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We prove a new upper bound on the number of $r$-rich lines (lines with atleast $r$ points) in a `truly' $d$-dimensional configuration of points$v_1,\ldots,v_n \in \mathbb{C}^d$. More formally, we show that, if the numberof $r$-rich lines is significantly larger than $n^2/r^d$ then there must exista large subset of the points contained in a hyperplane. We conjecture that thefactor $r^d$ can be replaced with a tight $r^{d+1}$. If true, this wouldgeneralize the classic Szemer\'edi-Trotter theorem which gives a bound of$n^2/r^3$ on the number of $r$-rich lines in a planar configuration. Thisconjecture was shown to hold in $\mathbb{R}^3$ in the seminal work of Guth andKatz \cite{GK10} and was also recently proved over $\mathbb{R}^4$ (under someadditional restrictions) \cite{SS14}. For the special case of arithmeticprogressions ($r$ collinear points that are evenly distanced) we give a boundthat is tight up to low order terms, showing that a $d$-dimensional gridachieves the largest number of $r$-term progressions.  The main ingredient in the proof is a new method to find a low degreepolynomial that vanishes on many of the rich lines. Unlike previousapplications of the polynomial method, we do not find this polynomial byinterpolation. The starting observation is that the degree $r-2$ Veroneseembedding takes $r$-collinear points to $r$ linearly dependent images. Hence,each collinear $r$-tuple of points, gives us a dependent $r$-tuple of images.We then use the design-matrix method of \cite{BDWY12} to convert these 'local'linear dependencies into a global one, showing that all the images lie in ahyperplane. This then translates into a low degree polynomial vanishing on theoriginal set.

On the number of rich lines in truly high dimensional sets