Distinct Distances on Algebraic Curves in the Plane

Pach, János and Zeeuw, Frank de (2016) Distinct Distances on Algebraic Curves in the Plane. COMBINATORICS PROBABILITY & COMPUTING. pp. 1-19. ISSN 0963-5483 (In Press)

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Abstract

Let S be a set of n points in (Formula presented.) contained in an algebraic curve C of degree d. We prove that the number of distinct distances determined by S is at least cdn 4/3, unless C contains a line or a circle. We also prove the lower bound cd ′ 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 [19]

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