KTH/SU Mathematics Colloquium

06-03-01

Per Salberger, CTH

Counting solutions to Diophantine equations

Let f_1,..., f_m be polynomials in n variables with integer coefficients and N(B) be the number of n-tuples of integers in [-B, B] such that f_1 = ... = f_m = 0. Several results in analytic number theory depend on finding good upper bounds for N(B). Vinogradov's mean value theorem gives a good bound for certain systems of diagonal equations. This can be be used to give a large zero-free region of Riemann's zeta-function and to give good bounds for G(k) in Waring's problem. Another example concerns the class group h(d) of the quadratic field Q(root(-d)). The n-torsion of h(d) is related to the density of solutions to the Diophantine equation X^2 +dY^2=Z^n.

Most of the estimates for N(B) so far have been obtained by means of sieve theory or the theory of exponential sums. The methods of Faltings are ineffective and esentially limited to cases where there are only finitely many solutions to the Diophantine equations. But recently a "determinant method" has been developed which gives very general results. It was initiated by Bombieri and Pila in the case of affine curves and then extended to a general method for hypersurfaces by Heath-Brown in an important paper in Annals of Math. 2002. Recently a new even more powerful version of the determinant method was developed by the speaker. It can be applied to very general classes of algebraic varieties and it gives better estimates for N(B) than could be obtained with the previous versions of the determinant method. An important feature is that a number of techniques from algebraic geometry appear like the theories of Hilbert schemes, Kodaira dimension and Seshadri constants. We shall in our talk present some new results obtained with this method like the solutions of two conjectures of Heath-Brown.