Has anyone actually compared them on a specific problem?
Dear Igor, I do not have personal experience with either but perhaps this MO discussion where some of the answers and comments touch upon your question could be useful:
http://mathoverflow.net/questions/23312/textbooks-on-singular-and-macaulay-2
One should also add CoCoA (see lin below) to the list of the freely available computer algebra systems specifically dedicated to commutative algebra (and non-free one could also add Magma). As written in the thread mentioned by Artur, there is not real point in asking which of the three is the "best" one, as this depends where much on what is important for you: breadth of the library, speed of computation, ease of programming. The next question is whether you have some specific problem in mind (for which perhaps only one of the three has some code available) or whether you are asking generally. But even for a very specific problem there is usually no simple answer. We developed recently a new algorithm for computing resolutions and implemented it in CoCoa. Then we run a lot of benchmarks comparing it with Singular and Macaulay2: each system had some examples where it was faster than the other one. Computational commutative algebra is a very complex field and there is usually not a "best" solution to a particular class of problems.
cocoa.dima.unige.it
I agree with prof. Seller, both trhee software are adapted to some kind of problems and some experience in computer algebra. If you are a beginner I advised you Cocoa, later you can easily use Singular or Macaulay.
... and talking about books on Computer Algebra Systems there are also the Kreuzer-Robbiano books based on CoCoA.
http://cocoa.dima.unige.it/kr/
I, too, agree with Prof. Seiler's reply; I would also like to add that it is a good idea to verify any (important) result produced by the computer, e.g. by computing the same result in two different ways using different algebra systems.
I'm a member of the CoCoA group; we do our best to make the software reliable, but we are only human, so such large and complex programs surely contain bugs. If two independent programs give the same result, you can be practically certain the result is correct (assuming the inputs were correct!)
Dear Igor
Singular without a doubt:
1) Frequently updated
2) Developers respond actively and fast to possible problem situations in Singular forum
3) Easy C/C++ like command structure
4) Excellent helps for each command
5) Largest library of sub commands related to commutative and non commutative algebra, singularity theory and algebraic geometry
Of course in a small set of some specific problems some other system might be better/faster but as a whole I would give my vote to Singular and if you want to get serious work done this is what I suggest. Of course if you are considering some academic examples for students for example then it possibly makes no difference if you use CoCoa, Macaulay2 or Singular.
If you know C language start with Singular even if you are a beginner in algebra/alg.geometry.
Of course this is my opinion and I am not paid to advertise Singular :-)
However you can look at my publications where I have used Singular successfully in real applications in every article whereas commercial programs like Maple failed every time !
regards
Samuli
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