Incidence bounds via extremal graph theory
arxiv(2024)
摘要
The study of counting point-hyperplane incidences in the d-dimensional
space was initiated in the 1990's by Chazelle and became one of the central
problems in discrete geometry. It has interesting connections to many other
topics, such as additive combinatorics and theoretical computer science.
Assuming a standard non-degeneracy condition, i.e., that no s points are
contained in the intersection of s hyperplanes, the currently best known
upper bound on the number of incidences of m points and n hyperplanes in
ℝ^d is
O_d, s((mn)^1-1/(d+1)+m+n).
This bound by Apfelbaum
and Sharir is based on geometrical space partitioning techniques, which apply
only over the real numbers.
In this paper, we propose a novel combinatorial approach to study such
incidence problems over arbitrary fields. Perhaps surprisingly, this approach
matches the best known bounds for point-hyperplane incidences in ℝ^d
for many interesting values of m, n, d, e.g. when m=n and d is odd.
Moreover, in finite fields our bounds are sharp as a function of m and n in
every dimension. We also study the size of the largest complete bipartite graph
in point-hyperplane incidence graphs with a given number of edges and obtain
optimal bounds as well.
Additionally, we study point-variety incidences and unit-distance problem in
finite fields, and give tight bounds for both problems under a similar
non-degeneracy assumption. We also resolve Zarankiewicz type problems for
algebraic graphs. Our proofs use tools such as induced Turán problems,
VC-dimension theory, evasive sets and Hilbert polynomials. Also, we extend the
celebrated result of Rónyai, Babai and Ganapathy on the number of
zero-patterns of polynomials to the context of varieties, which might be of
independent interest.
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