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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Magma. http://magma.maths.usyd.edu.au/magma/htmlhelp/text1282.htm#12832 You can also do it in Sage or Pari/GP with a bit of programming. 


There is a package for doing this in Macaulay 2 also: http://www.math.uiuc.edu/Macaulay2/doc/Macaulay21.3.1/share/doc/Macaulay2/RationalPoints/html/index.html 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. 

