Roger Heath-Brown abstract
Rational Points on Algebraic Varieties
Given a system of polynomial equations in many variables f_1(x_1,...,x_n)=...=f_k(x_1,...,x_n)=0 with rational coefficients, we ask whether there is a common solution with all the variables rational - and what one can say about such solutions. This is equivalent to asking about rational points on an algebraic variety. There are many interesting phenomena. Analytic methods provide some answers to the key questions, at least when the number of variables is large.