We propose an approach for the computation of multi-parameter families of Galois extensions with prescribed ramification type. More precisely, we combine existing deformation and interpolation techniques with recently developed strong tools for the computation of 3-point covers. To demonstrate the applicability of our method in relatively large degrees, we compute several families of polynomials with symplectic Galois groups, in particular obtaining the first totally real polynomials with Galois group PSp 6 (2).Note: Supplementary data are contained in an extra file, available at:https://arxiv.org/src/1803.08778/anc/anc.txt
Chapter 1. Introduction Chapter 2. Theoretical Background 2.1. Monodromy and ramification tuples 2.2. Function field setting 2.3. Belyi maps Chapter 3. Known methods for Belyi map computation 3.1. Gröbner basis method 3.2. Computing Shabat polynomials 3.3. Computing Belyi maps using modular forms Chapter 4. A new method for computing Belyi maps 4.1. Preparations 4.2. Fundamental domains 4.3. Obtaining an approximate dessin 4.4. Belyi map computation 4.5. Verification Chapter 5. Main results 5.1. Belyi maps defined over Q 5.2. A theorem of Magaard Chapter 6. Implementation 6.1. Instructions for use 6.2. Known issues and solutions 6.3. Codes Index of terms
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