Tessellations of rational complex functions and the Riemann's existence theorem
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866917039088599040 |
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| author | Alvarez-Parrilla, Alvaro Gutiérrez-Soto, Roberto Muciño-Raymundo, Jesús |
| author_facet | Alvarez-Parrilla, Alvaro Gutiérrez-Soto, Roberto Muciño-Raymundo, Jesús |
| contents | A complex rational function R, of degree n>1, on a compact Riemann surface M provided with a cyclic order of its q critical values, determines an homogeneous tessellation of the Riemann surface M, whose 2n tiles are topological q-gons with alternating colors.The tessellation provides a simple and straighforward visual description of the rational function R. Conversely, assume a possibly non homogeneous tessellation T of a compact differentiable surface M' with tiles of alternating colors and a suitable labelling in the vertices of its tiles. Non homogeneous means that the tiles of T are r-gons, for different values of r. Then there exists a Riemann surface structure M on M', a complex rational function R and a cyclic order of its critical values, such that the tessellation of R on M topologically coincides with the original T. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_21099 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tessellations of rational complex functions and the Riemann's existence theorem Alvarez-Parrilla, Alvaro Gutiérrez-Soto, Roberto Muciño-Raymundo, Jesús Complex Variables 30C10, 30A99, 30C25, 30F10 A complex rational function R, of degree n>1, on a compact Riemann surface M provided with a cyclic order of its q critical values, determines an homogeneous tessellation of the Riemann surface M, whose 2n tiles are topological q-gons with alternating colors.The tessellation provides a simple and straighforward visual description of the rational function R. Conversely, assume a possibly non homogeneous tessellation T of a compact differentiable surface M' with tiles of alternating colors and a suitable labelling in the vertices of its tiles. Non homogeneous means that the tiles of T are r-gons, for different values of r. Then there exists a Riemann surface structure M on M', a complex rational function R and a cyclic order of its critical values, such that the tessellation of R on M topologically coincides with the original T. |
| title | Tessellations of rational complex functions and the Riemann's existence theorem |
| topic | Complex Variables 30C10, 30A99, 30C25, 30F10 |
| url | https://arxiv.org/abs/2510.21099 |