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)(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-1(d) + ... + M-t(d) + Q, where M-1, ... , M-t are linear forms with t <= (d - 1)/2, and Q is a binary form such that Q = Sigma(q)(i=1) l(i)(d-di)m(i) with l(i)'s linear forms and m(i)'s forms of degree d(i) such that Sigma(d(i) + 1) = s - t.

Unique decomposition for a polynomial of low rank / Ballico, Edoardo; Bernardi, Alessandra. - In: ANNALES POLONICI MATHEMATICI. - ISSN 0066-2216. - 108:(2013), pp. 219-224. [10.4064/ap108-3-2]

Unique decomposition for a polynomial of low rank

Ballico, Edoardo;Bernardi, Alessandra
2013-01-01

Abstract

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)(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-1(d) + ... + M-t(d) + Q, where M-1, ... , M-t are linear forms with t <= (d - 1)/2, and Q is a binary form such that Q = Sigma(q)(i=1) l(i)(d-di)m(i) with l(i)'s linear forms and m(i)'s forms of degree d(i) such that Sigma(d(i) + 1) = s - t.
2013
Ballico, Edoardo; Bernardi, Alessandra
Unique decomposition for a polynomial of low rank / Ballico, Edoardo; Bernardi, Alessandra. - In: ANNALES POLONICI MATHEMATICI. - ISSN 0066-2216. - 108:(2013), pp. 219-224. [10.4064/ap108-3-2]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11572/134881
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