#### Vol. 9, No. 10, 2015

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On the normalized arithmetic Hilbert function

### Mounir Hajli

Vol. 9 (2015), No. 10, 2293–2302
##### Abstract

Let $X\subset {ℙ}_{\phantom{\rule{0.3em}{0ex}}‘\overline{ℚ}}^{N}$ be a subvariety of dimension $n$, and let ${\mathsc{ℋ}}_{norm}\left(X;\phantom{\rule{0.3em}{0ex}}\cdot \phantom{\rule{0.3em}{0ex}}\right)$ be the normalized arithmetic Hilbert function of $X$ introduced by Philippon and Sombra. We show that this function admits the asymptotic expansion

${\mathsc{ℋ}}_{norm}\left(X;D\right)=\frac{ĥ\left(X\right)}{\left(n+1\right)!}{D}^{n+1}+o\left({D}^{n+1}\right),\phantom{\rule{1em}{0ex}}\forall D\gg 1,$

where $ĥ\left(X\right)$ is the normalized height of $X$. This gives a positive answer to a question raised by Philippon and Sombra.

##### Keywords
arithmetic Hilbert function, height
##### Mathematical Subject Classification 2010
Primary: 14G40
Secondary: 11G50, 11G35