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Let $P(x_1, \ldots , x_n)$ be a homogeneous polynomials over a finite field with $q$ elements. Is there any way to count all the roots of $P$?

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Yes, see en.wikipedia.org/wiki/Local_zeta-function, but in what form are you seeking an answer? –  Charles Matthews Mar 28 '12 at 18:33
I look forms that are linear for all variables. For example to calculate the uember of matrices n-by-n with zero determinant in a finite field. This leads to calculate the zero of a form. In the case of the determinant is is easy. –  miguel Mar 29 '12 at 15:05
What you asked is certainly understandable, but it would have been nice if p hadn't been used to denote both the polynomial and the characteristic of the field. –  KConrad Mar 30 '12 at 3:38

4 Answers 4

The magic words are: Lang-Weil

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Thanks, helped me a lot –  miguel Mar 29 '12 at 14:54
L-W had helped me a lot in the past (your example of determinant is a very good example of a case where it is very convenient). –  Igor Rivin Mar 29 '12 at 15:21

This is really a comment on Igor Rivin's answer (I don't have enough rep. to comment, it seems), but one has to be attentive to whether or not the polynomial $p$ (not to be confused with the prime $p$...) is absolutely irreducible over the ground field, or at least whether its irreducible factors are absolutely irreducible. In general the answer is influenced by the degrees of the finite extensions of the ground field over which the various "geometric" irreducible factors of $p$ are defined. Lang and Weil would have known how to formulate this, though in modern terms it comes out from how the structure of the top-degree compactly supported etale cohomology is influenced by the "field of definition" (or field of constants in the function fields) of the various irreducible components.

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This is a big field and improvements are being made all the time. The key phrases to search on are "zeta function" and "hypersurface" -- the $\zeta$-function of $F=0$ is $\exp(\sum \frac{N_n}{n} t^n )$ where $N_n$ is the number of solutions to $F=0$ over $\mathbb{F}_{p^n}$.

I started learning the current approaches from Kedlaya's lecture notes (and consider myself far from fully understanding them). A quick mathscinet search suggests that the state of the art is this paper for smooth $F$. I don't know whether these are at your level or way too hard, but they might give you a sense for the sort of thing people are thinking about.

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Thank you. I am reading this paper –  miguel Mar 29 '12 at 14:56

One nice estimate on the number of roots of arbitrary polynomials over finite fields is given by the Schwartz-Zippel lemma which states that a degree $d$ polynomial in $\mathbb{F}_q[x_1, \ldots, x_n]$ has at most $d q^{n-1}$ roots in $\mathbb{F}_q^n$.

Another result, besides the Lang-Weil bound, that might be useful in certain situations is the Chevalley-Warning theorem.

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