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Unique decomposition for a polynomial of low rank

Edoardo Ballico, Alessandra Bernardi (2013)

Annales Polonici Mathematici

Let F be a homogeneous polynomial of degree d in m + 1 variables defined over an algebraically closed field of characteristic 0 and suppose that F belongs to the sth secant variety of the d-uple Veronese embedding of m into m + d d - 1 but that its minimal decomposition as a sum of dth powers of linear forms requires more than s summands. We show that if s ≤ d then F can be uniquely written as F = M d + + M t d + Q , where M , . . . , M t are linear forms with t ≤ (d-1)/2, and Q is a binary form such that Q = i = 1 q l i d - d i m i with l i ’s linear forms and m i ’s forms...

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