@phdthesis{Barth2022,
  author    = {Barth, Dominik},
  title     = {Computation of multi-branch-point covers and applications in Galois theory},
  doi       = {10.25972/OPUS-27702},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-277025},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2022},
  abstract  = {We present a technique for computing multi-branch-point covers with prescribed ramification and demonstrate the applicability of our method in relatively large degrees by computing several families of polynomials with symplectic and linear Galois groups. As a first application, we present polynomials over \(\mathbb{Q}(\alpha,t)\) for the primitive rank-3 groups \(PSp_4(3)\) and \(PSp_4(3).C_2\) of degree 27 and for the 2-transitive group \(PSp_6(2)\) in its actions on 28 and 36 points, respectively. Moreover, the degree-28 polynomial for \(PSp_6(2)\) admits infinitely many totally real specializations. Next, we present the first (to the best of our knowledge) explicit polynomials for the 2-transitive linear groups \(PSL_4(3)\) and \(PGL_4(3)\) of degree 40, and the imprimitive group \(Aut(PGL_4(3))\) of degree 80. Additionally, we negatively answer a question by K{\"o}nig whether there exists a degree-63 rational function with rational coefficients and monodromy group \(PSL_6(2)\) ramified over at least four points. This is achieved due to the explicit computation of the corresponding hyperelliptic genus-3 Hurwitz curve parameterizing this family, followed by a search for rational points on it. As a byproduct of our calculations we obtain the first explicit \(Aut(PSL_6(2))\)-realizations over \(\mathbb{Q}(t)\). At last, we present a technique by Elkies for bounding the transitivity degree of Galois groups. This provides an alternative way to verify the Galois groups from the previous chapters and also yields a proof that the monodromy group of a degree-276 cover computed by Monien is isomorphic to the sporadic 2-transitive Conway group \(Co_3\).},
  subject      = {Galois-Theorie},
  language  = {en}
}
@phdthesis{Wenz2021,
  author    = {Wenz, Andreas},
  title     = {Computation of Belyi maps with prescribed ramification and applications in Galois theory},
  doi       = {10.25972/OPUS-24083},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-240838},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2021},
  abstract  = {We compute genus-0 Belyi maps with prescribed monodromy and strictly verify the computed results. Among the computed examples are almost simple primitive groups that satisfy the rational rigidity criterion yielding polynomials with prescribed Galois groups over Q(t). We also give an explicit version of a theorem of Magaard, which lists all sporadic groups occurring as composition factors of monodromy groups of rational functions.},
  subject      = {Galois-Theorie},
  language  = {en}
}