# Choosing a fast computer algebra system that works in characteristic p?

Hi all,

I want to compute in $\mathbb{F}_q (x)((y))$ i.e. a Laurent series ring over the rational functions over $\mathbb{F}_q$. The computations are fairly basic, but they involve raising to the qth power a lot. I thought that this would be easy (I thought that it will merely shirt powers around), so I tried it in SAGE. I have to say that I am highly impressed with the ease of programming in SAGE, but I think it is too big (and slow) for the calculation I need (I know that SAGE has lots of components (PARI, GAP, etc.) some of them may be what I need).

So I wanted to ask the people who have more experience then me for a recommendation. Which algebra system is good at Laurent series over rational function fields in char p if you need to do a lot of raising to the qth power.

~AP

-

## 1 Answer

My personal experience is a few years old, but I don't think things have changed much. Sage is (or actually, was) more about ease of use then about performance. The only three CAS's you want to consider are

• Singular (Macaulay 2 uses Singular's engine)
• Cocoa.
• Magma.

Back then the fastest of the bunch was Magma, but not by much. Regarding ease of use, it was a tie between Macaulay 2 and Magma.

And now to some criticism: I never looked at Magma's code (proprietary), but I did look at both Singular and Cocoa. None of them uses SSE/GPGPU, which could probably give you an acceleration factor of 10-100.

-
I second Magma as being a good choice (I've never tried the others). Once I was using Magma to help with a problem in modular representation theory, and Magma worked fairly well for that, so I think it's good in characteristic $p$. –  Vinoth Dec 31 '09 at 11:55
I will try Macaulay 2 since I do not have access to Magma. Thanks for the advice. –  user2917 Dec 31 '09 at 21:14
To correct this here as well (for googlers), it is not true that Macaulay2 uses Singular's computational engine. "Macaulay2's engine code for polynomials, Groebner bases, and free resolutions is its own, written by Mike Stillman. What is true is that Singular and Macaulay2 both use two libraries written by the Singular group: Singular-Factory and Singular-Libfac. Macaulay2 uses those libraries for factoring of polynomials, gcd of polynomials, characteristic series (which is the core of the algorithm for computing minimal primes)." [quoted from groups.google.com/group/macaulay2 ] –  Graham Leuschke May 31 '10 at 21:48