On intersections of fields of rational functions
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866908925095313408 |
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| author | Pakovich, Fedor |
| author_facet | Pakovich, Fedor |
| contents | Let $X$ and $Y$ be rational functions of degree at least two with complex coefficients such that $\mathbb{C}(X,Y)=\mathbb{C}(z)$. We study the problem of determining when the field extension $[\mathbb{C}(z):\mathbb{C}(X)\cap\mathbb{C}(Y)]$ is finite and attains the minimal possible degree ${\rm deg X}\cdot{\rm deg Y}$. We give a complete characterization in the case where $X$ is a Galois covering. We also establish several related results concerning the functional equation $A \circ X = Y \circ B$ in rational functions, in the case where one of the functions involved is a Galois covering. Finally, we consider an analogous problem for holomorphic maps between compact Riemann surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_29609 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On intersections of fields of rational functions Pakovich, Fedor Algebraic Geometry Complex Variables Let $X$ and $Y$ be rational functions of degree at least two with complex coefficients such that $\mathbb{C}(X,Y)=\mathbb{C}(z)$. We study the problem of determining when the field extension $[\mathbb{C}(z):\mathbb{C}(X)\cap\mathbb{C}(Y)]$ is finite and attains the minimal possible degree ${\rm deg X}\cdot{\rm deg Y}$. We give a complete characterization in the case where $X$ is a Galois covering. We also establish several related results concerning the functional equation $A \circ X = Y \circ B$ in rational functions, in the case where one of the functions involved is a Galois covering. Finally, we consider an analogous problem for holomorphic maps between compact Riemann surfaces. |
| title | On intersections of fields of rational functions |
| topic | Algebraic Geometry Complex Variables |
| url | https://arxiv.org/abs/2603.29609 |