Real rank with respect to varieties

Blekherman, Grigoriy; Sinn, Rainer

doi:10.1016/j.laa.2016.04.035

Cited by 13 publications

(20 citation statements)

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“…This was also observed by Geramita when X is a Veronese variety, corresponding to the case of Waring rank [19, p. 60]. It is false in the positive characteristic case, see [4], and it is false over the real numbers, see [3,6]. (In the positive characteristic case and over the real numbers the bound is codim X + 2.…”

Section: Upper Bounds For Rankmentioning

confidence: 66%

“…Over the real numbers this bound is sharp, see [6,Theorem 2.10]. We show that, over a closed field k of characteristic zero, it can be improved to m 2g − 1 in some cases, such as when X is a curve or a homogeneous variety.…”

Section: Upper Bounds For Rankmentioning

confidence: 91%

“…6 we let X be a curve contained in a smooth quadric in P 3 . Then the generic rank is g = 2 and the maximal rank is m = 2 or 3.…”

Section: Introductionmentioning

confidence: 99%

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On the locus of points of high rank

Buczyński

Han

Mella

et al. 2017

European Journal of Mathematics

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Given a closed subvariety X in a projective space, the rank with respect to X of a point p in this projective space is the least integer r such that p lies in the linear span of some r points of X . Let W k be the closure of the set of points of rank with respect to X equal to k. For small values of k such loci are called secant varieties. This article studies the loci W k for values of k larger than the generic rank. We show they are nested, we bound their dimensions, and we estimate the maximal possible rank with respect to X in special cases, including when X is a homogeneous space or a curve. The theory is illustrated by numerous examples, including Veronese varieties, the Segre product of dimensions (1, 3, 3), and curves. An intermediate result provides a lower bound on the dimension of any GL n orbit of a homogeneous form.

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Section: Upper Bounds For Rankmentioning

confidence: 66%

Section: Upper Bounds For Rankmentioning

confidence: 91%

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On the locus of points of high rank

Buczyński

Han

Mella

et al. 2017

European Journal of Mathematics

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“…The Waring rank of binary forms has been studied extensively [1,2,6,13,15]. Recently the real Waring rank of binary forms has been investigated [3,5,7,8,9]. The relative Waring rank of binary forms over some intermediate fields of C/Q was analyzed in [15,16].…”

Section: Introductionmentioning

confidence: 99%

On the waring rank of binary forms

Tokcan

2017

Linear Algebra and its Applications

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“…(iii) Fix integers n ≥ 1 and d ≥ 3. Assume (n, d) / ∈ {(2,6),(3,4),(5,3)}. Set r = n+d n − 1 and g = ⌈(r + 1)/(n + 1)⌉.…”

mentioning

confidence: 99%

Labels of real projective varieties

Ballico

Ventura

2020

Boll Unione Mat Ital

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Let X be a complex projective variety defined over R. Recently, Bernardi and the first author introduced the notion of admissible rank with respect to X. This rank takes into account only decompositions that are stable under complex conjugation. Such a decomposition carries a label, i.e., a pair of integers recording the cardinality of its totally real part. We study basic properties of admissible ranks for varieties, along with special examples of curves; for instance, for rational normal curves admissible and complex ranks coincide. Along the way, we introduce the scheme theoretic version of admissible rank. Finally, analogously to the situation of real ranks, we analyze typical labels, i.e., those arising as labels of a full-dimensional Euclidean open set. We highlight similarities and differences with typical ranks.Let X be a projective variety defined over an arbitrary field K. For a subfield L ⊆ K, the set of L-points of X is denoted X(L). The set of smooth L-points of X is X reg (L) and the singular locus X sing (L). In the following, K = C and L = R.Let X(C) ⊂ P r (C) be an integral and nondegenerate complex projective variety. The pair consisting of X(C) and the given embedding X(C) ֒→ P r (C) is defined over R if and

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Real rank with respect to varieties

Cited by 13 publications

References 13 publications

On the locus of points of high rank

On the locus of points of high rank

On the waring rank of binary forms

Labels of real projective varieties

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