Proceedings of the Thirtieth Annual Symposium on Computational Geometry 2014
DOI: 10.1145/2582112.2582135
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Distinct distances on algebraic curves in the plane

Abstract: 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 betw… Show more

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Cited by 20 publications
(47 citation statements)
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“…This was proved by Elekes and Rónyai [7], improved by Elekes [4] to Ω(n 5/4 ) distinct distances, and recently improved to Ω(n 4/3 ) distinct distances by Sharir, Sheffer, and Solymosi [20]. The approach of [20] was very recently generalized to any two algebraic curves by Pach and De Zeeuw [18], who also deduced the following theorem for a single algebraic curve. This theorem improves the planar case of a recent result by Charalambides [1], who proved the bound Ω d,D (n 5/4 ) for the number of distinct distances between n points on a common algebraic curve of degree d in R D , with an approach based on that of [4].…”
Section: Introductionmentioning
confidence: 93%
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“…This was proved by Elekes and Rónyai [7], improved by Elekes [4] to Ω(n 5/4 ) distinct distances, and recently improved to Ω(n 4/3 ) distinct distances by Sharir, Sheffer, and Solymosi [20]. The approach of [20] was very recently generalized to any two algebraic curves by Pach and De Zeeuw [18], who also deduced the following theorem for a single algebraic curve. This theorem improves the planar case of a recent result by Charalambides [1], who proved the bound Ω d,D (n 5/4 ) for the number of distinct distances between n points on a common algebraic curve of degree d in R D , with an approach based on that of [4].…”
Section: Introductionmentioning
confidence: 93%
“…We conclude the introduction with a few words about the proofs of Theorems 1.1 and 1.5. Both proofs rely on the approach used in [20] (and subsequently in [18] and [19]), which is based on the Elekes-Sharir framework from [8] and [13]. More precisely, this approach defines a set Q of quadruples of points and then applies a double counting argument to |Q|.…”
Section: Theorem 13 (Pach and De Zeeuwmentioning
confidence: 99%
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“…A few weeks after an initial preprint of this paper was released, Pach and de Zeeuw [17] improved the exponent in Theorem 1.8 from 1 + 1 4 to 1 + 1 3 in the special case where the ambient dimension is 2 (and the polynomial in question is the square distance function). Their method is mostly algebraic and simpler than our argument for that particular case; when the curve Γ is planar, there is no need to appeal to more advanced tools from algebraic geometry such as the Thom-Milnor Theorem or the theory of Gröbner bases.…”
Section: Further Remarksmentioning
confidence: 99%
“…More precisely, they show that if P 1 and P 2 are two sets of N points in the plane so that P 1 is contained in a line L 1 , P 2 is contained in a line L 2 , and L 1 and L 2 are neither parallel nor orthogonal, then the number of distinct distances determined by the pairs P 1 × P 2 is N 1+ 1 3 . In both [22] and [17], the problem considered is, in fact, a bipartite problem: there are two curves Γ 1 , Γ 2 and two finite subsets P 1 , P 2 and the aim is to obtain a lower bound on the cardinality of the set {D(p, q) | p ∈ P 1 , q ∈ P 2 }.…”
Section: Further Remarksmentioning
confidence: 99%