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Unique decomposition for a polynomial of low rank

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Ballico E., Bernardi A.

Annales Polonici Mathematici

2013

tom 108

nr 3

219-224

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 $ℙ^m$ into $ℙ^{{m+d \atop d}-1}$ 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 $F = M₁^d + ⋯ + M_t^d + Q$, where $M₁,. .., M_t$ 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 of degree $d_i$ such that $∑(d_i + 1) = s - t.$

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1 Annales Polonici Mathematici

1 Ballico E.

1 Bernardi A.

1 2013

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Unique decomposition for a polynomial of low rank