11. Point-plane incidences and some applications in positive characteristic
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Misha Rudnev
Abstract
The point-plane incidence theorem states that the number of incidences between n points and m ≥ n planes in the projective three-space over a field F, is O(m√n + mk), where k is the maximum number of collinear points, with the extra condition n < p2 if F has characteristic p > 0. This theorem also underlies a state-of-the-art Szemerédi- Trotter-type bound for point-line incidences in F2, due to Stevens and de Zeeuw. This review focuses on some recent, as well as new, applications of these bounds that lead to progress in several open geometric questions in Fd, for d = 2, 3, 4. These are the problem of the minimum number of distinct nonzero values of a nondegenerate bilinear form on a point set in d = 2, the analogue of the Erdős distinct distance problem in d = 2, 3 and additive energy estimates for sets, supported on a paraboloid and sphere in d = 3, 4. It avoids discussing sum-product type problems (corresponding to the special case of incidences with Cartesian products), which have lately received more attention.
Abstract
The point-plane incidence theorem states that the number of incidences between n points and m ≥ n planes in the projective three-space over a field F, is O(m√n + mk), where k is the maximum number of collinear points, with the extra condition n < p2 if F has characteristic p > 0. This theorem also underlies a state-of-the-art Szemerédi- Trotter-type bound for point-line incidences in F2, due to Stevens and de Zeeuw. This review focuses on some recent, as well as new, applications of these bounds that lead to progress in several open geometric questions in Fd, for d = 2, 3, 4. These are the problem of the minimum number of distinct nonzero values of a nondegenerate bilinear form on a point set in d = 2, the analogue of the Erdős distinct distance problem in d = 2, 3 and additive energy estimates for sets, supported on a paraboloid and sphere in d = 3, 4. It avoids discussing sum-product type problems (corresponding to the special case of incidences with Cartesian products), which have lately received more attention.
Kapitel in diesem Buch
- Frontmatter I
- Introduction V
- Contents VII
- 1. Tiling rings with “precious” differences 1
- 2. q-analogs of group divisible designs 21
- 3. A survey on bent functions and their duals 39
- 4. Construction of Bent functions using covering extended building sets 57
- 5. Results on permutation polynomials of shape xt + γTrqn/q(xd) 67
- 6. A new structure for difference matrices over abelian p-groups 79
- 7. Weil sums of binomials: properties, applications and open problems 109
- 8. A survey on iterations of mappings over finite fields 135
- 9. Cyclotomy, difference sets, sequences with low correlation, strongly regular graphs and related geometric substructures 173
- 10. Pseudorandomness of large sets in finite fields 199
- 11. Point-plane incidences and some applications in positive characteristic 211
- 12. A survey of group invariant Butson matrices and their relation to generalized bent functions and various other objects 241
- 13. MRD codes: constructions and connections 255
- 14. Differential fault attack on hardware stream ciphers—a technical survey 287
- 15. Polynomials over finite fields: an index approach 319
Kapitel in diesem Buch
- Frontmatter I
- Introduction V
- Contents VII
- 1. Tiling rings with “precious” differences 1
- 2. q-analogs of group divisible designs 21
- 3. A survey on bent functions and their duals 39
- 4. Construction of Bent functions using covering extended building sets 57
- 5. Results on permutation polynomials of shape xt + γTrqn/q(xd) 67
- 6. A new structure for difference matrices over abelian p-groups 79
- 7. Weil sums of binomials: properties, applications and open problems 109
- 8. A survey on iterations of mappings over finite fields 135
- 9. Cyclotomy, difference sets, sequences with low correlation, strongly regular graphs and related geometric substructures 173
- 10. Pseudorandomness of large sets in finite fields 199
- 11. Point-plane incidences and some applications in positive characteristic 211
- 12. A survey of group invariant Butson matrices and their relation to generalized bent functions and various other objects 241
- 13. MRD codes: constructions and connections 255
- 14. Differential fault attack on hardware stream ciphers—a technical survey 287
- 15. Polynomials over finite fields: an index approach 319
