Distinct distances on algebraic curves in the plane

Let S be a set of n points in R2 contained in an algebraic curve C of degree d. We prove that the number of distinct distances determined by S is at least cdn4/3, unless C contains a line or a circle. We also prove the lower bound c'd min{m2/3n2/3, m2, n2} 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].

[1]  S. Basu,et al.  Algorithms in real algebraic geometry , 2003 .

[2]  Micha Sharir,et al.  On the Number of Incidences Between Points and Curves , 1998, Combinatorics, Probability and Computing.

[3]  Orit E. Raz,et al.  Polynomials vanishing on grids: The Elekes-Rónyai problem revisited , 2014, SoCG.

[4]  L. Guth,et al.  On the Erdős distinct distances problem in the plane , 2015 .

[5]  Micha Sharir,et al.  Incidences in three dimensions and distinct distances in the plane , 2010, Combinatorics, Probability and Computing.

[6]  Ryan Schwartz,et al.  Extensions of a result of Elekes and Rónyai , 2012, J. Comb. Theory, Ser. A.

[7]  P. Erdös.  On Sets of Distances of n Points , 1946 .

[8]  Micha Sharir,et al.  Distinct Distances from Three Points , 2013, Combinatorics, Probability and Computing.

[9]  David A. Cox,et al.  Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics) , 2007 .

[10]  Joe W. Harris,et al.  Algebraic Geometry: A First Course , 1995 .

[11]  János Pach,et al.  Research problems in discrete geometry , 2005 .

[12]  Haim Kaplan,et al.  Simple Proofs of Classical Theorems in Discrete Geometry via the Guth–Katz Polynomial Partitioning Technique , 2011, Discret. Comput. Geom..

[13]  László A. Székely,et al.  Crossing Numbers and Hard Erdős Problems in Discrete Geometry , 1997, Combinatorics, Probability and Computing.

[14]  Marcos Charalambides.  Exponent gaps on curves via rigidity , 2013, ArXiv.

[15]  S. Basu,et al.  On a real analog of Bezout inequality and the number of connected components of sign conditions , 2013, 1303.1577.

[16]  Joshua Zahl,et al.  Few distinct distances implies no heavy lines or circles , 2016, Comb..

[17]  Nolan R. Wallach.  On a theorem of Milnor and Thom , 1996 .

[18]  József Solymosi,et al.  An Incidence Theorem in Higher Dimensions , 2012, Discret. Comput. Geom..

[19]  Lajos Rónyai,et al.  A Combinatorial Problem on Polynomials and Rational Functions , 2000, J. Comb. Theory, Ser. A.

[20]  Micha Sharir,et al.  Repeated Angles in the Plane and Related Problems , 1992, J. Comb. Theory, Ser. A.

[21]  G. Elekes,et al.  A Note on the Number of Distinct Distances , 1999 .

[22]  L. A. Oa,et al.  Crossing Numbers and Hard Erd} os Problems in Discrete Geometry , 1997 .

[23]  Micha Sharir,et al.  Distinct distances on two lines , 2013, J. Comb. Theory, Ser. A.