An approach for computing families of multi-branch-point covers and applications for symplectic Galois groups

TL;DR

This paper introduces a new method combining deformation, interpolation, and recent techniques to compute multi-parameter Galois extensions, successfully producing families of polynomials with symplectic Galois groups, including the first totally real examples with Galois group PSp(6,2).

Contribution

It presents a novel approach for computing families of Galois extensions with prescribed ramification, integrating existing methods with new tools for 3-point covers.

Findings

Computed several families of polynomials with symplectic Galois groups.

First totally real polynomials with Galois group PSp(6,2) obtained.

Demonstrated applicability to relatively large degrees.

Abstract

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).