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On intersections of fields of rational functions

Fedor Pakovich

Abstract

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.

On intersections of fields of rational functions

Abstract

Let and be rational functions of degree at least two with complex coefficients such that . We study the problem of determining when the field extension is finite and attains the minimal possible degree . We give a complete characterization in the case where is a Galois covering. We also establish several related results concerning the functional equation 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.

Paper Structure

This paper contains 9 sections, 17 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Cofibred products of holomorpic map
  3. Fiber products and normalizations
  4. Good solutions of $A \circ X = B \circ Y$
  5. Good solutions of $A \circ X = B \circ Y$ with $X$ being a Galois covering
  6. Cofibred products of rational functions
  7. Good solutions of $A\circ X = B\circ Y$ involving a Galois covering
  8. Cofibred products of rational functions sharing a critical point
  9. Some examples

Key Result

Theorem 1.1

Let $X$ and $Y$ be rational functions of degree at least two satisfying $\mathbb{C}(X,Y) = \mathbb{C}(z)$. Assume that $X$ is a Galois covering. Then the equality holds if and only if $Y$ admits a factorization $Y = U \circ V$, where $V$ is a $G_X$-equivariant rational function and $U$ is a rational Galois covering such that the group $\langle G_X, G_U \rangle$ is finite of order ${\rm deg\,} X \

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Theorem 2.6
  • ...and 8 more