A Szemerédi–Trotter Type Theorem in R[superscript 4]
We show that m points and n two-dimensional algebraic surfaces in R[superscript 4] can have at most O(m[superscript k/(2k−1)n(2k−2)/(2k−1)]+m+n) incidences, provided that the algebraic surfaces behave like pseudoflats with k degrees of freedom, and that m≤n[superscript (2k+2)/3k]. As a special c...
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| Format: | Article |
| Language: | English |
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Springer US
2017
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| Online Access: | http://hdl.handle.net/1721.1/106902 https://orcid.org/0000-0001-5129-8300 |
| _version_ | 1826210374118014976 |
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| author | Zahl, Joshua |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Zahl, Joshua |
| author_sort | Zahl, Joshua |
| collection | MIT |
| description | We show that m points and n two-dimensional algebraic surfaces in R[superscript 4] can have at most O(m[superscript k/(2k−1)n(2k−2)/(2k−1)]+m+n) incidences, provided that the algebraic surfaces behave like pseudoflats with k degrees of freedom, and that m≤n[superscript (2k+2)/3k]. As a special case, we obtain a Szemerédi–Trotter type theorem for 2-planes in R[superscript 4], provided m≤n and the planes intersect transversely. As a further special case, we obtain a Szemerédi–Trotter type theorem for complex lines in C[superscript 2] with no restrictions on m and n (this theorem was originally proved by Tóth using a different method). As a third special case, we obtain a Szemerédi–Trotter type theorem for complex unit circles in C2. We obtain our results by combining several tools, including a two-level analogue of the discrete polynomial partitioning theorem and the crossing lemma. |
| first_indexed | 2024-09-23T14:48:29Z |
| format | Article |
| id | mit-1721.1/106902 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T14:48:29Z |
| publishDate | 2017 |
| publisher | Springer US |
| record_format | dspace |
| spelling | mit-1721.1/1069022022-10-01T22:37:44Z A Szemerédi–Trotter Type Theorem in R[superscript 4] A Szemerédi–Trotter Type Theorem in R4 Zahl, Joshua Massachusetts Institute of Technology. Department of Mathematics Zahl, Joshua We show that m points and n two-dimensional algebraic surfaces in R[superscript 4] can have at most O(m[superscript k/(2k−1)n(2k−2)/(2k−1)]+m+n) incidences, provided that the algebraic surfaces behave like pseudoflats with k degrees of freedom, and that m≤n[superscript (2k+2)/3k]. As a special case, we obtain a Szemerédi–Trotter type theorem for 2-planes in R[superscript 4], provided m≤n and the planes intersect transversely. As a further special case, we obtain a Szemerédi–Trotter type theorem for complex lines in C[superscript 2] with no restrictions on m and n (this theorem was originally proved by Tóth using a different method). As a third special case, we obtain a Szemerédi–Trotter type theorem for complex unit circles in C2. We obtain our results by combining several tools, including a two-level analogue of the discrete polynomial partitioning theorem and the crossing lemma. United States. Department of Defense (National Defense Science & Engineering Graduate Fellowship (NDSEG) Program) 2017-02-10T19:05:00Z 2017-02-10T19:05:00Z 2015-08 2015-06 2016-05-23T12:14:21Z Article http://purl.org/eprint/type/JournalArticle 0179-5376 1432-0444 http://hdl.handle.net/1721.1/106902 Zahl, Joshua. “A Szemerédi–Trotter Type Theorem in $$\mathbb {R}^4$$ R 4.” Discrete Comput Geom 54, no. 3 (August 14, 2015): 513–572. https://orcid.org/0000-0001-5129-8300 en http://dx.doi.org/10.1007/s00454-015-9717-7 Discrete & Computational Geometry Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Springer Science+Business Media New York application/pdf Springer US Springer US |
| spellingShingle | Zahl, Joshua A Szemerédi–Trotter Type Theorem in R[superscript 4] |
| title | A Szemerédi–Trotter Type Theorem in R[superscript 4] |
| title_full | A Szemerédi–Trotter Type Theorem in R[superscript 4] |
| title_fullStr | A Szemerédi–Trotter Type Theorem in R[superscript 4] |
| title_full_unstemmed | A Szemerédi–Trotter Type Theorem in R[superscript 4] |
| title_short | A Szemerédi–Trotter Type Theorem in R[superscript 4] |
| title_sort | szemeredi trotter type theorem in r superscript 4 |
| url | http://hdl.handle.net/1721.1/106902 https://orcid.org/0000-0001-5129-8300 |
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