On Sat, Apr 18, 2009 at 2:46 PM, Maurizio <maurizio.gran...@gmail.com> wrote: > Could you be clearer? As I told, I'm not familiar with rings. I don't > even know the meaning of the argument of GF (I took the number 5 from > an example I see in sage-support group, I think). Do you think that QQ > [] could fit in this case? Moreover, what's the difference between QQ > and QQbar?
GF(5) means the Galois field of characteristic 5, a.k.a. the integers modulo 5. (So in GF(5), you have 2*3 = 1, 3+4 = 2, etc.) It's probably quite irrelevant for computing integrals. QQ is the rational numbers (fractions). QQbar is the algebraic closure of QQ; this means it includes every complex number which is the root of a polynomial with rational coefficients. So it includes things like sqrt(2) (which is a root of x^2-2), and sqrt(-1) (a root of x^2+1), as well as more exotic numbers like the roots of x^5-x-1, which can't be expressed using radicals (roots). (QQbar does not include all complex numbers, though; for instance, it does not include pi or e, which are transcendental rather than algebraic.) > Now let's go to Carl's help... > >> Taking a quick look at that page, it looks like they want the exact >> roots in CC of a polynomial with algebraic coefficients. In Sage, we >> can get this with QQbar: >> >> sage: K.<x> = QQbar[] >> sage: (x^5-x-1).roots(ring=QQbar) >> > > First problem with QQbar: it seems that resultant() doesn't like it, > because it is not able to convert it to a Singular ring (this is the > error, I'm not attaching all the output, tell me if you need it) > > TypeError: no conversion of this ring to a Singular ring defined Looks like this hasn't been implemented yet. > On the contrary, QQ[] seems to work fine with resultant (but it > doesn't have roots() ) Univariate polynomials over QQ definitely have roots(); were you using a multivariate polynomial ring? > Moreover, it seems that QQbar roots() is not working for multivariate > polynomials ring... is it true or am I just missing something else? In > that case, is possible to let it work in multivariate polynomials? As > you can imagine, I would like to think about this as a method of > solving integrals, so it is very likely to have a symbolic expression > with more than just a single symbolic variable. > > Apart from this, is there another way to solve an equation (with more > than a single symbolic variable) obtaining exact roots? It seems that > maxima would do the work (with algebraic numbers...), is it possible > that it is the only symbolic equation solver within SAGE? What about > SymPy or anything else? What does this even mean? .roots() gives a list of all the solutions of a univariate polynomial equation. But a multivariate polynomial equation will usually have an infinite number of solutions; for instance, x^2+y^2-1=0 has an infinite number of solutions over the rationals (or the reals, or the algebraic reals, etc.) If you have a system of multivariate polynomial equations, then the system might have only finitely many solutions. > Finally, I still would like to know which is the best way to translate > the output of a calculation with polynomial rings into a symbolic > expression, that can be carried on with maxima or pynac. Can you help > me? If p is a polynomial, then SR(p) is a symbolic expression. Carl --~--~---------~--~----~------------~-------~--~----~ 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 For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
