Distinct distances on algebraic curves in the plane

Abstract

Let $P$ be a set of $n$ points in the real plane contained in an algebraic curve $C$ of degree $d$. We prove that the number of distinct distances determined by $P$ is at least $c_d{n}^{4/3}$, unless $C$ contains a line or a circle. We also prove the lower bound $c_d^{\prime }\min (m^{2/3}{n}^{2/3},m^2,n^2)$ 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 arXiv:1302.3081.

Final version. To appear in Combinatorics, Probability and Computing

References in corpus (3)

• Distinct Distances on Curves via Rigidity
• Polynomials vanishing on grids: The Elekes-Rónyai problem revisited
• Distinct distances from three points

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• Exposition of Elekes Szabo paper
• Improved Elekes-Szabó type estimates using proximity
• Schwartz-Zippel bounds for two-dimensional products
• Sets with few distinct distances do not have heavy lines
• A note on distinct distances
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• Distinct values of bilinear forms on algebraic curves
• A Structural Theorem for Sets With Few Triangles
• Distinct distances between a collinear set and an arbitrary set of points
• Distinct Distances Between a Circle and a Generic Set