Timeline for Computational Question about finite local rings:
Current License: CC BY-SA 2.5
14 events
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| Aug 22, 2010 at 16:19 | comment | added | Noah Snyder | One thing I've heard, but can't vouch for myself, is that the original Macaulay is actually more powerful in terms of generality than Macaulay2, but much harder to use. | |
| Aug 22, 2010 at 14:17 | vote | accept | CommunityBot | moved from User.Id=631 by developer User.Id=35318 | |
| Aug 22, 2010 at 14:16 | answer | added | user631 | timeline score: 5 | |
| Aug 22, 2010 at 3:05 | comment | added | user2490 | Regarding Magma: anything that Macaulay 2 and Singular can do for you (for (1)-(3)), Magma can do as well. The quotient rings that you are looking for are called "affine algebras." In Magma (as in the other packages), affine algebras work best when the coefficient ring is a field. Incidentally: if Macaulay 2 favors computations with graded rings and ideals (as suggested in one of Sam's comments), you may be better off using Singular (or perhaps Magma---which I have never tried). If you are having trouble finding the commands for (1)-(3) in Singular, I can provide them. | |
| Aug 22, 2010 at 2:30 | history | edited | user631 | CC BY-SA 2.5 |
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| Aug 21, 2010 at 21:32 | comment | added | Sam Lichtenstein | As for (1) and (3), working over k = Z/3, if R is a quotient of a polynomial k-algebra and M is a finite dimensional graded R-module, Macaulay can compute dim_k M with the command "degree M". This should give you some handle on the lengths of I/I^2 and I/mI for certain R and I. | |
| Aug 21, 2010 at 21:25 | comment | added | user2490 | If you set 3 = 0, then you can do everything you want directly using Singular. (That is what Sage uses for its commutative algebra, but Sage does not wrap most of the relevant Singular functions.) If you are still interested in the case where you only start with 9 = 0, then since Singular can compute Groebner bases over Z, you can probably still do everything, but you will have to do many steps by hand, as the built-in algorithms may assume that the coefficient ring is a field. Maple, Mathematica, Magma, and Macaulay 2 also compute Groebner bases over Z, and so they are also options. | |
| Aug 21, 2010 at 21:21 | comment | added | Sam Lichtenstein | Also, Macaulay will complain if you work over Z/9 for example, so stick with char = 3. | |
| Aug 21, 2010 at 21:19 | comment | added | Sam Lichtenstein | To clarify (I didn't realize comments had no formatting), the lines starting with R, I, and ann are the three inputs, in order. | |
| Aug 21, 2010 at 21:19 | comment | added | Sam Lichtenstein | I had no luck with SAGE. I'm convinced the right program for this problem is MACAULAY2, but I'm not very experienced with it. However, it can certainly handle (2) with no problem. Here's some prototypical code: R = (ZZ/3)[a,b]/(a^3,a^2*b,ab^2,b^3) I = ideal (a+2*b, b^2) ann I The output is (b^2, ab, a^2). | |
| Aug 21, 2010 at 19:12 | comment | added | Pooja Singla | Have you tried asking at Gap forum? gap-system.org/Contacts/Forum/forum.html I think you will get useful answers there. | |
| Aug 21, 2010 at 18:48 | comment | added | Sam Lichtenstein | In SAGE 4.4.4 it is at least possible to create ideals in this ring. The code is something like: R = ZZ.quo(9); S.<x,y>=PolynomialRing(R,2); I = S.ideal(x^2,y^2). Now SAGE knows that I is an ideal in S. Unfortunately, some experimentation shows that very few operations on ideals in polynomial algebras have been implemented when the base is not a field. (I'm sure people would be thrilled if you implement these!) Similarly, you can form the ring T = S.quo(I). But again the standard methods SAGE has implemented for algebras over fields don't seem to be implemented for T. | |
| Aug 21, 2010 at 18:27 | history | edited | user631 | CC BY-SA 2.5 |
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| Aug 19, 2010 at 21:26 | history | asked | user631 | CC BY-SA 2.5 |