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I have a set of polynomial equations for which I want to know the solutions (actually really the number of solutions). It would be great if I could get a computer to do it, but I'm not sure exactly which programs will do things over finite fields. I surmise that Macaulay 2 may be able to do something like this, but I'm not quite sure how.

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You can also do it in Sage or Pari/GP with a bit of programming.

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While you're here, let me ask: Is there a command to count points without listing them? Of course, I can get the length of a list, but I am thinking about situations where the list would be very long and I could save time by not generating it. For a trivial example, something that could tell me that A^n has q^n points without taking time proportional to q^n. – David Speyer Jan 7 '10 at 19:42
I think you do this for curves but not for general varieties. – Felipe Voloch Jan 7 '10 at 19:56
Thank you very much! – David Speyer Jan 7 '10 at 20:29

There is a package for doing this in Macaulay 2 also:

You might have to download the file and load it.

It doesn't seem to have any shortcuts for just producing the number of points though, so I guess you can compare this with the one from magma to see which is faster.

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