On the ranks and border ranks of symmetric tensors. (English summary) Found. Comput. Math. 10 (2010), no. 3, 339–366.1615-3383 This paper deals with sums of powers decompositions of homogenous polynomials. The authors study two quantities related to a degree d form F. The rank (also called Waring rank in this paper) R(F) represents the minimal number of linear forms xi such that one has F = ∑R(F) 1 xd i. The border rank R(F) represents the minimal number such that F is in the Zariski closure of forms of rank R(F). Thus, for example, R(x2y) = 3 and R(x2y) = 2. Among the many results in the paper, the authors compute R for special monomials. They also produce a lower bound for R(F), F any form, depending on the singularities of the hypersurface F = 0. Explicit lower bounds are presented for the determinant and the permanent polynomials. Finally, they present an upper bound for R(F), for any form F
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