Thanks for the answer. As the time goes, I get more understanding of the complexity of the problem (much more than I expected at first).
On 19 Apr, 02:27, 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. > Thank you, I should probably have looked for it on the internet rather than annoying you :) > 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.) > I understand. This means (as I was kind of suspecting) that QQbar could have been a good candidate to try to work with (I am just trying to solve examples 1.4 and 1.5 of that web page). Unfortunately, QQbar doesn't provide resultant() yet, so it should jump in after the resultant has been calculated over QQ[]. I think that is not really an elegant solution (and probably very suboptimal...). > > 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. Unfortunately... > > > 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? > Actually, I was using a multivariate polynomial ring to let SAGE understand the expression: tores = A - z*B.derivative(x) (by the way...wasn't there a trac opened to let derivative syntax unify to diff or something like that? it seems that singular doesn't like B.diff(x) yet... but that's very off-topic) if I don't define a ring containing x and z, how do I inject z into the workspace? certainly not as a symbolic variable, so what else? I know that at the very end this is a univariate polynomial ring root finding problem, I'm just wondering how to correctly manage it. > > 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. > I understand, but let's consider the problem I'm coming from: indefinite integral. I'm not using polynomials because I'm well aware of the implications, rather because I found resultant() implemented there :) What I intended with my sentences was: imagine a polynomial like: x^2 + a^2. The solutions of x^2 - a^2 = 0 with respect to x over any field wouldn't always coincide with +/- a? That is the kind of reasoning I would make to solve the indefinite integral, like when I work with symbolic variables that have a precise meaning (i.e. they are not really variables in the problem, it's just that I want to keep that degree of freedom with respect to the solution, especially if I want to see the effect of that value over the solution) into my problem. > > 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. > Funny, the next minutes I spent, I recognized how to do that in this same way :) By the way, is there a SR equivalent for the new pynac symbolic? Something like NSR (new symbolic ring)? I can see that pynac already has gcd implemented, although I can already (unfortunately) see: RuntimeError: gcd: arguments must be polynomials over the rationals when I try to do a gcd over quantities containing square roots and complex numbers... hopefully pynac gcd could be ported to work with QQbar :) Jokes apart, I have seen a very long thread (which took place around april 2008) about multivariate gcd, and it seems a VERY big issue, in the sense that it is not trivial to get an algorithm for gcd robust, fast, and working over fields like QQbar (and possibly over functionals)... Basically, my aim was to see whether we did already have the tools to implement the most basic indefinite integrals of rational functions, but it seems that I already need: resultant(), solve(), gcd() and these three are not that trivial to get... (probably I wasn't paying attention to how important this basic functions are, when I took them for granted, working with MatLab [Maple]...) > Carl Thanks and keep up the good work! Maurizio --~--~---------~--~----~------------~-------~--~----~ 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 -~----------~----~----~----~------~----~------~--~---
