On the proportion of transverse-free plane curves
| Autor: | Shamil Asgarli, Brian Freidin |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
High probability
Algebra and Number Theory Plane curve Applied Mathematics 010102 general mathematics Mathematical analysis General Engineering Tangent 14H50 (Primary) 14N05 14N10 05B25 15A15 (Secondary) 0102 computer and information sciences 01 natural sciences Theoretical Computer Science Transverse plane Mathematics - Algebraic Geometry Finite field 010201 computation theory & mathematics Line (geometry) FOS: Mathematics Bipartite graph Mathematics - Combinatorics Combinatorics (math.CO) 0101 mathematics Algebraic Geometry (math.AG) Mathematics |
| Popis: | We study the asymptotic proportion of smooth plane curves over a finite field $\mathbb{F}_q$ which are tangent to every line defined over $\mathbb{F}_q$. This partially answers a question raised by Charles Favre. Our techniques include applications of Poonen's Bertini theorem and Schrijver's theorem on perfect matchings in regular bipartite graphs. Our main theorem implies that a random smooth plane curve over $\mathbb{F}_q$ admits a transverse $\mathbb{F}_q$-line with very high probability. 19 pages |
| Databáze: | OpenAIRE |
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