Real rank two geometry

Seigal, Anna; Sturmfels, Bernd

doi:10.1016/j.jalgebra.2017.04.014

Cited by 6 publications

(10 citation statements)

References 16 publications

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“…Note that (4) is isomorphic to (5). This reflects the isomorphism between ( 9) and (10) in [37]. All of these computations can be reversed.…”

Section: Ruled Surfaces and Developable Surfacesmentioning

confidence: 82%

“…The edge surface E(X) was already discussed in [33,37]. We therefore focus on the other two ruled surfaces in Theorem 3.1.…”

Section: Views Of Curvesmentioning

confidence: 99%

“…This was shown by Ranestad and Sturmfels in [33, §2], and in [33, §3] they describe a method for computing E(X) when X is rational. This theme was picked up by Seigal and Sturmfels in their study of real tensor decompositions [37]. According to [37, §3], the real rank two boundary of X is the union T (X) ∪ E(X), so it is part of the visual event surface of X.…”

Section: Views Of Curvesmentioning

confidence: 99%

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Changing Views on Curves and Surfaces

2017

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Visual events in computer vision are studied from the perspective of algebraic geometry. Given a sufficiently general curve or surface in 3-space, we consider the image or contour curve that arises by projecting from a viewpoint. Qualitative changes in that curve occur when the viewpoint crosses the visual event surface. We examine the components of this ruled surface, and observe that these coincide with the iterated singular loci of the coisotropic hypersurfaces associated with the original curve or surface. We derive formulas, due to Salmon and Petitjean, for the degrees of these surfaces, and show how to compute exact representations for all visual event surfaces using algebraic methods. arXiv:1707.01877v2 [math.AG] 11 Nov 2017 Views of SurfacesWe now turn to the visual events for a general surface X in projective 3-space P 3 . The six visual events associated with X were mentioned in the Introduction in items (L) and (M). We shall explain these events and how they give rise to the following five irreducible surfaces:1. The flecnodal surface F(X) is the union of all lines L with contact of order 4 at a point of X. In other words, the equation of X restricted to L has a root of multiplicity 4.2. The cusp crossing surface C(X) is the union of all lines L with contact of order 3 + 2 at two points of X, i.e., the equation for X ∩ L on L has a triple root and a double root.3. The tritangent surface T (X) is the union of all lines L with contact of order 2 + 2 + 2 at three points of X, i.e., the equation for X ∩ L on L has three double roots.4. The edge surface E(X) is the envelope of the bitangent planes of X. It is the union of all bitangent lines arising from these planes. This surface was denoted (X [2] ) ∨ in [33].

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“…Note that (4) is isomorphic to (5). This reflects the isomorphism between ( 9) and (10) in [37]. All of these computations can be reversed.…”

Section: Ruled Surfaces and Developable Surfacesmentioning

confidence: 82%

“…The edge surface E(X) was already discussed in [33,37]. We therefore focus on the other two ruled surfaces in Theorem 3.1.…”

Section: Views Of Curvesmentioning

confidence: 99%

Section: Views Of Curvesmentioning

confidence: 99%

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Changing Views on Curves and Surfaces

2017

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“…Such cubic surfaces lie in the real rank two locus, shown in [20] to be defined by the non-negativity of the hyperdeterminant of all 2 × 2 × 2 blocks. The locus of real symmetric border rank two tensors is contained in this set, being described by the non-negativity of the diagonal (symmetric) 2 × 2 × 2 blocks [20]. All diagonal combinations occur among the non-symmetric inequalities, hence the two sets are equal.…”

Section: Real Ranks Of Cubic Surfacesmentioning

confidence: 99%

“…The monomial x 2 1 x 2 has (complex or real) border rank two, and (complex or real) rank three. Evaluating the hyperdeterminant of x 3 3 −x 3 x 2 4 shows that it has complex rank two and real rank three [20], and its rank and border rank are equal. Since the variables from the two parts of the sum are disjoint, and Strassen's Conjecture [16, §5.7] holds here, the cubic surface has complex border rank four, real border rank five, complex rank five and real rank six.…”

Section: Introductionmentioning

confidence: 99%

Ranks and symmetric ranks of cubic surfaces

Seigal

2020

Journal of Symbolic Computation

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We study cubic surfaces as symmetric tensors of format 4 × 4 × 4. We consider the non-symmetric tensor rank and the symmetric Waring rank of cubic surfaces, and show that the two notions coincide over the complex numbers. The corresponding algebraic problem concerns border ranks. We show that the non-symmetric border rank coincides with the symmetric border rank for cubic surfaces. As part of our analysis, we obtain minimal ideal generators for the symmetric analogue to the secant variety from the salmon conjecture. We also give a test for symmetric rank given by the non-vanishing of certain discriminants. The results extend to order three tensors of all sizes, implying the equality of rank and symmetric rank when the symmetric rank is at most seven, and the equality of border rank and symmetric border rank when the symmetric border rank is at most five. We also study real ranks via the real substitution method.

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