Quantitative Results on Diophantine Equations in Many Variables

TL;DR

This paper extends Birch's work to provide quantitative asymptotics and strong approximation bounds for integer solutions of systems of polynomial equations with many variables and non-singular local zeros.

Contribution

It generalizes Birch's results by deriving explicit asymptotic formulas and bounds for integer solutions in systems of polynomials with many variables.

Findings

Quantitative asymptotics for the number of integer solutions.

Upper bounds on the smallest integer zero.

Application of a quantitative Nullstellensatz.

Abstract

We consider a system of integer polynomials of the same degree with non-singular local zeros and in many variables. Generalising the work of Birch (1962) we find quantitative asymptotics (in terms of the maximum of the absolute value of the coefficients of these polynomials) for the number of integer zeros of this system within a growing box. Using a quantitative version of the Nullstellensatz, we obtain a quantitative strong approximation result, i.e. an upper bound on the smallest integer zero provided the system of polynomials is non-singular.