Distinct Distances on Curves Via Rigidity

Charalambides, Marcos

doi:10.1007/s00454-014-9586-5

Cited by 14 publications

(43 citation statements)

References 24 publications

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“…In [4], Charalambides establishes a version of Theorem 1.1 with the weaker lower bound c d n 5/4 . He combines the technique of [6] with analytic as well as algebraic tools, and even extends it to higher dimensions, with a more complicated set of exceptions.…”

Section: Introductionmentioning

confidence: 97%

“…This makes it possible to extend their methods to parametrizable curves, but makes it harder to extend to general algebraic curves, which are defined by an implicit equation. In [4], this is overcome using the Implicit Function Theorem, which allows implicit curves to be "parametrized" analytically. One important new element of our proofs is that we construct the new curves in an implicit and algebraic way (see in particular (1) in Section 3), making parametrization unnecessary.…”

Section: Introductionmentioning

confidence: 99%

“…In [6] and [18], this is relatively easy because the curves have low degree. In [4], it is done using concepts from the theory of rigidity. We observe instead that, if some of the constructed curves have large intersection, this must be due to some kind of symmetry of the original curve.…”

Section: Introductionmentioning

confidence: 99%

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Distinct distances on algebraic curves in the plane

Pach

Zeeuw

2014

Proceedings of the Thirtieth Annual Symposium on Computational Geometry

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Let S be a set of n points in R 2 contained in an algebraic curve C of degree d. We prove that the number of distinct distances determined by S is at least c d n 4/3 , unless C contains a line or a circle.We also prove the lower bound c d min{m 2/3 n 2/3 , m 2 , n 2 } for the number of distinct distances between m points on one irreducible plane algebraic curve and n points on another, unless the two curves are parallel lines, orthogonal lines, or concentric circles. This generalizes a result on distances between lines of Sharir, Sheffer, and Solymosi in [18].

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Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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Distinct distances on algebraic curves in the plane

Pach

Zeeuw

2014

Proceedings of the Thirtieth Annual Symposium on Computational Geometry

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“…The condition on the curve is tight in the sense that for any excluded curve there is a bilinear form that can take a linear number of values on that curve. To the authors, it was surprising that such curves can have large degree, whereas previous evidence (namely [8] and [20]) suggested that the exceptional curves in such problems have degree at most three. The description of the exceptional curves in Theorem 3.3 is very succinct but requires some clarification.…”

Section: Other Polynomials On Curvesmentioning

confidence: 97%

“…Charalambides [8] also considered the function A(p, q) = p x q y − p y q x , which (in R 2 ) gives twice the signed area of the triangle spanned by p, q, and the origin. He proved that, for P contained in an irreducible algebraic curve C in R 2 , we have |A(P ×P )| = Ω d (|P | 5/4 ), unless C is a line, an ellipse centered at the origin, or a hyperbola centered at the origin.…”

Section: Other Polynomials On Curvesmentioning

confidence: 99%

A Survey of Elekes-Rónyai-Type Problems

Zeeuw

2018

Bolyai Society Mathematical Studies

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We give an overview of recent progress around a problem introduced by Elekes and Rónyai. The prototype problem is to show that a polynomial f ∈ R[x, y] has a large image on a Cartesian product A×B ⊂ R 2 , unless f has a group-related special form. We discuss this problem and a number of variants and generalizations. This includes the Elekes-Szabó problem, which generalizes the Elekes-Rónyai problem to a question about an upper bound on the intersection of an algebraic surface with a Cartesian product, and curve variants, where we ask the same questions for Cartesian products of finite subsets of algebraic curves.These problems lie at the crossroads of combinatorics, algebra, and geometry: They ask combinatorial questions about algebraic objects, whose answers turn out to have applications to geometric questions involving basic objects like distances, lines, and circles, as well as to sum-product-type questions from additive combinatorics. As part of a recent surge of algebraic techniques in combinatorial geometry, a number of quantitative and qualitative steps have been made within this framework. Nevertheless, many tantalizing open questions remain.

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Few distinct distances implies no heavy lines or circles

2014

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We study the structure of planar point sets that determine a small number of distinct distances. Specifically, we show that if a set P of n points determines o(n) distinct distances, then no line contains Ω(n 7/8 ) points of P and no circle contains Ω(n 5/6 ) points of P.We rely on the partial variant of the Elekes-Sharir framework that was introduced by Sharir, Sheffer, and Solymosi in [19] for bipartite distinct distance problems. To prove our bound for the case of lines we combine this framework with a theorem from additive combinatorics, and for our bound for the case of circles we combine it with some basic algebraic geometry and a recent incidence bound for plane algebraic curves by Wang, Yang, and Zhang [20].A significant difference between our approach and that of [19] (and of other related results) is that instead of dealing with distances between two point sets that are restricted to one-dimensional curves, we consider distances between one set that is restricted to a curve and one set with no restrictions on it.

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Distinct Distances on Curves Via Rigidity

Cited by 14 publications

References 24 publications

Distinct distances on algebraic curves in the plane

Distinct distances on algebraic curves in the plane

A Survey of Elekes-Rónyai-Type Problems

Few distinct distances implies no heavy lines or circles

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