Point-curve incidences in the complex plane

Sheffer, Ádám and Szabó, Endre and Zahl, Joshua (2018) Point-curve incidences in the complex plane. COMBINATORICA, 38 (2). pp. 487-499. ISSN 0209-9683

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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 (Formula presented.). We establish the slightly weaker bound (Formula presented.) 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. © 2017 János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg

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