Speaker
Northey (online), Durham
Abstract
In 1961 Birch verified the Hasse principle for any non-singular system of R forms of degree d, provided that there are at least n>=R(R+1)(d-1)2^(d-1)+R-1 variables. Since then, many people have to worked to reduce the number of variables required for specific R and d, and have successfully done so in all cases except for two cubic forms (d=3, R=2), and three quadratic forms (d=2, R=3). In this talk, I will discuss the key steps that can be taken to improve on Birch for two cubic forms (from n>=49 to n>=39), which will involve a new path to Kloosterman refinement via the van der Corput differencing method. No knowledge beyond a basic understanding of the circle method will be assumed in order to make this talk accessible to anyone unfamiliar with these techniques.