Intercity Number Theory Seminar
May 16

PhD defense Christiaan van de Woestijne in Leiden

11:00-11:40 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 so-called Stauduhar method and is not restricted to polynomials of small degree.
11:50-12:30 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 y2=f(x) with f a polynomial of degree 6 with integral coefficients of absolute value at most 3.
13:00-13:20 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 abc-formula. 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 abc-formula: just fill in the coefficients of the equation and a solution comes out.
14:15-15:15PhD defense of Christiaan van de Woestijne
Akademiegebouw, Rapenburg, downtown Leiden