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Elementary incidence theorems for complex numbers and quaternions (2007).
| Content Provider | CiteSeerX |
|---|---|
| Author | Solymosi, József Konrad Swanepoel, J. |
| Abstract | Abstract. We present some elementary ideas to prove the following Sylvester-Gallai type theorems involving incidences between points and lines in the planes over the complex numbers and quaternions. (1) Let A and B be finite sets of at least two complex numbers each. Then there exists a line ℓ in the complex affine plane such that (A × B) ∩ ℓ = 2. (2) Let S be a finite noncollinear set of points in the complex affine plane. Then there exists a line ℓ such that 2 ≤ S ∩ ℓ ≤ 5. (3) Let A and B be finite sets of at least two quaternions each. Then there exists a line ℓ in the quaternionic affine plane such that 2 ≤ (A × B) ∩ ℓ ≤ 5. (4) Let S be a finite noncollinear set of points in the quaternionic affine plane. Then there exists a line ℓ such that 2 ≤ S ∩ ℓ ≤ 24. 1. |
| File Format | |
| Publisher Date | 2007-01-01 |
| Access Restriction | Open |
| Subject Keyword | Finite Noncollinear Set Complex Number Finite Set Complex Affine Plane Quaternionic Affine Plane Following Sylvester-gallai Type Elementary Idea |
| Content Type | Text |
| Resource Type | Article |
