Decomposition of homogeneous polynomials with low rank

作者：Ballico Edoardo^*; Bernardi Alessandra

来源：Mathematische Zeitschrift, 2012, 271(3-4): 1141-1149.

DOI：10.1007/s00209-011-0907-6

• Waring problem
• Polynomial decomposition
• Symmetric rank
• Symmetric tensors
• Veronese varieties
• Secant varieties

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 s-th secant variety of the d-uple Veronese embedding of into but that its minimal decomposition as a sum of d-th powers of linear forms M (1), . . . , M (r) is with r %26gt; s. We show that if s + r a parts per thousand currency sign 2d + 1 then such a decomposition of F can be split in two parts: one of them is made by linear forms that can be written using only two variables, the other part is uniquely determined once one has fixed the first part. We also obtain a uniqueness theorem for the minimal decomposition of F if r is at most d and a mild condition is satisfied.

• 出版日期2012-8
• 单位INRIA