New Trends in Intuitive Geometry pp 385-405 | Cite as

# Incidence Bounds for Complex Algebraic Curves on Cartesian Products

## Abstract

We prove bounds on the number of incidences between a set of algebraic curves in \({\mathbb C}^2\) and a Cartesian product \(A\times B\) with finite sets \(A,B\subset {\mathbb C}\). Similar bounds are known under various restrictive conditions, but we show that the Cartesian product assumption leads to a simpler proof and lets us remove these conditions. This assumption holds in a number of interesting applications, and with our bound these applications can be extended from \({\mathbb R}\) to \({\mathbb C}\). We also obtain more precise information in the bound, which is used in several recent papers (Raz et al., 31st international symposium on computational geometry (SoCG 2015), pp 522–536, 2015, [17], Valculescu, de Zeeuw, Distinct values of bilinear forms on algebraic curves, 2014, [25]). Our proof works via an incidence bound for surfaces in \({\mathbb R}^4\), which has its own applications (Raz, Sharir, 31st international symposium on computational geometry (SoCG 2015), pp 569–583, 2015, [15]). The proof is a new application of the polynomial partitioning technique introduced by Guth and Katz (Ann Math 181: 155–190, 2015, [11]).

## References

- 1.S. Barone, S. Basu, On a real analogue of Bezout inequality and the number of connected components of sign conditions, in Proceedings of the London Mathematical Society, vol. 112, n. 1 (1 January 2016), pp. 115–145, arXiv:1303.1577
- 2.S. Basu, R. Pollack, M.-F. Roy,
*Algorithms in Real Algebraic Geometry*(Springer, Berlin, 2003)zbMATHGoogle Scholar - 3.S. Basu, M. Sombra, Polynomial partitioning on varieties and point-hypersurface incidences in four dimensions, Discrete Comput. Geom.
**55**(1), 158–184 (January 2016), arXiv:1406.2144 - 4.J. Bochnak, M. Coste, M.-F. Roy,
*Real Algebraic Geometry*(Springer, Berlin, 1998)CrossRefGoogle Scholar - 5.B. Bollobás,
*Modern Graph Theory*(Springer, Berlin, 1998)CrossRefGoogle Scholar - 6.G. Elekes,
*On the Dimension of Finite Point Sets II. “Das Budapester Programm"*(2011), arXiv:1109.0636 - 7.G. Elekes, SUMS versus PRODUCTS in Number Theory. Algebra and Erdős Geometry, Paul Erdős and his Mathematics II, Bolyai Society Mathematical Studies
**11**, 241–290 (2002)zbMATHGoogle Scholar - 8.G. Elekes, L. Rónyai, A combinatorial problem on polynomials and rational functions. J. Comb. Theory, Ser. A
**89**, 1–20 (2000)MathSciNetCrossRefGoogle Scholar - 9.G. Elekes, M. Nathanson, I.Z. Ruzsa, Convexity and sumsets. J. Number Theory
**83**, 194–201 (2000)MathSciNetCrossRefGoogle Scholar - 10.G. Fischer,
*Plane Algebraic Curves*(American Mathematical Society, Providence, 2001)zbMATHGoogle Scholar - 11.L. Guth, N.H. Katz, On the Erdős distinct distances problem in the plane. Ann. Math.
**181**, 155–190 (2015)MathSciNetCrossRefGoogle Scholar - 12.J. Harris,
*Algebraic Geometry: A First Course*(Springer, Berlin, 1992)CrossRefGoogle Scholar - 13.J. Heintz, Definability and fast quantifier elimination in algebraically closed fields. Theor. Comput. Sci.
**24**, 239–277 (1983)MathSciNetCrossRefGoogle Scholar - 14.J. Pach, M. Sharir, On the number of incidences between points and curves. Comb. Probab. Comput.
**7**, 121–127 (1998)MathSciNetCrossRefGoogle Scholar - 15.O.E. Raz, M. Sharir, The number of unit-area triangles in the plane: Theme and variations. Combinatorica
**37**(6), 1221–1240 (December 2017). Also in arXiv:1501.00379MathSciNetCrossRefGoogle Scholar - 16.O.E. Raz, M. Sharir, J. Solymosi, Polynomials vanishing on grids: The Elekes-Rónyai problem revisited, in
*Proceedings of the Thirtieth Annual Symposium on Computational Geometry*(2014), pp. 251–260, arXiv:1401.7419 - 17.O.E. Raz, M. Sharir, F. de Zeeuw, Polynomials vanishing on Cartesian products: The Elekes-Szabó Theorem revisited, in
*31st International Symposium on Computational Geometry (SoCG 2015)*(2015), pp. 522–536. Also in arXiv:1504.05012 - 18.M. Sharir, A. Sheffer, J. Solymosi, Distinct distances on two lines. J. Comb. Theory, Ser. A
**120**, 1732–1736 (2013)MathSciNetCrossRefGoogle Scholar - 19.A. Sheffer, E. Szabó, J. Zahl, Point-curve incidences in the complex plane. Combinatorica
**38**(2), 487–499 (April 2018), arXiv:1502.07003MathSciNetCrossRefGoogle Scholar - 20.J. Solymosi, On the number of sums and products. Bull. Lond. Math. Soc.
**37**, 491–494 (2005)MathSciNetCrossRefGoogle Scholar - 21.J. Solymosi, T. Tao, An incidence theorem in higher dimensions. Discrete Comput. Geom.
**48**, 255–280 (2012)MathSciNetCrossRefGoogle Scholar - 22.J. Solymosi, G. Tardos, On the number of k-rich transformations, in
*Proceedings of the Twenty-Third Annual Symposium on Computational Geometry*(2007), pp. 227–231Google Scholar - 23.J. Solymosi, V. Vu, Distinct distances in high dimensional homogeneous sets, Towards a theory of geometric graphs. Contemp. Math.
**342**, 259–268 (American Mathematical Society, 2004)Google Scholar - 24.C.D. Tóth, The Szemerédi-Trotter theorem in the complex plane. Combinatorica
**35**, 95–126 (2015)MathSciNetCrossRefGoogle Scholar - 25.C. Valculescu, F. de Zeeuw,
*Distinct values of bilinear forms on algebraic curves*. Contributions to Discrete Mathematics,**11**(1), (July 2016). Also in arXiv:1403.3867 - 26.J. Zahl, A Szemerédi-Trotter type theorem in \({\mathbb{R}}^4\). Discrete Comput. Geom.
**54**, 513–572 (2015)MathSciNetCrossRefGoogle Scholar