A General Incidence Bound in ${\mathbb R}^d$ and Related Problems

We derive a general upper bound for the number of incidences with $k$-dimensional varieties in ${\mathbb R}^d$. The leading term of this new bound generalizes previous bounds for the special cases of $k=1, k=d-1,$ and $k= d/2$, to every $1\le k <d$. We derive lower bounds showing that this leading term is tight in various cases. We derive a bound for incidences with transverse varieties, generalizing a result of Solymosi and Tao. Finally, we derive a bound for incidences with hyperplanes in ${\mathbb C}^d$, which is also tight in some cases. (In both ${\mathbb R}^d$ and ${\mathbb C}^d$, the bounds are tight up to sub-polynomial factors.) To prove our incidence bounds, we define the \emph{dimension ratio} of an incidence problem. This ratio provides an intuitive approach for deriving incidence bounds and isolating the main difficulties in each proof. We rely on the dimension ratio both in ${\mathbb R}^d$ and in ${\mathbb C}^d$, and also in some of our lower bounds.