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Niareh
Niareh
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The Lang-Weil bounds imply that for a geometrically irreducible variety $X$ of dimension d over $\overline{\mathbb{F}_p}$ we have, $$|N_q(X)-q^{d}|\le O(q^{d-1/2}),$$

where $N_q(X)$ is the number of $\mathbb{F}_q$ points on $X$.

Are there any explicit expressions for what $O(q^{d-1/2})$ can be in general? I can find several references for the case of geometrically irreducible hypersurfaces (such as this question on math overflow). I just wanted a reference to a general explicit expression (I imagine I could calculate it from some inductive arguments, but I was wondering if there are known citable references).

Edit: Found an answer here $$|N_q(X)-q^{d}|\le (\delta-1)(\delta-2)q^{d-1/2}+5\delta^{13/3}q^{d-1},$$ where $\delta$ is the degree of $X$.

The Lang-Weil bounds imply that for a geometrically irreducible variety $X$ of dimension d over $\overline{\mathbb{F}_p}$ we have, $$|N_q(X)-q^{d}|\le O(q^{d-1/2}),$$

where $N_q(X)$ is the number of $\mathbb{F}_q$ points on $X$.

Are there any explicit expressions for what $O(q^{d-1/2})$ can be in general? I can find several references for the case of geometrically irreducible hypersurfaces (such as this question on math overflow). I just wanted a reference to a general explicit expression (I imagine I could calculate it from some inductive arguments, but I was wondering if there are known citable references).

The Lang-Weil bounds imply that for a geometrically irreducible variety $X$ of dimension d over $\overline{\mathbb{F}_p}$ we have, $$|N_q(X)-q^{d}|\le O(q^{d-1/2}),$$

where $N_q(X)$ is the number of $\mathbb{F}_q$ points on $X$.

Are there any explicit expressions for what $O(q^{d-1/2})$ can be in general? I can find several references for the case of geometrically irreducible hypersurfaces (such as this question on math overflow). I just wanted a reference to a general explicit expression (I imagine I could calculate it from some inductive arguments, but I was wondering if there are known citable references).

Edit: Found an answer here $$|N_q(X)-q^{d}|\le (\delta-1)(\delta-2)q^{d-1/2}+5\delta^{13/3}q^{d-1},$$ where $\delta$ is the degree of $X$.

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Niareh
Niareh
  • 145
  • 4

Effective Lang-Weil bounds for higher codimension varities

The Lang-Weil bounds imply that for a geometrically irreducible variety $X$ of dimension d over $\overline{\mathbb{F}_p}$ we have, $$|N_q(X)-q^{d}|\le O(q^{d-1/2}),$$

where $N_q(X)$ is the number of $\mathbb{F}_q$ points on $X$.

Are there any explicit expressions for what $O(q^{d-1/2})$ can be in general? I can find several references for the case of geometrically irreducible hypersurfaces (such as this question on math overflow). I just wanted a reference to a general explicit expression (I imagine I could calculate it from some inductive arguments, but I was wondering if there are known citable references).