4 research outputs found
Computation of Hurwitz spaces and new explicit polynomials for almost simple Galois groups
We compute the first explicit polynomials with Galois groups , , and over .
Furthermore we compute the first examples of totally real polynomials with
Galois groups , , and over
. All these examples make use of families of covers of the
projective line ramified over four or more points, and therefore use techniques
of explicit computations of Hurwitz spaces. Similar techniques were used
previously e.g. by Malle, Couveignes, Granboulan and Hallouin. Unlike previous
examples, however, some of our computations show the existence of rational
points on Hurwitz spaces that would not have been obvious from theoretical
arguments
A family of 4-branch-point covers with monodromy group PSL(6,2)
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. K\"onig 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).Comment: 13 pages, the polynomials are contained in the ancillary file
The Grunwald problem and specialization of families of regular Galois extensions
We investigate specializations of infinite families of regular Galois
extensions over number fields. The problem to what extent the local behaviour
of specializations of one single regular Galois extension can be prescribed has
been investigated by D\`ebes and Ghazi in the unramified case, and by Legrand,
Neftin and the author in general. Here, we generalize these results and give a
partial solution to Grunwald problems using Galois extensions arising as
specializations of a family of regular Galois extensions.Comment: 21
An approach for computing families of multi-branch-point covers and applications for symplectic Galois groups
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 -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).Comment: 19 pages, 5 figures, the polynomials are contained in an ancillary
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