A. Since, we are in the case of Proposition 9 where we get that r X m,d (P ) = 2d. (III2.3) The scheme A is the union of a simple point and of a degree 3 curvilinear 0-dimensional scheme supported on one point. Proposition 11 gives us that r X m,d (P ) = 2d, III3) If (Supp(A)) = 3 then A can only be the union of two simple points and a degree 2

L. Albera, P. Chevalier, P. Comon, and A. Ferreol, On the virtual array concept for higher order array processing, IEEE Trans. Sig. Proc, vol.53, issue.4, pp.1254-1271, 2005.

J. Alexander and A. Hirschowitz, Polynomial interpolation in several variables, J. Algebraic Geom, vol.4, issue.2, pp.201-222, 1995.

J. Alexander and A. Hirschowitz, An asymptotic vanishing theorem for generic unions of multiple points, Inventiones Mathematicae, vol.140, issue.2, pp.3003-325, 2000.
DOI : 10.1007/s002220000053

E. Ballico and A. Bernardi, Decomposition of homogeneous polynomials with low rank. arXiv:1003.5157v2 [math, pp.10-1007
URL : https://hal.archives-ouvertes.fr/hal-00645978

M. C. Brambilla and G. Ottaviani, On the Alexander???Hirschowitz theorem, Journal of Pure and Applied Algebra, vol.212, issue.5, pp.1229-1251, 2008.
DOI : 10.1016/j.jpaa.2007.09.014

A. Bernardi, A. Gimigliano, and M. Idà, Computing symmetric rank for symmetric tensors, Journal of Symbolic Computation, vol.46, issue.1, pp.34-53, 2011.
DOI : 10.1016/j.jsc.2010.08.001
URL : https://hal.archives-ouvertes.fr/hal-00645973

J. Brachat, P. Comon, B. Mourrain, and E. P. Tsigaridas, Symmetric tensor decomposition, Linear Algebra and its Applications, vol.433, issue.11-12, pp.11-12, 2010.
DOI : 10.1016/j.laa.2010.06.046
URL : https://hal.archives-ouvertes.fr/inria-00355713

W. Buczy´nskabuczy´nska and J. Buczy´nskibuczy´nski, Secant varieties to high degree veronese reembeddings, catalecticant matrices and smoothable Gorenstein schemes

J. Buczy´nskibuczy´nski, A. Ginensky, and J. M. Landsberg, Determinantal equations for secant varieties and the Eisenbud- Koh-Stillman conjecture

J. Buczy´nskibuczy´nski and J. M. Landsberg, Rank of tensors and a generalization of secant varieties

P. G. Chevalier and M. Seiguer, Optimal separation of independent narrow-band sources -concept and performance On the rank of a binary form, Signal Processing Found. Comp. Math, vol.11, issue.1, pp.73-100, 1999.

P. Comon, Independent Component Analysis, Higher Order Statistics, pp.29-38, 1992.
URL : https://hal.archives-ouvertes.fr/hal-00346684

P. Comon, G. H. Golub, L. Lim, and B. Mourrain, Symmetric Tensors and Symmetric Tensor Rank, SIAM Journal on Matrix Analysis and Applications, vol.30, issue.3, pp.1254-1279, 2008.
DOI : 10.1137/060661569
URL : https://hal.archives-ouvertes.fr/hal-00327599

D. Lathauwer, L. Castaing, and J. , Tensor-based techniques for the blind separation of DS???CDMA signals, Signal Processing, vol.87, issue.2, pp.322-336, 2007.
DOI : 10.1016/j.sigpro.2005.12.015

M. C. Dog?andog?an and J. M. Mendel, Applications of cumulants to array processing. I. aperture extension and array calibration, IEEE Trans. Sig. Proc, vol.43, issue.5, pp.1200-1216, 1995.

D. Eisenbud, Commutative Algebra. With a view toward algebraic geometry, Graduate Texts in Mathematics, vol.150, 1995.

D. Eisenbud and J. Harris, Finite projective schemes in linearly general position, J. Algebraic Geom, vol.1, issue.1, pp.15-30, 1992.

W. Fulton, Intersection Theory, 1984.

T. Jiang and N. D. Sidiropoulos, Kruskal's Permutation Lemma and the Identification of CANDECOMP/PARAFAC and Bilinear Models with Constant Modulus Constraints, IEEE Transactions on Signal Processing, vol.52, issue.9, pp.2625-2636, 2004.
DOI : 10.1109/TSP.2004.832022

V. Kanev, Chordal varieties of Veronese varieties and catalecticant matrices Algebraic geometry, J. Math. Sci, vol.9, issue.941, pp.1114-1125, 1999.

J. M. Landsberg and Z. Teitler, On the Ranks and Border Ranks of Symmetric Tensors, Foundations of Computational Mathematics, vol.54, issue.2, pp.339-366, 2010.
DOI : 10.1007/s10208-009-9055-3

L. Lim and V. De-silva, Tensor rank and the ill-posedness of the best low-rank approximation problem, SIAM J. Matrix Anal. Appl, vol.30, issue.3, pp.1084-1127, 2008.

P. Mccullagh, Tensor Methods in Statistics, Monographs on Statistics and Applied Probability. Chapman and Hall, 1987.

G. Ottaviani, Abstract, Nagoya Mathematical Journal, vol.525, pp.95-110, 2009.
DOI : 10.1016/0024-3795(92)90437-F