Incidences with Curves in R

We prove that the number of incidences between m points and n bounded-degree curves with k degrees of freedom in R is

[1]  Joshua Zahl,et al.  Improved bounds for incidences between points and circles , 2012, SoCG '13.

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

[3]  Micha Sharir,et al.  Incidences between Points and Lines in Three Dimensions , 2015, SoCG.

[4]  Zeev Dvir,et al.  On the Number of Rich Lines in Truly High Dimensional Sets , 2014, SoCG.

[5]  Jiri Matousek,et al.  Lectures on discrete geometry , 2002, Graduate texts in mathematics.

[6]  Micha Sharir,et al.  Incidences between Points and Lines in R^4 , 2014, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[7]  Terence Tao,et al.  From rotating needles to stability of waves; emerging connections between combinatorics, analysis and PDE , 2000 .

[8]  Márton Hablicsek,et al.  On the Number of Rich Lines in High Dimensional Real Vector Spaces , 2014, Discret. Comput. Geom..

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

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

[11]  Micha Sharir,et al.  Incidences between points and circles in three and higher dimensions , 2002, SCG '02.

[12]  Saugata Basu,et al.  Polynomial Partitioning on Varieties of Codimension Two and Point-Hypersurface Incidences in Four Dimensions , 2014, Discret. Comput. Geom..

[13]  Haim Kaplan,et al.  On lines, joints, and incidences in three dimensions , 2009, J. Comb. Theory, Ser. A.

[14]  Leonidas J. Guibas,et al.  Combinatorial complexity bounds for arrangements of curves and spheres , 1990, Discret. Comput. Geom..

[15]  Larry Guth,et al.  Distinct Distance Estimates and Low Degree Polynomial Partitioning , 2014, Discret. Comput. Geom..

[16]  Endre Szemerédi,et al.  Extremal problems in discrete geometry , 1983, Comb..

[17]  J. Pach Towards a Theory of Geometric Graphs , 2004 .

[18]  Joshua Zahl,et al.  An improved bound on the number of point-surface incidences in three dimensions , 2011, Contributions Discret. Math..

[19]  Haim Kaplan,et al.  Unit Distances in Three Dimensions , 2012, Comb. Probab. Comput..

[20]  Larry Guth,et al.  Algebraic curves, rich points, and doubly-ruled surfaces , 2015, ArXiv.

[21]  E. Szemerédi,et al.  Unit distances in the Euclidean plane , 1984 .

[22]  Mikhail Skopenkov,et al.  A surface containing a line and a circle through each point is a quadric , 2013 .

[23]  Larry Guth,et al.  Algebraic methods in discrete analogs of the Kakeya problem , 2008, 0812.1043.

[24]  L. A S Z L,et al.  Crossing Numbers and Hard Erdős Problems in Discrete Geometry , 1997 .

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

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

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

[28]  J. Pach,et al.  A semi-algebraic version of Zarankiewicz's problem , 2014, 1407.5705.