Mathematics Colloquia and Seminars

Return to Colloquia & Seminar listing

Description

We aim to describe how the Hardy-Littlewood (circle) method may be applied to obtain an asymptotic formula for the number of solutions to a system of Diophantine equations. Although the method is particularly successful when the underlying polynomials have diagonal structure, quite general results are available at the cost of requiring more variables. We discuss some recent work with Julia Brandes and Anna Theorin Johannson, in which we intersect a diagonal hypersurface of degree k with an arbitrary hypersurface of smaller degree d. By combining ideas from the study of general forms with techniques adapted to the diagonal case, we obtain bounds on the number of variables growing exponentially in d but only quadratically in k, reflecting the growth rates typically obtained for both problems separately.