Lachaud's point-count bound for large-dimensional complete intersections

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Let X⊂PnX\subset\mathbb{P}^{n} be a projective algebraic set defined over Fq\mathbb{F}_{q}, of dimension m≥n/2m\geq n/2 and degree d≤q+1d\leq q+1, which is an i.t. complete intersection. Writing πj=∣Pj(Fq)∣\pi_j=|\mathbb{P}^{j}(\mathbb{F}_q)|, Lachaud's conjecture.

∣X(Fq)∣≤dπm−(d−1)π2m−n=d(πm−π2m−n)+π2m−n.|X(\mathbb{F}_{q})|\leq d\pi_m-(d-1)\pi_{2m-n}=d(\pi_m-\pi_{2m-n})+\pi_{2m-n}.

This bound was proposed for point counts of projective complete intersections in the small-codimension range. The source states that the conjecture has been proved by Couvreur.

References

Primary source

Gilles Lachaud and Robert Rolland, “On the Number of Points of Algebraic Sets over Finite Fields”, arXiv:1405.3027 (2014).

Additional references

2 papers in this index state this conjecture (2008–2014). The statement above is taken from the most recent of them; the others are arXiv:0808.2169.

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