Are you asking me? I'm not even sure I understand all the different options Magma offers yet!!
Basically I don't see any problem with doing the computation the way Pari does it, whatever that equates to in Magma. *IF* I understand it correctly, from Magma's point of view, Pari uses the bound BachBound/20 or BachBound/40 (depending on whether the field is quadratic or not) and uses the "subgroup" level of proof. If you do demand "proof = true" then I believe you can do better than Minkowski's bound. Zimmert's bounds are also unconditional and better than Minkowski. My reason for feeling that Pari's approach is OK is that in many cases you know that hR is at least correct from the way the algorithm works, if not it has checked up to a fairly conservative bound. You really can't go too wrong (though obviously it is not excluded theoretically). Furthermore there is a conjecture which gives an even lower bound than the bound used by Pari. There are also no known exceptions. The thing that surprises me the most about this subject is that it probably isn't possible to compute the class number faster than the class group structure, especially in the generic case. One could get hR from the analytic class number formula, but to compute R one would need fundamental units, but computing those is essentially equivalent to what we are doing. Anything else I think of seems to have the same complexity except in very special cases where one knows something else in advance. Bill. On 6 Sep, 03:52, "William Stein" <[EMAIL PROTECTED]> wrote: > On 9/5/07, Bill Hart <[EMAIL PROTECTED]> wrote: > > > > > Degree 4: > > > 4, 5, 1000 : 10s 10s > > 4, 10, 100 : 20-30s 21-30s > > x^4 - 19668*x^3 + 17560*x^2 + 26384*x + 30412 : 7.89s .... > > x^4 - 956*x^3 - 7384*x^2 + 2639*x - 17854 : 1.01s 100s > > x^4 - 1059*x^3 + 30230*x^2 + 10713*x + 25155 : 0.521s 13.5s > > x^4 + 27001*x^3 + 7255*x^2 + 16695*x + 23797 : 16.3s .... > > x^4 - 27623*x^3 + 14668*x^2 + 31695*x - 22705 : 6.01s .... > > > The obvious question - don't ask, I have absolutely no idea.\ > > Yeah, OK, that sort of behavior in MAGMA is really annoying. > I remember running into problems like that when computing > Selmer groups... > > Anyways, just out of curiosity do you think SAGE should default > to "proof = True" (like Magma claims to do, but I've heard doesn't > really) or "proof = False" (like PARIA) when computing class groups? > Right now, SAGE defaults to proof = True for that, but maybe it > shouldn't? Please share your thoughts. > > The difference in speed between proof True and proof=False is > vast in this case. > > William --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-devel@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://sage.scipy.org/sage/ and http://modular.math.washington.edu/sage/ -~----------~----~----~----~------~----~------~--~---
