Incidences between Points and Lines in R^4
IEEE Symposium on Foundations of Computer Science(2015)
摘要
We show that the number of incidences between m distinct points and n distinct lines in R4 is O(2{c sqrt{logm}} (m{2/5}n{4/5}+m) + m{1/2}n{1/2}q{1/4} +m{2/3}n{1/3}s{1/3} + n), for a suitable absolute constantc, provided that no 2-plane contains more than s input lines, and no hyper plane or quadric contains more than q lines. The bound holds without the extra factor 2{c sqrt{log m}} when mlen{6/7} or m ge n{5/3}. Except for this possible factor, the bound is tight in the worst case. The context of this work is incidence geometry, a topic that has been widely studied for more than three decades, with strong connections to a variety of topics, from range searching in computational geometry to the Kakeya problem in harmonic analysis and geometric measure theory. The area has picked up considerable momentum in the past seven years, following the seminal works of Guth and Katz, where the later work solves the point-line incidence problem in three dimensions, using new tools and techniques from algebraic geometry. This work extends their result to four dimensions. In doing so, it had to overcome many new technical hurdles that arise from the higher-dimensional context, by developing and adapting more advanced tools from algebraic geometry.
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关键词
Combinatorial geometry,incidences,the polynomial method,algebraic geometry,ruled surfaces
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