Abstract
We prove an incidence theorem for points and curves in the complex plane. Given a set of m points in ℝ2 and a set of n curves with k degrees of freedom, Pach and Sharir proved that the number of point-curve incidences is \(O\left( {{m^{\frac{k}{{2k - 1}}}}{n^{\frac{{2k - 2}}{{2k - 1}}}} + m + n} \right)\). We establish the slightly weaker bound \({O_\varepsilon }\left( {{m^{\frac{k}{{2k - 1}} + \varepsilon }}{n^{\frac{{2k - 2}}{{2k - 1}}}} + m + n} \right)\) on the number of incidences between m points and n (complex) algebraic curves in ℂ2 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.
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Sheffer, A., Szabó, E. & Zahl, J. Point-Curve Incidences in the Complex Plane. Combinatorica 38, 487–499 (2018). https://doi.org/10.1007/s00493-016-3441-7
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DOI: https://doi.org/10.1007/s00493-016-3441-7