Diophantine equations in moderately many variables

Abstract

We give upper bounds for the number of integral solutions of bounded height to a system of equations $f_i(x_1,\dots ,x_n)=0$, $1\le i\le r$, where the $f_i$ are polynomials with integer coefficients. The estimates are obtained by generalising an approach due to Heath-Brown, using a certain $q$-analogue of van der Corput's method, to the case of systems of polynomials of differing degree. Our results apply for a wider range of $n$, in terms of the degrees of the polynomials $f_i$, than bounds obtained with the circle method.

22 pages, to appear in Michigan Mathematical Journal