The asymptotic leading term for maximum rank of ternary forms of a given degree

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 n, d, 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 forms of degree d ≥ 3 can be annihilated by the product of d − 1 pairwise independent linear forms.

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“…Moreover, note that a condition bℓ ∂ l 0 g 1 = · · · = bℓ ∂ l 0 g t = s+1 is used in the proof of [14, Lemma 2.7], but only to get [14,Eq. (8)].…”

Section: Maximum Open Rank For Ternary Quarticsmentioning

confidence: 99%

Generic Power Sum Decompositions and Bounds for the Waring Rank

Ballico

Paris

2017

Discrete Comput Geom

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“…Of course, it is tempting also to ask a more precise information about r max (n, d) for d 0. In the case n = 2, De Paris proved in [55,56] …”

Section: Questions On the Case Of Veronese Varietiesmentioning

confidence: 96%

“…Let r max (n, d) denote the maximum of all X n,d -ranks (in [55,56] it is denoted with r max (n + 1, d)). The integer r max (n, d) depends on two variables, n and d. In this section, we ask some question on the asymptotic behavior of r max (n, d) when we fix one variable, while the other one goes to +∞.…”

Section: Questions On the Case Of Veronese Varietiesmentioning

confidence: 99%

Set Evincing the Ranks with Respect to an Embedded Variety (Symmetric Tensor Rank and Tensor Rank

Ballico

2018

Mathematics

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Let X ⊂ P r be an integral and non-degenerate variety. We study when a finite set S ⊂ X evinces the X-rank of the general point of the linear span of S. We give a criterion when X is the order d Veronese embedding X n,d of P n and |S| ≤ (For the tensor rank, we describe the cases with |S| ≤ 3. For X n,d , we raise some questions of the maximum rank for d 0 (for a fixed n) and for n 0 (for a fixed d).

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“…In a few cases in small numbers of variables and small degrees, exact values of maximal ranks have been given. We resume them in the following [165] We want to underline the fact that it is very difficult to find examples of forms having high rank, in the sense higher than the generic rank. Thanks to the complete result on monomials in [166] (see Theorem 5.15), we can easily see that in the case of binary and ternary forms, we can find monomials having rank higher than the generic one.…”

Section: Which Is the Minimum Integer R Such That Any Form Can Be Wrimentioning

confidence: 99%

The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition

et al. 2018

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We consider here the problem, which is quite classical in Algebraic geometry, of studying the secant varieties of a projective variety X. The case we concentrate on is when X is a Veronese variety, a Grassmannian or a Segre variety. Not only these varieties are among the ones that have been most classically studied, but a strong motivation in taking them into consideration is the fact that they parameterize, respectively, symmetric, skew-symmetric and general tensors, which are decomposable, and their secant varieties give a stratification of tensors via tensor rank. We collect here most of the known results and the open problems on this fascinating subject.

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The asymptotic leading term for maximum rank of ternary forms of a given degree

Cited by 10 publications

References 13 publications

Generic Power Sum Decompositions and Bounds for the Waring Rank

Generic Power Sum Decompositions and Bounds for the Waring Rank

Set Evincing the Ranks with Respect to an Embedded Variety (Symmetric Tensor Rank and Tensor Rank

The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition

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