On Sat, Apr 18, 2009 at 5:27 PM, Carl Witty <carl.wi...@gmail.com> wrote: > > 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.
I think Singular doesn't have QQbar as a base ring, so I don't think this is likely to be implemented for a while. > >> 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. The "variety" command is relevant here. > >> 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 > > > > -- William Stein Associate Professor of Mathematics University of Washington http://wstein.org --~--~---------~--~----~------------~-------~--~----~ 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 -~----------~----~----~----~------~----~------~--~---
