We are motivated by the following problem. Let , where is irreducible of degree over . We look at the group
Let , and let
Proposition. Let . Let . Then
Therefore we consider the problem: Can we compute in time polynomial in , , and ?
The general setup: Let , and . We want to count
For example, we consider the Artin-Schreier hypersurface. Let
Theorem. [Deligne] Write , where are homogeneous of degree . Assume defines a smooth projective hypersurface in , and that , . Then
What about ?
Definition. If , we define the th fibred sum of to be
Theorem. [Fu-W] Write , and assume that is smooth in and . Then
Example. In the case that we can write
Since the condition that the fibred sum be smooth is Zariski open, we have shown it is nonempty if and therefore there exist many examples of such to which the theorem applies.
Definition. Let be the set of over such that is smooth. Then is Zariski open in the set of all over with .
Theorem. [Gao-W] is Zariski dense if and only if . In fact,
Problem. What about Kummer hypersurfaces
Remark. We expect , but one can get the weaker estimate in many cases (Katz).
Now we consider partial zeta functions over . Let , . Define
By exponentiation to a root of unity, we mean the formal binomial expansion. From a counting point of view, this is `as good as rational'.
Theorem. In all cases, .
Now . We refine the above argument as follows. First, for , you have
Open problem: can you bound the total degree of ? The best bound we have is , . Can this be improved to ? Yes, if and (Fu-W).
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