Quantitative results on Diophantine equations in many variables

Abstract

We consider a system of polynomials f 1 , . . . , f R ∈ Z[x 1 , . . . , x n ] of the same degree with non-singular local zeros and in many variables. Generalising the work of Birch [Bir62] 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.

On zeros of multilinear polynomials

“…The subject of search bounds for quadratic polynomials has been started by a classical theorem of Cassels [6]; see [8] for a detailed overview of a large body of work on various extensions and generalizations of this important theorem. Additionally, there are search bounds for integral cubic forms in a sufficient number of variables [4], as well as for systems of integral forms under certain technical non-singularity conditions [13].…”

“…Pick some I 1 ∈ J (n, t) and let m 1 be the dimension of the subspace V ∩ H I1 V . Since dim V + dim H I1 > n, we must have 1 ≤ m 1 < m. Let x 1 be a vector of smallest height from a basis for V ∩ H I1 guaranteed by Theorem 3.1, then x 1 is an (n − m + 1)-sparse vector satisfying (13) h…”