I agree with Nils! John
On 9/6/07, Nils Bruin <[EMAIL PROTECTED]> wrote: > > I would be very disappointed if this approach would be taken. Sure, it > has to be straightforward and convenient to choose the level of rigour > desired in class group computations, but by default results should be > unconditional. It would be a real shame if SAGE, which tries to up the > level of trustworthiness of computational proofs by having the source > open for inspection would at the same time lower the level by tacitly > assuming conjectures. > > Would you advocate that ranks of elliptic curves should be computed > analytically because BSD is a well-established conjecture without > known counter examples? Should we be computing ranks of arbitrary > abelian varieties analytically because BSD doubtlessly holds for these > too? > > It makes a big difference whether you prove "subject to GRH, this > curve does not have rational points" or "this curve does not have > rational points". > > By allowing unproved default behaviour in Sage for this, you will set > a very bad precedent and you risk getting a bad word-of-mouth > reputation of the kind "Oh you used SAGE for that? Better verify those > results. They assume conjectures". Pari definitely suffered from that > effect due to their "bachbound/40" and made me avoid it. > > I also don't agree that class group computations are virtually > infeasible without conjectures. If the minkowskibound (or if someone > wants to work out an improvement, the improved bound) is around, say > 10^6, one can easily check the factorisation of all ideals up to that > norm, even for degree 12 fields. This actually does happen for certain > applications. > > On Sep 5, 10:00 pm, "William Stein" <[EMAIL PROTECTED]> wrote: > > On 9/5/07, Bill Hart <[EMAIL PROTECTED]> wrote: > > > > > 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. > > > > I'm specifically asking you, but would like to get feedback from > > others. It seems to me that with most class number calculations, > > one either assumes conjectural bounds or they all become > > virtually impractical. > > > > Unless anybody seriously objects, I'm going to set the default > > to False in SAGE, based on your comments below and my own > > personal inclinations. > > > > > > > > > *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. > > > > > -- John Cremona --~--~---------~--~----~------------~-------~--~----~ 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/ -~----------~----~----~----~------~----~------~--~---
