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Section: New Results

A new bound on the number of rational points of arbitrary projective varieties

In [38] , the authors asked for a general upper bound on the number of rational points of a (possibly reducible) equidimensional variety X𝐏n of dimension d and degree δ. They conjectured that

| X ( 𝐅 q ) | δ ( π d - π 2 d - n ) + π 2 d n , (1)

where for all positive integer , π is defined as the number of rational points of the projective space of dimension over 𝐅q. That is to say, π=q+1-1q-1.

By combining algebraic geometric methods with a combinatorial method of double counting, A. Couvreur proved this conjecture [32] 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.