On Nests and Large Components of Random Real Algebraic Curves
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866914492063940608 |
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| author | Kişisel, Ali Ulaş Özgür Bayraktar, Turgay |
| author_facet | Kişisel, Ali Ulaş Özgür Bayraktar, Turgay |
| contents | We develop a variant of the barrier method in order to address questions about topology of Kostlan random real algebraic plane curves. In particular we prove that the expected number of connected components of the curve of length at least $\displaystyle{O\left(\sqrt{d^{-1}\log \log d}\right)}$ grows to infinity with $d$, and likewise, the expected number of nests of the curve of depth at least $\displaystyle{O\left(\log\log d\right)}$ grows to infinity with $d$. In another direction, we adapt an $L^{\infty}$-norm bound result of Shifmann and Zelditch to subspaces and employ it to obtain a lower bound for the probability that a finite number of points remain all in different components of the complement of a large degree random curve. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_18350 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Nests and Large Components of Random Real Algebraic Curves Kişisel, Ali Ulaş Özgür Bayraktar, Turgay Algebraic Geometry Probability 14P25, 32A60, 60D05 We develop a variant of the barrier method in order to address questions about topology of Kostlan random real algebraic plane curves. In particular we prove that the expected number of connected components of the curve of length at least $\displaystyle{O\left(\sqrt{d^{-1}\log \log d}\right)}$ grows to infinity with $d$, and likewise, the expected number of nests of the curve of depth at least $\displaystyle{O\left(\log\log d\right)}$ grows to infinity with $d$. In another direction, we adapt an $L^{\infty}$-norm bound result of Shifmann and Zelditch to subspaces and employ it to obtain a lower bound for the probability that a finite number of points remain all in different components of the complement of a large degree random curve. |
| title | On Nests and Large Components of Random Real Algebraic Curves |
| topic | Algebraic Geometry Probability 14P25, 32A60, 60D05 |
| url | https://arxiv.org/abs/2604.18350 |