
May 16

PhD defense Christiaan van de Woestijne in
Leiden

11:0011:40

Jürgen Klüners, Computation of Galois groups:
degree 24 and beyond
Room 401, Snellius building
Abstract.
In this talk I give an overview about the computation of Galois groups
of rational polynomials. The new algorithm, which is implemented in
Magma, extends the socalled Stauduhar method and is not restricted to
polynomials of small degree.

11:5012:30 
Michael Stoll, Can we decide existence of
rational points on curves?
Room 401, Snellius building
Abstract.
It is a fundamental question whether we can, for a given projective
curve (over the rationals, say), decide if it has rational points or
not. In this talk, I will try to give some evidence for a positive
answer, focusing on a computational experiment carried out jointly
with Nils Bruin. In this project, we attempted to decide for all
"small" genus 2 curves whether theyhave rational points or not.
"Small" here means given by an equation
y^{2}=f(x) with f a polynomial of
degree 6 with integral coefficients of absolute value at most 3.

13:0013:20 
Christiaan van de Woestijne, The other ABC formula,
This weeks discoveries in the Sitter room of the Lorentz Building.
Abstract.
In classical analysis, quadratic equations in one variable are easily
solved using the good old abcformula. In number theory, however,
quadratic equations are much more difficult, and usually even
unsolvable if we have only one variable. This is because we want the
solution to be in integers, or an otherwise restricted class of
numbers, without having to take square roots. In this talk, we will
consider quadratic equations in two variables. To see if solutions
in, say, integers exist, we may ask ourselves if the components of a
solution will be even or odd. It is often possible to decide this
beforehand. More generally, we may ask what the possible solutions
will be modulo some other prime number than 2, and even what the
solutions are with components in an arbitrary finite field. In the
talk, we will discuss the algorithmic difficulties that one encounters
when computing the solutions of a bivariate quadratic equation over a
finite field. As our own contribution, we will show how to avoid the
need to use a randomised solution algorithm. The deterministic
algorithm that we found may be considered as another abcformula: just
fill in the coefficients of the equation and a solution comes out.

14:1515:15  PhD defense of Christiaan van de
Woestijne
Akademiegebouw, Rapenburg, downtown Leiden



