Ilya Zakharevich on Thu, 4 May 2000 20:58:39 -0400 |
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Re: Bug in Mod (2.0.15) |
On Fri, May 05, 2000 at 12:53:40AM +0200, Bill Allombert wrote: > >But is a "much more correct" answer. What I see is the following: > >simplify() assumes that any ring in which POLMODs live is a ring of > >principal ideals. (This assumption is hidden in the algorithm of > >converting to the lowest common denominator when combining POLMODs > >with different moduli). > > Not really. PARI assume that all polynomial have coefficient in a _field_. This is not hardwired into the internal representation. Thus the only problem is with the assumption used in algorithms, which should not be many, right? > This is reasonnable, since there is currently no ways to specify rings. > (this is a major problem in the cursed gdivexact function). Did not find any mention of the problem in my archive. And it looks like a very recent function... And it is not documented what it does (in 2.0.15 I have here). > PARI works in the ring K(y,z,t)[X] in which the result is true. I see. But I do not see any reason why it would/should ;-) do it... > > a) a way to represent an element in a quotient by a non-principal > > ideal in terms of the PARI type system; > > Yes, but we have to add support for rings first, and then add ideals. Do not see what you mean. Which other places (in addition to simplify()) depend-on/use the K(y,z,t)[X] hack? > > I can be wrong, but do not Groebner bases give a canonical system of > > generators of any ideal? > > Not truly. You get a reduced basis which nice reduction property but > it still depend of lot of choice. Given marked generators (x,y,z etc) of the ring, could not we get something canonical? > simplify() only handle "stupid" conversion (5+0*I -->5 ; 5*X^0 -->5). > It is probably gmod that would need to be modified. Well, if it is two places to modify, it is still not that bad as adding a new type to PARI... > Saying that there exists canonical generators for ideals is not the > same thing that saying there are canonical generators for cosets, but > it is probably not a big problem here. All one needs is the ability to check that an element is in the ideal... > Currently the way PARI implement multivariate polynomial is not > efficient and need to be changed before considering to implement > Groebner basis. Why this obsession with efficiency? It is good when possible, but having *some* pilot implementation will clear the way for future improvements... > Moreover writing efficient Groebner basis programs is not easy. Do not people have them written? Cannot we use IPC with some other program which can do them? Ilya