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- W2808084711 abstract "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 $1le k <d$. 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." @default.
- W2808084711 created "2018-06-21" @default.
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- W2808084711 date "2018-06-11" @default.
- W2808084711 modified "2023-09-24" @default.
- W2808084711 title "A General Incidence Bound in ${mathbb R}^d$ and Related Problems" @default.
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- W2808084711 doi "https://doi.org/10.48550/arxiv.1806.04230" @default.
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