arXiv subjects
Publications and source records attributed to Dominik Barth.
In 2013 Elkies described a method for bounding the transitivity degree of Galois groups. Our goal is to give additional applications of this technique, in particular verifying that the monodromy group of the degree-276 cover defined over a degree-12 number field computed by Monien is isomorphic to the sporadic Conway group Co3.
We describe the explicit computation of a family of 4-branch-point rational functions of degree 63 with monodromy group PSL(6,2). This, in particular, negatively answers a question by J. König whether there exists a such a function with rational coefficients. The computed family also gives rise to non-regular degree-126 realizations of Aut(PSL(6,2)) over Q(t).
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).
We present a one-parameter family of degree 36 polynomials with the symplectic 2-transitive group PSp(6,2) as Galois group over Q(t).
We present a genus 0 Belyi map for the sporadic Janko group J2 of degree 280. As a consequence we obtain a polynomial having Aut(J2) as a Galois group over K(t) where K is a number field of degree 10.
We compute the genus 0 Belyi map for the sporadic Janko group J1 of degree 266 and describe the applied method. This yields explicit polynomials having J1 as a Galois group over K(t), [K:Q] = 7.
We present all Belyi maps P^1(C) -> P^1(C) having almost simple primitive monodromy groups (not isomorphic to A_n or S_n) containing rigid and rational generating triples of degree between 50 and 250. This also leads to new polynomials having almost simple Galois groups over Q(t).
We compute explicit polynomials having the sporadic Higman-Sims group HS and its automorphism group Aut(HS) as Galois groups over the rational function field Q(t).
We use Müller and Nagy's method of contradicting subsets to give a new proof for the non-existence of sharply $2$-transitive subsets of the symplectic groups $\mathrm{Sp}(2d,2)$ in their doubly-transitive actions of degrees $2^{2d-1}\pm 2^{d-1}$. The original proof by Grundhöfer and Müller was rather complicated and used some results from modular representation theory, whereas our new proof requires only simple counting arguments.