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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Abstract. 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 Sp(2d, 2) in their doubly-transitive actions of degrees 2 2d−1 ± 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. Sharply 2-transitive sets in finite permutation groupsLet Ω be a finite set. A set S ⊆ Sym Ω of permutations is called sharply transitive if for each α, β ∈ Ω there is a unique g ∈ S with α g = β. We call S sharply 2-transitive on Ω if it is sharply transitive on the set of pairs (ω 1 , ω 2 ) ∈ Ω 2 with ω 1 = ω 2 .Sharply transitive and sharply 2-transitive sets of permutations correspond to Latin squares and projective planes, respectively, and are therefore relevant to the fields of combinatorics and geometry.In the 1970s, Lorimer started the classification of all finite permutation groups containing a sharply 2-transitive subset, see [6] for a summary of his work. In 1984, O'Nan continued Lorimer's research and mentions in [9] that "work on the groups Sp(2d, 2) of degrees 2 2d−1 ±2 d−1 is in progress" -these are the permutation groups our paper will mainly focus on. Later in 2009, Grundhöfer and Müller [5] extended O'Nan's character-theoretic methods and finally showed -among many non-existence results for other almost simple finite permutation groups -that there are no sharply 2-transitive sets in Sp(2d, 2) of degree 2 2d−1 ± 2 d−1 . Their complicated proof was based on results by Sastry and Sin [10] regarding characteristic 2 permutation representations for symplectic groups.In 2010, Müller and Nagy [8] introduced the so-called method of contradicting subsets and were -in addition to proving new results -also able to give simple combinatorial proofs for most of the previously known results by Lorimer, O'Nan, Grundhöfer and Müller.However, they didn't reprove the non-existence of sharply 2-transitive sets in the symplectic groups Sp(2d, 2) of degrees 2 2d−1 ± 2 d−1 . This is done in the present paper. More precisely, we construct contradicting subsets for the orthogonal groups O ± (2d, 2), d ≥ 4, in their natural actions on singular vectors -which happen to be the point stabilizers of the symplectic groups.

We study the computational complexity of emptiness problems for circuits over sets of natural numbers with the operations union, intersection, complement, addition, and multiplication. For most settings of allowed operations we precisely characterize the complexity in terms of completeness for classes like NL, NP, and PSPACE. The case where intersection, addition, and multiplication is allowed turns out to be equivalent to the complement of polynomial identity testing (PIT).Our results imply the following improvements and insights on problems studied in earlier papers. We improve the bounds for the membership problem MC(∪, ∩, , +, ×) studied by McKenzie and Wagner 2007 and for the equivalence problem EQ(∪, ∩, , +, ×) studied by Glaßer et al. 2010. Moreover, it turns out that the following problems are equivalent to PIT, which shows that the challenge to improve their bounds is just a reformulation of a major open problem in algebraic computing complexity: membership problem MC(∩, +, ×) studied by McKenzie and Wagner 2007 integer membership problems MC Z (+, ×) and MC Z (∩, +, ×) studied by Travers 2006 equivalence problem EQ(+, ×) studied by Glaßer et al. 2010