Diophantine equations in moderately many variables

TL;DR

This paper establishes upper bounds on the number of integral solutions of polynomial systems with bounded height, extending Heath-Brown's approach to systems with polynomials of varying degrees.

Contribution

It generalizes Heath-Brown's method using a q-analogue of van der Corput's technique for systems of polynomials with different degrees, applicable to a broader range of variables.

Findings

Provides new upper bounds for solutions of polynomial systems

Extends Heath-Brown's approach to systems with varying degrees

Applicable to a wider range of variables than circle method bounds

Abstract

We give upper bounds for the number of integral solutions of bounded height to a system of equations $f_i(x_1,\ldots,x_n) = 0$, $1 \leq i \leq r$, 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.