That was my first idea when i encountered these problems. But then, things like primary decomposition rely on factorization of polynomials... which will differ a lot from QQbar to an algebraic extension of Q.
I also thought of the aproach of translating my polynomials to a common extension and then computing the groebner basis there... but the problem here is that, in order to pass that to singular, we need to compute a primitive element of the extension.... which is again very expensive. On 20 oct, 20:25, Nils Bruin <nbr...@sfu.ca> wrote: > On Oct 19, 4:45 am, mmarco <mma...@unizar.es> wrote: > > > Do you think it would be worth implementing all these features > > ourselves? Is there some movement in the Singular community to > > implement these rings? > > My first reaction is "no". For any non-trivial linear algebra or > polynomial computations, you'll be testing whether things are 0, which > is exactly what can be extremely expensive in QQbar and AA. I'd > normally expect to do computations in an algebraic extension, do some > root finding in QQbar to see what algebraic extension is required, and > then base change to that extension to continue computation. > > However, if you're able to improve algorithms involved to such a > degree that working in QQbar directly is viable, go for it! I suspect > you'll need to seriously adjust various algorithms, however. > > In a way, QQbar really shines in "algebraically sparse" settings (you > use many different algebraic extensions, but only very few relations > between them). Non-trivial polynomial operations don't tend to > preserve sparseness. The main problem with QQbar is that not all > algebraic relations are given explicitly. (i.e., if you find the roots > of x^2-2x-1 and of x^2-2, some serious work needs to be done to find > the relations between them in QQbar, whereas in the number field > Q[sqrt(2)] these relations are explicit upon finding the roots). > > If you're in a situation where all pivots and leading coefficients > encountered are "easy", you might get some mileage out of working over > QQbar, but in general for things like groebner bases, QQbar is > irrelevant. You work in a fixed algebraic extension anyway (in fact, > since you can just adjoin the appropriate variables and algebraic > relations, being able to work over QQ is really enough for > characteristic 0). -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To post to this group, send email to sage-devel@googlegroups.com. To unsubscribe from this group, send email to sage-devel+unsubscr...@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel?hl=en.
