Probabilities of incidence between lines and a plane curve over finite fields |
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| Affiliation: | 1. Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Linz, Austria;2. Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Linz, Austria;3. International School for Advanced Studies/Scuola Internazionale Superiore di Studi Avanzati (ISAS/SISSA), Via Bonomea 265, 34136 Trieste, Italy;1. University of Napoli “Federico II”, Italy;2. University of Udine, Italy;3. RWTH Aachen University, Germany;1. School of Mathematics and Statistics, Shandong University of Technology, Zibo, Shandong 255091, China;2. Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan 430062, China;3. School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, Hunan 410114, China;4. Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Viet Nam;5. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Viet Nam;6. Chern Institute of Mathematics and LPMC, and Tianjin Key Laboratory of Network and Data Security Technology, Nankai University, Tianjin 300071, China;1. Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai, 200062, China;2. State Key Laboratory of Cryptology, P.O. Box 5159, Beijing 100878, China;3. School of Computer Science and Software Engineering, East China Normal University, Shanghai 200062, China;1. Department of Electronic Engineering and Information Technologies, University of Napoli “Federico II”, Italy;2. Department of Mathematics, Computer Science, and Physics, University of Udine, Italy |
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| Abstract: | ![]()
We study the probability for a random line to intersect a given plane curve, over a finite field, in a given number of points over the same field. In particular, we focus on the limits of these probabilities under successive finite field extensions. Supposing absolute irreducibility for the curve, we show how a variant of the Chebotarev density theorem for function fields can be used to prove the existence of these limits, and to compute them under a mildly stronger condition, known as simple tangency. Partial results have already appeared in the literature, and we propose this work as an introduction to the use of the Chebotarev theorem in the context of incidence geometry. Finally, Veronese maps allow us to compute similar probabilities of intersection between a given curve and random curves of given degree. |
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| Keywords: | Sylvester-Gallai theorem Galois group Finite fields Chebotarev density theorem |
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