Welcome to MathOverflow

Question

2

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

flag	asked Mar 28 2012 at 18:31 miguel 168●4

1

Yes, see en.wikipedia.org/wiki/Local_zeta-function, but in what form are you seeking an answer? – Charles Matthews Mar 28 2012 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 2012 at 15:05

1

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 2012 at 3:38

quasi-coherent · Answer 1 · 2012-03-28 19:10:58Z UTC

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.

Igor Rivin · Answer 2 · 2012-03-28 18:43:35Z UTC

2

The magic words are: Lang-Weil

answered Mar 28 2012 at 18:43

Igor Rivin
31.3k●3●37●71

	Thanks, helped me a lot – miguel Mar 29 2012 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 2012 at 15:21

David Speyer · Answer 3 · 2012-03-28 20:29:09Z UTC

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.

homogeneous polynomials over a finite field

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

3 Answers

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.