On Oct 19, 4:45 am, mmarco <[email protected]> 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 [email protected]. To unsubscribe from this group, send email to [email protected]. Visit this group at http://groups.google.com/group/sage-devel?hl=en.
