An upper bound for the Waring rank of a form

Jelisiejew, Joachim

doi:10.1007/s00013-014-0632-6

Cited by 21 publications

(34 citation statements)

References 5 publications

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“…Apply m − 1 times [1], proof of Theorem 1 up to line 12 of page 6, and the proof of Lemma 2. In the case m = 1 and char(K) = 0, then this is the proof of [2], Lemma 11.…”

Section: The Proofsmentioning

confidence: 80%

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Open Rank (Or Widerank) for Reducible Projective Sets: A Classification of an Extremal Case

Ballico¹

2014

Int. J. of Pure and Appl. Math.

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Let X ⊂ P r , r ≥ 2, be a reduced projective set such that X = P r , where denote the linear span of X. For any P ∈ P r the X-widerank w X (P ) of P is the minimal integer t > 0 such that for each closed set B ⊂ X containing no irreducible component of X there is S ⊂ X \ B with P ∈ S . Here we classify all (r, X) such that w X (P ) ≥ r + 1 for a general P (r is odd and X is the union of (r + 1)/2 linearly independent lines). We give conditions on X which imply that w X (P ) ≤ r (or w X (P ) ≤ r+1−dim(X)) for every P ∈ P r \X. AMS Subject Classification: 14N05, 14Q05, 15A69Key Words: widerank, open rank, symmetric tensor rank, reducible varieties, lines, strange variety The StatementsFor each set B ⊆ P r let B denote its linear span.Let X ⊂ P r , r ≥ 2, be a reduced projective set such that X = P r . For each P ∈ P r the X-rank r X (P ) of P is the minimal cardinality of a set S ⊂ X such that P ∈ S . Now assume that no component of X is a single point. The X-widerank w X (P ) of P is the minimal integer t > 0 such that for each closed set B ⊂ X containing no irreducible component of X there is S ⊂ X \ B with

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“…Apply m − 1 times [1], proof of Theorem 1 up to line 12 of page 6, and the proof of Lemma 2. In the case m = 1 and char(K) = 0, then this is the proof of [2], Lemma 11.…”

Section: The Proofsmentioning

confidence: 80%

“…If X is a Veronese embedding of P m , then the definition of w X (P ) is related to the open rank or(P ) defined in [2]: when or(P ) is defined, then or(P ) = w X (P ), but or(P ) is defined only for points associated to forms depending on all m + 1 homogeneous variables.…”

Section: Theoremmentioning

confidence: 99%

Open Rank (Or Widerank) for Reducible Projective Sets: A Classification of an Extremal Case

Ballico¹

2014

Int. J. of Pure and Appl. Math.

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“…In the classical case of Waring ranks, the latter bound is (almost) sharp for binary forms, but in many other cases it seems rather crude. At present, better bounds are known only in few special cases of low degrees Ballico and Paris (2017), Jelisiejew (2014). To the best of our knowledge, the exact values of the maximal Waring rank are only known for binary forms (classical, see ), quadrics (classical), ternary cubics [see Segre (1942), Landsberg and Teitler (2010)], ternary quartics, see Kleppe (1999), ternary quintics, see De Paris (2015) and quaternary cubics, see Segre (1942).…”

Section: Problem B Given a Triple Of Positive Integersmentioning

confidence: 99%

Algebraic Stories from One and from the Other Pockets

et al. 2018

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“…For some recent progress on upper bounds for Waring rank, see [4], [15], [3], and [5]. But there are only a few cases in which the actual maximum rank, or even explicit forms of greater than generic rank, are known.…”

Section: Introductionmentioning

confidence: 99%