Polynomial Inscriptions

We prove that for every smooth Jordan curve \(\gamma \subset \mathbb{C}\) 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 \(p(Q) \subset \gamma\). The proof relies on a theorem of Fukaya and Irie....

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Bibliographic Details
Published in:arXiv.org 2024-12
Main Authors: Greene, Joshua Evan, Lobb, Andrew
Format: Article
Language:English
Subjects:
Homology
Polynomials
Vertex sets
Online Access:Get full text
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Summary:We prove that for every smooth Jordan curve \(\gamma \subset \mathbb{C}\) 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 \(p(Q) \subset \gamma\). 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 \(p(Q) \subset \gamma\). 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 \(\gamma\).
ISSN:2331-8422