Computational Algebra Seminar: Bruin -- Mordell-Weil sieving

Speaker: Nils Bruin
Title: Mordell-Weil sieving
Time & Place: 3-4pm, Thursday 25 June, Carslaw LT 173

Abstract: It would probably have been a big surprise for Hilbert to see
that one of his famous problems was resolved with a negative answer:

As we know by the work of Davis, Putnam, Robinson and Matyasevitch, there
is no algorithm that takes as input a multivariate polynomial
f(x1,...,xr) over the integers and gives as output whether or not the
the equation

f(x1,...,xr) = 0

has any integer solutions.

This result establishes that number theory, and in particular
the study of diophantine equations, is generally hard.

If instead of integer points on hypersurfaces, we consider rational points
on curves, the picture changes dramatically. In the last six years, a
suprisingly simple method, now commonly referred to as Mordell-Weil
sieving, has been developed. A heuristic argument by Bjorn Poonen
indicates that this method should always be able to decide if a projective
curve has any rational points.

I will discuss the method and give some experimental evidence of its
efficacy.