MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $d$?

share|cite|improve this question

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.

share|cite|improve this answer
    
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.