分类： Incidence geometry

incidence combinatorics

hxypqr留下评论

the method from algebraic geometry and algebraic topology have a effect on incidence combinatorics this years.espatialy on finite field case.there is some examples of the achievement follow this idea.

Dvir-Finite Kakeya conjecture

Guth-Katz-Erdos Distance problem

there is a example with classical algebraic geometry,cubic curve in fact.

Ben Green:

P\subset R^2,P is a set consist with n points.

A k-rich line is a line in R^2 which  contain k points of P

N_k=#k-rich lines,k\geq 2.we call 2-rich line as original line.

there is a classical theorem:

Sylvester-Gallai theorem:if the points in P is not collinear,then N_2\geq 2.

this theorem is not true in other fields.

there is a lots of counterexample.

the original proof of sylvester-Galli theorem:

find the pair of point and line minimize the distance from the point to the line.if the line is not original we can get a contradiction!

this proof is very pretty,but too clever to extend to a system method to deal with similar problem in incidence geometry…

 

 

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