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Polynomial Inscriptions
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Paper
Polynomial Inscriptions
2024
Overview
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\\).
Publisher
Cornell University Library, arXiv.org
Subject
