The number of real eigenvectors of a real polynomial

Maccioni, Mauro

doi:10.1007/s40574-016-0112-y

Cited by 3 publications

(2 citation statements)

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“…A theoretical study of real eigenvectors was undertaken by Maccioni in [26]. He proved that the number of real eigenvectors of a ternary form is bounded below by 2ω + 1, where ω is the number of ovals.…”

Section: Experiments and K3 Surfacesmentioning

confidence: 99%

Sixty-Four Curves of Degree Six

Kaihnsa

Kummer

Plaumann

et al. 2017

Experimental Mathematics

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We present a computational study of smooth curves of degree six in the real projective plane. In the Rokhlin-Nikulin classification, there are 56 topological types, refined into 64 rigid isotopy classes. We developed software that determines the topological type of a given sextic and used it to compute empirical probability distributions on the various types. We list 64 explicit representatives with integer coefficients, and we perturb these to draw many samples from each class. This allows us to explore how many of the bitangents, inflection points and tensor eigenvectors are real. We also study the real tensor rank, the construction of quartic surfaces with prescribed topology, and the avoidance locus, which is the locus of all real lines that do not meet a given sextic. This is a union of up to 46 convex regions, bounded by the dual curve.

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Section: Experiments and K3 Surfacesmentioning

confidence: 99%

Sixty-Four Curves of Degree Six

Kaihnsa

Kummer

Plaumann

et al. 2017

Experimental Mathematics

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“…In the case of symmetric tensors, these critical rank-one tensors correspond to the so-called eigenvectors of f [11], while in the case of ordinary tensors, they correspond to singular vector tuples [10]. In the case n = 1 of binary forms, Corollary 1.3 was proved in [16].…”

Section: Introductionmentioning

confidence: 99%

Best rank-k approximations for tensors: generalizing Eckart–Young

2018

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Given a tensor f in a Euclidean tensor space, we are interested in the critical points of the distance function from f to the set of tensors of rank at most k, which we call the critical rank-at-most-k tensors for f . When f is a matrix, the critical rank-one matrices for f correspond to the singular pairs of f . The critical rank-one tensors for f lie in a linear subspace H f , the critical space of f . Our main result is that, for any k, the critical rank-at-most-k tensors for a sufficiently general f also lie in the critical space H f . This is the part of Eckart-Young Theorem that generalizes from matrices to tensors. Moreover, we show that when the tensor format satisfies the triangle inequalities, the critical space H f is spanned by the complex critical rank-one tensors. Since f itself belongs to H f , we deduce that also f itself is a linear combination of its critical rank-one tensors.2000 Mathematics Subject Classification. 15A69, 15A18, 14M17, 14P05.

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