Polynomial Inscriptions
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913609700868096 |
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| author | Greene, Joshua Evan Lobb, Andrew |
| author_facet | Greene, Joshua Evan Lobb, Andrew |
| contents | We prove that for every smooth Jordan curve $γ\subset \mathbb{C}$ and for every set $Q \subset \mathbb{C}$ of six concyclic points, there exists a non-constant quadratic polynomial $p \in \mathbb{C}[z]$ such that $p(Q) \subset γ$. The proof relies on a theorem of Fukaya and Irie. We also prove that if $Q$ is the union of the vertex sets of two concyclic regular $n$-gons, there exists a non-constant polynomial $p \in \mathbb{C}[z]$ of degree at most $n-1$ such that $p(Q) \subset γ$. The proof is based on a computation in Floer homology. These results support a conjecture about which point sets $Q \subset \mathbb{C}$ admit a polynomial inscription of a given degree into every smooth Jordan curve $γ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_09546 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Polynomial Inscriptions Greene, Joshua Evan Lobb, Andrew Symplectic Geometry Algebraic Geometry Combinatorics Geometric Topology Metric Geometry We prove that for every smooth Jordan curve $γ\subset \mathbb{C}$ and for every set $Q \subset \mathbb{C}$ of six concyclic points, there exists a non-constant quadratic polynomial $p \in \mathbb{C}[z]$ such that $p(Q) \subset γ$. The proof relies on a theorem of Fukaya and Irie. We also prove that if $Q$ is the union of the vertex sets of two concyclic regular $n$-gons, there exists a non-constant polynomial $p \in \mathbb{C}[z]$ of degree at most $n-1$ such that $p(Q) \subset γ$. The proof is based on a computation in Floer homology. These results support a conjecture about which point sets $Q \subset \mathbb{C}$ admit a polynomial inscription of a given degree into every smooth Jordan curve $γ$. |
| title | Polynomial Inscriptions |
| topic | Symplectic Geometry Algebraic Geometry Combinatorics Geometric Topology Metric Geometry |
| url | https://arxiv.org/abs/2412.09546 |