---
abstract: |
    We prove a new upper bound on the number of $r$-rich lines (lines with at least $r$ points) in a ‘truly’ $d$-dimensional configuration of points $v_1,\ldots,v_n \in \C^d$. More formally, we show that, if the number of $r$-rich lines is significantly larger than $n^2/r^d$ then there must exist a large subset of the points contained in a hyperplane. We conjecture that the factor $r^d$ can be replaced with a tight $r^{d+1}$. If true, this would generalize the classic Szemerédi-Trotter theorem which gives a bound of $n^2/r^3$ on the number of $r$-rich lines in a planar configuration. This conjecture was shown to hold in $\R^3$ in the seminal work of Guth and Katz [@GK10] and was also recently proved over $\R^4$ (under some additional restrictions) [@SS14]. For the special case of arithmetic progressions ($r$ collinear points that are evenly distanced) we give a bound that is tight up to low order terms, showing that a $d$-dimensional grid achieves the largest number of $r$-term progressions.

    The main ingredient in the proof is a new method to find a low degree polynomial that vanishes on many of the rich lines. Unlike previous applications of the polynomial method, we do not 