Implementing FreeAlgebra using GAP?

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Simon King

Oct 1, 2010, 7:37:06 AM10/1/10
to sage-devel,

My colleague Robert Müller pointed me to the fact that free algebras
in Sage seem to be highly inefficient:

sage: R.<a,b,c> = FreeAlgebra(GF(3),3)
sage: f = a^2+b
sage: timeit('g=f^12')
5 loops, best of 3: 1.14 s per loop

GAP does much better:

sage: RG = gap.FreeAlgebra(GF(3),'["a","b","c"]')
sage: fG = RG.1^2+RG.2
sage: timeit('g=fG^12')
5 loops, best of 3: 54.4 ms per loop

And LetterPlace algebra (I think in Singular since version 3-1-0, but
the example below was done with 3-1-1) is even faster. The
disadvantage: It requires a degree bound.

sage: singular.LIB("freegb.lib")
sage: R0 = singular.ring(0,'(a,b,c)','dp')
sage: RS = singular.makeLetterplaceRing(50)
sage: RS.set_ring()
sage: fS = singular('a(1)*a(2)+b(1)')
sage: fS.lpPower(2)
sage: timeit('g=fS.lpPower(12)')
25 loops, best of 3: 23.7 ms per loop

In the near future (probably at least until december) I will not have
the time to this myself, but it seems rather obvious that in addition
to FreeAlgebra_generic there should also be a FreeAlgebra_gap and a
FreeAlgebra_letterplace, wrapping the functionality provided by GAP
and Singular. The constructor "FreeAlgebra" would then choose the
appropriate implementation or let the user choose.

Does someone volunteer to implement it?

Best regards,
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