Quantitative results on Diophantine equations in many variables
Tom 194 / 2020
Acta Arithmetica 194 (2020), 219-240 MSC: Primary 11D72; Secondary 11P55, 14G12. DOI: 10.4064/aa171212-24-9 Opublikowany online: 16 March 2020
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 a quantitative asymptotic formula (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 non-trivial integer zero provided the system of polynomials is non-singular.