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Article overview
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Incidences with curves and surfaces in three dimensions, with applications to distinct and repeated distances | | Micha Sharir
; Noam Solomon
; | | Date: |
5 Oct 2016 | | Abstract: | We study a wide spectrum of incidence problems involving points and curves or
points and surfaces in $mathbb R^3$. The current (and in fact the only viable)
approach to such problems, pioneered by Guth and Katz [2010,2015], requires a
variety of tools from algebraic geometry, most notably (i) the polynomial
partitioning technique, and (ii) the study of algebraic surfaces that are ruled
by lines or, in more recent studies [Guth-Zahl 2016], by algebraic curves of
some constant degree. By exploiting and refining these tools, we obtain new and
improved bounds for numerous incidence problems in $mathbb R^3$.
In broad terms, we consider two kinds of problems, those involving points and
constant-degree algebraic emph{curves}, and those involving points and
constant-degree algebraic emph{surfaces}. In some variants we assume that the
points lie on some fixed constant-degree algebraic variety, and in others we
consider arbitrary sets of points in 3-space.
The case of points and curves has been considered in several previous
studies, our results, which are based on a recent work of Guth and Zahl
concerning surfaces that are doubly ruled by curves, provide a grand
generalization of all previous results.
In the case of points and surfaces, our results provide a "grand
generalization" of most of the previous studies of (special instances of) this
problem.
As applications of our point-surface incidence bounds, we consider the
problems of distinct and repeated distances determined by a set of $n$ points
in $mathbb R^3$, two of the most celebrated open problems in combinatorial
geometry. We obtain new and improved bounds for two special cases, one in which
the points lie on some algebraic variety of constant degree, and one involving
incidences between pairs in $P_1 imes P_2$, where $P_1$ is contained in a
variety and $P_2$ is arbitrary. | | Source: | arXiv, 1610.1560 | | Services: | Forum | Review | PDF | Favorites | |
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