Abstract
We prove an incidence theorem for points and planes in the projective space P^{3} over any field F, whose characteristic p ≠ 2 An incidence is viewed as an intersection along a line of a pair of twoplanes from two canonical rulings of the Klein quadric. The Klein quadric can be traversed by a generic hyperplane, yielding a lineline incidence problem in a threequadric, the Klein image of a regular line complex. This hyperplane can be chosen so that at most two lines meet. Hence, one can apply an algebraic theorem of Guth and Katz, with a constraint involving p if p > 0.
This yields a bound on the number of incidences between m points and n planes in P^{3}, with m ≥ n as O (m√n + mk),where k is the maximum number of collinear planes, provided that n=O(p^{2}) if p>0. Examples show that this bound cannot be improved without additional assumptions. This gives one a vehicle to establish geometric incidence estimates when p>0. For a noncollinear point set S ⊆ F^{2} and a nondegenerate symmetric or skewsymmetric bilinear form ω, the number of distinct values of ω on pairs of points of S is Ω[min(S^{2/3},p)]. This is also the best known bound over R, where it follows from the SzemerédiTrotter theorem. Also, a set S ⊆ F^{3}, not supported in a single semiisotropic plane contains a point, from which Ω[min(S^{½},p)] distinct distances to other points of S are attained.
This yields a bound on the number of incidences between m points and n planes in P^{3}, with m ≥ n as O (m√n + mk),where k is the maximum number of collinear planes, provided that n=O(p^{2}) if p>0. Examples show that this bound cannot be improved without additional assumptions. This gives one a vehicle to establish geometric incidence estimates when p>0. For a noncollinear point set S ⊆ F^{2} and a nondegenerate symmetric or skewsymmetric bilinear form ω, the number of distinct values of ω on pairs of points of S is Ω[min(S^{2/3},p)]. This is also the best known bound over R, where it follows from the SzemerédiTrotter theorem. Also, a set S ⊆ F^{3}, not supported in a single semiisotropic plane contains a point, from which Ω[min(S^{½},p)] distinct distances to other points of S are attained.
Original language  English 

Pages (fromto)  219–254 
Number of pages  36 
Journal  Combinatorica 
Volume  38 
Issue number  1 
Early online date  10 Jan 2017 
DOIs  
Publication status  Published  Feb 2018 
Bibliographical note
Accepted author manuscript a 25pp. revised version: a remark, a lemma, and a theorem have been added.Keywords
 math.CO
 68R05
 11B75
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Dr Misha Rudnev
 School of Mathematics  Associate Professor in Mathematics
 Number theory and combinatorics
 Pure Mathematics
Person: Academic , Member