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Incidences between points and curves with almost two degrees of freedom | | Micha Sharir
; Oleg Zlydenko
; | | Date: |
4 Mar 2020 | | Abstract: | We study incidences between points and algebraic curves in three dimensions,
taken from a family $C$ of curves that have almost two degrees of freedom,
meaning that every pair of curves intersect in $O(1)$ points, for any pair of
points $p$, $q$, there are only $O(1)$ curves of $C$ that pass through both
points, and a pair $p$, $q$ of points admit a curve of $C$ that passes through
both of them iff $F(p,q)=0$ for some polynomial $F$.
We study two specific instances, one involving unit circles in $R^3$ that
pass through some fixed point (so called anchored unit circles), and the other
involving tangencies between directed points (points and directions) and
circles in the plane; a directed point is tangent to a circle if the point lies
on the circle and the direction is the tangent direction. A lifting
transformation of Ellenberg et al. maps these tangencies to incidences between
points and curves in three dimensions. In both instances the curves in $R^3$
have almost two degrees of freedom.
We show that the number of incidences between $m$ points and $n$ anchored
unit circles in $R^3$, as well as the number of tangencies between $m$ directed
points and $n$ arbitrary circles in the plane, is $O(m^{3/5}n^{3/5}+m+n)$.
We derive a similar incidence bound, with a few additional terms, for more
general families of curves in $R^3$ with almost two degrees of freedom.
The proofs follow standard techniques, based on polynomial partitioning, but
face a novel issue involving surfaces that are infinitely ruled by the
respective family of curves, as well as surfaces in a dual 3D space that are
infinitely ruled by the respective family of suitably defined dual curves.
The general bound that we obtain is $O(m^{3/5}n^{3/5}+m+n)$ plus additional
terms that depend on how many curves or dual curves can lie on an
infinitely-ruled surface. | | Source: | arXiv, 2003.2190 | | Services: | Forum | Review | PDF | Favorites | |
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