Algorithms for Galois group computations over multivariate function fields

Abstract

In this thesis, two separate algorithms are described to determine the Galois group of a polynomial f defined over a function field of the form Q(w1,...,wa), both of which use Stauduhar’s method for polynomials over Q, with various techniques used to make them compatible for function fields as the base field. In particular, both algorithms construct the resolvent of f. The first algorithm uses a special form of Hensel lifting to express the roots as a multivariate power series of sufficient precision to determine the resolvent exactly. This precision is derived through the use of Newton polygons and by obtaining a bound on the size of the coefficients of the resolvent. The second algorithm constructs the resolvents of a set of specialisations of f, which are then interpolated. Computational complexity and timing of implementations in MAGMA for polynomials of degree up to 8 are compared and discussed.