Let $X$ be an algebraic variety over $\mathbb{F}_q$ with dimensional $n$. We know that if $X$ is smooth than $X$ has about $q^{nk}$ rational points over $\mathbb{F}_{q^k}$ (Weil hypothesis). Is there an analogous statement for non-smooth varieties?
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1$\begingroup$ Yes a similar property holds whenever $X$ is geometrically integral. Search google for the "Lang-Weil estimate". $\endgroup$– Daniel LoughranCommented Apr 23, 2016 at 18:49
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2$\begingroup$ You can assume $X$ reduced, and then $X$ has a smooth dense open $U$. Let $Z=X\setminus U$, which has dimension $\leq n-1$. If you accept the asymptotic for smooth varieties, by induction on $n$ you get that $\# X(\mathbb{F}_{q^k}) = \# U(\mathbb{F}_{q^k}) + \# Z(\mathbb{F}_{q^k}) = (1+o(1))q^{nk} + O(q^{(n-1)k}) = (1+o(1)) q^{nk}$. $\endgroup$– Piotr AchingerCommented Apr 23, 2016 at 19:09
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As per Daniel Loughran's comment, I found this very nice exposition by Terence Tao which seems to answer the question in detail.
answered Apr 23, 2016 at 19:39