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##### GRACE - 2014

Overall Objectives
New Software and Platforms
Bilateral Contracts and Grants with Industry
Partnerships and Cooperations
Bibliography

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

 $|X\left({𝐅}_{q}\right)|\le \delta \left({\pi }_{d}-{\pi }_{2d-n}\right)+{\pi }_{2{d}_{n}},$ (1)

where for all positive integer $\ell$, ${\pi }_{\ell }$ is defined as the number of rational points of the projective space of dimension $\ell$ over ${𝐅}_{q}$. 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.