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Published April 2018 | Submitted
Journal Article Open

Point-Curve Incidences in the Complex Plane

Abstract

We prove an incidence theorem for points and curves in the complex plane. Given a set of mpoints in ℝ^2 and a set of n curves with k degrees of freedom, Pach and Sharir proved that the number of point-curve incidences is O(m k/2k−1 n 2k−2/2k−1+m+n). We establish the slightly weaker bound O_ε(m k/2k−1 + ε n 2k−2/2k−1+m+n) on the number of incidences between m points and n (complex) algebraic curves in ℂ^2 with k degrees of freedom. We combine tools from algebraic geometry and differential geometry to prove a key technical lemma that controls the number of complex curves that can be contained inside a real hypersurface. This lemma may be of independent interest to other researchers proving incidence theorems over ℂ.

Additional Information

© 2017 János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature. Received April 24, 2015; Revised June 7, 2016; Online First February 13, 2017. The authors would like to thank Orit Raz and Frank de Zeeuw for a discussion that pushed us to work on this problem, and László Lempert for finding an error in an earlier version of the proof. We would like to thank the anonymous referee for numerous suggestions and recommendations. Part of this research was performed while the authors were visiting the Institute for Pure and Applied Mathematics (IPAM) in Los Angeles, which is supported by the National Science Foundation. The second author was supported by National Research, Development and Innovation Office (NKFIH) Grants K115799, K120697, ERC_HU_ 15 118286. The third author was supported in part by an NSF Postdoctoral Fellowship.

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August 19, 2023
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