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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.

flag	asked Jan 7 2010 at 19:17 Daniel Moseley 93●4

Felipe Voloch · Answer 1 · 2010-01-07 19:26:21Z UTC

6

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.

answered Jan 7 2010 at 19:26

Felipe Voloch
16.6k●2●36●61

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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 2010 at 19:42

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I think you do this for curves but not for general varieties. – Felipe Voloch Jan 7 2010 at 19:56

Thank you very much! – David Speyer Jan 7 2010 at 20:29

Steven Sam · Answer 2 · 2010-01-07 20:53:21Z UTC

There is a package for doing this in Macaulay 2 also: http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.3.1/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.

Computer program to solve a system of polynomial equations over a finite field

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