On Jun 1, 6:08 am, Henryk Trappmann <[EMAIL PROTECTED]> wrote: > > there is an "obvious" convention that by default we mean the positive > > root. > > We have to distinguish between solutions of polynomials and roots. > Roots are clearly defined mono-valued functions: > z.nth_root(n)=e^(log(z)/n) > however this function is not continuous in z, as log is not continuous > at the negative real axis. This makes things complicated. > > > It's a lot more complicated when you deal with general > > algebraic numbers which have several ways of being embedded into CC. > > Even for square roots of negative reals: you might suggest taking the > > root with positive imaginary part, but then sqrt(-2)*sqrt(-3) equals > > -sqrt(6) and not +sqrt(6). > > by the above definition this can easily be computed: > sqrt(-2)*sqrt(-3)=e^(log(-2)/2+log(-3)/2)=sqrt(6)e^(-pi*i/2-pi*i/ > 2)=sqrt(6)*(-1) > > I never said it is simple but I am sure that there are equality > deciding algorithms. > And I really want to learn about those.
Sage has such a decision procedure built in to its implementation of QQbar, the algebraic numbers. For example: sage: QQbar(sqrt(-2)*sqrt(-3) + sqrt(6)) == 0 True In QQbar, radicals are defined to mean the principal root (with a branch cut just below the negative x axis). This works by (lazily) building number fields (with embeddings into CC) that contain the numbers in question. Computation in QQbar is fairly efficient if you only want to prove inequalities (QQbar automatically attempts to do this using interval arithmetic) or if the number field is of low degree, but it can be very inefficient to construct these number fields if they are of moderate or high degree. Carl --~--~---------~--~----~------------~-------~--~----~ 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://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
