Classification (2010): 14N05, 15A69.
Keywords: Tensor rank, real tensor rank, real symmetric tensor rank, additive decomposition of polynomials, typical rank.
Author address:
[a]: Department of Mathematics, University of Trento, 38123 Povo (TN), Italy

The author was partially supported by MIUR and GNSAGA of INdAM (Italy)

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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.

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