## 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 $X\subseteq {\mathbf{P}}^{n}$ of dimension $d$ and degree $\delta $. They conjectured that

where for all positive integer $\ell $, ${\pi}_{\ell}$ is defined as the number of rational points of the projective space of dimension $\ell $ over ${\mathbf{F}}_{q}$. That is to say, ${\pi}_{\ell}=\frac{{q}^{\ell +1}-1}{q-1}.$

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.