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Generic Power Sum Decompositions and Bounds for the Waring Rank


A notion of open rank, related with generic power sum decompositions of forms, has recently been introduced in the literature. The main result here is that the maximum open rank for plane quartics is eight. In particular, this gives the first example of nd, such that the maximum open rank for degree d forms that essentially depend on n variables is strictly greater than the maximum rank. On one hand, the result allows to improve the previously known bounds on open rank, but on the other hand indicates that such bounds are likely quite relaxed. Nevertheless, some of the preparatory results are of independent interest, and still may provide useful information in connection with the problem of finding the maximum rank for the set of all forms of given degree and number of variables. For instance, we get that every ternary form of degree \(d\ge 3\) can be annihilated by the product of \(d-1\) pairwise independent linear forms.

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    Sometimes the term “Waring decomposition” has been used to indicate simply a power sum decomposition, without the minimality hypothesis. We also mention that the symmetric rank is sometimes called polar rank: see [25].

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Some improvements are due to an anonymous referee, in particular Example 3.4 and a simplification in the proof of Lemma 4.2. Financial support by MIUR (IT), GNSAGA of INdAM (IT) and Università degli Studi di Napoli Federico II (IT).

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Correspondence to Alessandro De Paris.

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Ballico, E., De Paris, A. Generic Power Sum Decompositions and Bounds for the Waring Rank. Discrete Comput Geom 57, 896–914 (2017). https://doi.org/10.1007/s00454-017-9886-7

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  • Power sum
  • Waring rank
  • Tensor rank
  • Symmetric tensor

Mathematics Subject Classification

  • 15A21
  • 15A69
  • 15A72
  • 14A25
  • 14N05
  • 14N15