Diophantine equations in moderately many variables

2016/07/06 by Marmon, Oscar
#11D45 #11D72 (Secondary) #11G35 (Primary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1607.01588

Abstract

We give upper bounds for the number of integral solutions of bounded height to a system of equations f_i(x₁,…,x_n) = 0, 1 ≤ i ≤ 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.

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