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Let $V\subset \mathbb{A}^n_{\mathbb{F}_q}$ be a closed subvariety defined by simultaneously vanishing of $r$ polynomials $f_1,\cdots,f_r\in \mathbb{F}_q[x_1,\cdots,x_n]$, each of degree at most $d$. Set $N=\dim(V)$ and fix a prime $\ell$ not dividing $q$.

It can be shown that $m$ the number of top dimensional irreducible components of $V_{\overline{\mathbb{F}_q}}$ is just the dimension of the $\ell$-adic cohomology group with compact support $H_c^{2N}(V_{\overline{\mathbb{F}_q}},\mathbb{Q}_\ell)$.

So my question is: is there a way to bound $m$ using only $n,r$ and $d$?

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Yes, indeed, the sum of the dimensions of all cohomology groups can be bounded effectively, see the statements in Katz's paper "Sums of Betti numbers in arbitrary characteristic", Finite Fields Appl. 7 (2001) (these are based on earlier bounds of Bombieri and Adolphson--Sperber). The paper is also available on Katz's web page.

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Does $d^r$ work? That would be the naive guess based on degree arguments/Bezout etc, so I'd be interested to see an example where this failed. – wrigley Feb 28 at 12:36

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