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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.00374 |
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| _version_ | 1866917601455636480 |
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| author | Kişisel, Ali Ulaş Özgür Welschinger, Jean-Yves |
| author_facet | Kişisel, Ali Ulaş Özgür Welschinger, Jean-Yves |
| contents | We prove that the expected area of the amoeba of a complex plane curve of degree $d$ is less than $\displaystyle{3\ln(d)^2/2+9\ln(d)+9}$ and once rescaled by $\ln(d)^2$, is asymptotically bounded from below by $3/4$. In order to get this lower bound, given disjoint isometric embeddings of a bidisc of size $1/\sqrt{d}$ in the complex projective plane, we lower estimate the probability that one of them is a submanifold chart of a complex plane curve. It exponentially converges to one as the number of bidiscs grow to $+\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_00374 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Amoeba Measures of Random Plane Curves Kişisel, Ali Ulaş Özgür Welschinger, Jean-Yves Algebraic Geometry Probability We prove that the expected area of the amoeba of a complex plane curve of degree $d$ is less than $\displaystyle{3\ln(d)^2/2+9\ln(d)+9}$ and once rescaled by $\ln(d)^2$, is asymptotically bounded from below by $3/4$. In order to get this lower bound, given disjoint isometric embeddings of a bidisc of size $1/\sqrt{d}$ in the complex projective plane, we lower estimate the probability that one of them is a submanifold chart of a complex plane curve. It exponentially converges to one as the number of bidiscs grow to $+\infty$. |
| title | Amoeba Measures of Random Plane Curves |
| topic | Algebraic Geometry Probability |
| url | https://arxiv.org/abs/2403.00374 |