Geodesic
Quantitative results on Diophantine equations in many variables
Jan-Willem M. van Ittersum 1
1 Mathematisch Instituut Universiteit Utrecht Postbus 80.010 3508 TA Utrecht, the Netherlands
Acta Arithmetica, Tome 194 (2020) no. 3, pp. 219-240 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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.
DOI : 10.4064/aa171212-24-9
Keywords: consider system integer polynomials degree non singular local zeros many variables generalising work birch quantitative asymptotic formula terms maximum absolute value coefficients these polynomials number integer zeros system within growing box using quantitative version nullstellensatz obtain quantitative strong approximation result upper bound smallest non trivial integer zero provided system polynomials non singular

Jan-Willem M. van Ittersum  1

1 Mathematisch Instituut Universiteit Utrecht Postbus 80.010 3508 TA Utrecht, the Netherlands 
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Jan-Willem M. van Ittersum. Quantitative results on Diophantine equations in many variables. Acta Arithmetica, Tome 194 (2020) no. 3, pp. 219-240. doi: 10.4064/aa171212-24-9

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