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 $m$ points in ${\mathbb{R}}^2$ and a set of $n$ curves with $k$ degrees of freedom, Pach and Sharir proved that the number of point-curve incidences is $O(m^{\frac{k}{2k-1}}{n}^{\frac{2k-2}{2k-1}}+m+n)$. We establish the slightly weaker bound $O_{\epsilon }(m^{\frac{k}{2k-1}+\epsilon }{n}^{\frac{2k-2}{2k-1}}+m+n)$ on the number of incidences between $m$ points and $n$ (complex) algebraic curves in ${\mathbb{C}}^2$ 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 $\mathbb{C}$.

The proof was significantly simplified, and now relies on the Picard-Lindelof theorem, rather than on foliations

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Cited by in corpus (5)

• Polynomials vanishing on Cartesian products: The Elekes-Szabó Theorem revisited
• Distinct distances in the complex plane
• Schwartz-Zippel bounds for two-dimensional products
• The Multivariate Schwartz-Zippel Lemma
• A note on rich lines in truly high dimensional sets