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Diophantine equations in two variables
This is an expository lecture on the subject of the title delivered at the
Park-IAS mathematical institute in Princeton (July, 2000).Comment: Not for separate publicatio
Ill-distributed sets over global fields and exceptional sets in Diophantine Geometry
Let be a number field. Using techniques of discrete
analysis, we prove that for definable sets in of
dimension at most a conjecture of Wilkie about the density of rational
points is equivalent to the fact that is badly distributed at the level of
residue classes for many primes of . This provides a new strategy to prove
this conjecture of Wilkie. In order to prove this result, we are lead to study
an inverse problem as in the works \cite{Walsh2, Walsh}, but in the context of
number fields, or more generally global fields. Specifically, we prove that if
is a global field, then every subset
consisting of rational points of projective height bounded by , occupying
few residue classes modulo for many primes of
, must essentially lie in the solution set of a polynomial equation of
degree , for some constant
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