On the real \(X\)-ranks of points of \(\mathbb{P}^n(\mathbb{R})\) with respect to a real variety \(X\subset\mathbb{P}^n\)

Edoardo Ballico


Let \(X\subset\mathbb{P}^n\) be an integral and non-degenerate \(m\)-dimensional variety defined over \(\mathbb{R}\). For any \(P \in \mathbb{P}^n(\mathbb{R})\) the real \(X\)-rank \(r_{X,\mathbb{R}}(P)\) is the minimal cardinality of \(S\subset X(\mathbb{R})\) such that \(P\in \langle S\rangle\). Here we extend to the real case an upper bound for the \(X\)-rank due to Landsberg and Teitler.


Ranks; real variety; structured rank

Full Text:



Albera, L., Chevalier, P., Comon, P. and Ferreol, A., On the virtual array concept for higher order array processing, IEEE Trans. Signal Process. 53(4) (2005), 1254-1271.

Arbarello, E., Cornalba, M., Griffiths, P. and Harris, J., Geometry of Algebraic Curves. I, Springer-Verlag, New York, 1985.

Ballico, E., Ranks of subvarieties of Pn over non-algebraically closed fields, Int. J. Pure Appl. Math. 61(1) (2010), 7-10.

Ballico, E., Subsets of the variety (X subset mathbb{P}^n) computing the (X)-rank of a point of (mathbb{P}^n), preprint.

Bernardi, A., Gimigliano, A. and Ida, M., Computing symmetric rank for symmetric tensors, J. Symbolic Comput. 46 (2011), 34-55.

Bochnak, J., Coste, M. and Roy, F.-M., Real Algebraic Geometry, Translated from the 1987 French original. Revised by the authors. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 36, Springer-Verlag, Berlin, 1998.

Buczynski, J., Landsberg, J. M., Ranks of tensors and a generalization of secant varieties, arXiv:0909.4262v1 [math.AG].

Comas, G., Seiguer, M., On the rank of a binary form, arXiv:math.AG/0112311.

Comon, P., Golub, G., Lim, L.-H. and Mourrain, B., Symmetric tensors and symmetric tensor rank, SIAM J. Matrix Anal. Appl. 30(3) (2008), 1254-1279.

Hartshorne, R., Algebraic Geometry, Springer-Verlag, Berlin, 1977.

Landsberg, J. M., Teitler, Z., On the ranks and border ranks of symmetric tensors, Found. Comput. Math. 10 (2010), 339-366.

Lang, S., Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965.

DOI: http://dx.doi.org/10.17951/a.2010.54.2.15-19
Data publikacji: 2016-07-29 10:39:54
Data złożenia artykułu: 2016-07-28 18:46:17


Total abstract view - 120
Downloads (from 2020-06-17) - PDF - 2



  • There are currently no refbacks.

Copyright (c) 2010 Edoardo Ballico