A Tale of Two Cubics
by
Trevor Wooley
University of Michigan

In 1957 a space race was underway, and the outcome was almost a dead heat. Consider a homogeneous cubic polynomial with rational integral coefficients. The competition was to show, assuming that the cubic has "sufficiently many" variables, that the polynomial necessarily possesses a non-trivial integral solution (and more generally, linear spaces of rational solutions). The three contestants in the photo-finish (Lewis, Birch and Davenport) applied quite different methods, and later developments demonstrated these methods to be applicable in a wider context. In this talk we discuss what can be said for the analogous problem for pairs of cubic equations. The methods we present are based on simple geometry, linear algebra and harmonic analysis, and should be largely accessible to those less familiar with number theory.

Date received: April 12, 2005