Unique decomposition for a polynomial of low rank

Edoardo Ballico; Alessandra Bernardi

Annales Polonici Mathematici, Tome 107 (2013), p. 219-224 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $ℙ^{\binom{m + d}{d} - 1}$ 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 $F = M ₁^{d} + \dots + M_{t}^{d} + Q$ , where $M ₁, . . ., M_{t}$ are linear forms with t ≤ (d-1)/2, and Q is a binary form such that $Q = \sum_{i = 1}^{q} l_{i}^{d - d_{i}} m_{i}$ with $l_{i}$ ’s linear forms and $m_{i}$ ’s forms of degree $d_{i}$ such that $\sum (d_{i} + 1) = s - t .$

Publié le : 2013-01-01

Zbl 1297.14058

EUDML-ID : urn:eudml:doc:280239

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     author = {Edoardo Ballico and Alessandra Bernardi},
     title = {Unique decomposition for a polynomial of low rank},
     journal = {Annales Polonici Mathematici},
     volume = {107},
     year = {2013},
     pages = {219-224},
     zbl = {1297.14058},
     language = {en},
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Edoardo Ballico; Alessandra Bernardi. Unique decomposition for a polynomial of low rank. Annales Polonici Mathematici, Tome 107 (2013) pp. 219-224. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap108-3-2/