Section: New Results
A new bound on the number of rational points of arbitrary projective varieties
In  , the authors asked for a general upper bound on the number of rational points of a (possibly reducible) equidimensional variety of dimension and degree . They conjectured that
where for all positive integer , is defined as the number of rational points of the projective space of dimension over . That is to say,
By combining algebraic geometric methods with a combinatorial method of double counting, A. Couvreur proved this conjecture  and got a more general upper bound on the number of rational points of arbitrary varieties (possibly non-equidimensional). In addition, he proved that (1 ) is sharp by providing examples of varieties reaching this bound.