Point-curve incidences in the complex plane

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 $O\big(m^{\frac{k}{2k-1}}n^{\frac{2k-2}{2k-1}}+m+n\big)$. We establish the slightly weaker bound $O_\varepsilon\big(m^{\frac{k}{2k-1}+\varepsilon}n^{\frac{2k-2}{2k-1}}+m+n\big)$ 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 ${\mathbb C}$.Comment: The proof was significantly simplified, and now relies on the Picard-Lindelof theorem, rather than on foliation

Point-curve incidences in the complex plane

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 Heidelber

Counting joints with multiplicities

Let $\mathfrak{L}$ be a collection of $L$ lines in $\R^3$ and $J$ the set of joints formed by $\mathfrak{L}$, i.e. the set of points each of which lies in at least 3 non-coplanar lines of $\mathfrak{L}$. It is known that $|J| \lesssim L^{3/2}$ (first proved by Guth and Katz). For each joint $x \in J$, let the multiplicity $N(x)$ of $x$ be the number of triples of non-coplanar lines through $x$. We prove here that $\sum_{x \in J}N(x)^{1/2} \lesssim L^{3/2}$, while in the last section we extend this result to real algebraic curves of uniformly bounded degree in $\R^3$, as well as to curves in $\R^3$ parametrised by real polynomials of uniformly bounded degree.Comment: More details in section 4. Typos corrected. The main results are unchange