LAUSR

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 $$O\left( {{m^{\frac{k}{{2k - 1}}}}{n^{\frac{{2k - 2}}{{2k - 1}}}} + m + n} \right)$$O(mk2k−1n2k−22k−1+m+n). We establish the slightly weaker bound $${O_\varepsilon }\left( {{m^{\frac{k}{{2k - 1}} + \varepsilon }}{n^{\frac{{2k - 2}}{{2k - 1}}}} + m + n} \right)$$Oε(mk2k−1+εn2k−22k−1+m+n) 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.