Distinct Distances on Pfaffian Curves
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866916990906531840 |
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| author | Natarajan, Abhiram Sheffer, Adam |
| author_facet | Natarajan, Abhiram Sheffer, Adam |
| contents | We generalize Pach and de Zeeuw's bound for distinct distances between points on two curves, from algebraic curves to Pfaffian curves. Pfaffian curves include those that can be defined by any combination of elementary functions, including exponential and logarithmic functions, rational and irrational powers, trigonometric functions and their inverses, integration, and more. The bound remains $Ω(\min\{m^{3/4}n^{3/4},m^2,n^2\})$, as obtained from the proximity technique of Solymosi and Zahl. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_04337 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Distinct Distances on Pfaffian Curves Natarajan, Abhiram Sheffer, Adam Metric Geometry Discrete Mathematics Combinatorics Logic We generalize Pach and de Zeeuw's bound for distinct distances between points on two curves, from algebraic curves to Pfaffian curves. Pfaffian curves include those that can be defined by any combination of elementary functions, including exponential and logarithmic functions, rational and irrational powers, trigonometric functions and their inverses, integration, and more. The bound remains $Ω(\min\{m^{3/4}n^{3/4},m^2,n^2\})$, as obtained from the proximity technique of Solymosi and Zahl. |
| title | Distinct Distances on Pfaffian Curves |
| topic | Metric Geometry Discrete Mathematics Combinatorics Logic |
| url | https://arxiv.org/abs/2510.04337 |