Algebraic and analytic structure of Morikawa's sangaku problem
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866912911296823296 |
|---|---|
| author | Krumm, David |
| author_facet | Krumm, David |
| contents | Let $μ(r)$ denote the minimal side length of a square inscribed in the curvilinear triangular region formed by two tangent circles of radii $1$ and $r \ge 1$ together with their common tangent line. The problem of finding a closed-form expression for $μ(r)$ was posed in early nineteenth-century Japan by Morikawa. It was proved by Holly and Krumm (2021) that no expression in radicals exists for $μ(r)$. In this article we show that $μ$ is an algebraic function, and consequently real-analytic on $[1,\infty)$ outside a finite explicitly computable set. In particular, although no expression in radicals exists, the function admits convergent Taylor expansions at all non-exceptional values of $r$, whose coefficients may be computed by Newton iteration from the defining algebraic equation. We illustrate the method by explicitly computing the Taylor expansion of $μ(r)$ centered at $r=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_16115 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Algebraic and analytic structure of Morikawa's sangaku problem Krumm, David Algebraic Geometry History and Overview 14P05 Let $μ(r)$ denote the minimal side length of a square inscribed in the curvilinear triangular region formed by two tangent circles of radii $1$ and $r \ge 1$ together with their common tangent line. The problem of finding a closed-form expression for $μ(r)$ was posed in early nineteenth-century Japan by Morikawa. It was proved by Holly and Krumm (2021) that no expression in radicals exists for $μ(r)$. In this article we show that $μ$ is an algebraic function, and consequently real-analytic on $[1,\infty)$ outside a finite explicitly computable set. In particular, although no expression in radicals exists, the function admits convergent Taylor expansions at all non-exceptional values of $r$, whose coefficients may be computed by Newton iteration from the defining algebraic equation. We illustrate the method by explicitly computing the Taylor expansion of $μ(r)$ centered at $r=1$. |
| title | Algebraic and analytic structure of Morikawa's sangaku problem |
| topic | Algebraic Geometry History and Overview 14P05 |
| url | https://arxiv.org/abs/2602.16115 |