Quantitative results on Diophantine equations in many variables

Volume 194 / 2020

Jan-Willem M. van Ittersum Acta Arithmetica 194 (2020), 219-240 MSC: Primary 11D72; Secondary 11P55, 14G12. DOI: 10.4064/aa171212-24-9 Published 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.


  • Jan-Willem M. van IttersumMathematisch Instituut
    Universiteit Utrecht
    Postbus 80.010
    3508 TA Utrecht, the Netherlands

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