I confess I'm not a mathematician (I'm an economist) and it's been almost 25 years since I took a basic course in abstract algebra. But this is interesting. From the wikipedia page on Grobner bases, it seems I should be able to compute solutions to the system based on the Grobner basis, but I can't figure out how. I don't suppose sage has a function that maps these bases to solutions?
Also, the fact that there are three bases (each corresponding to a solution?) makes me worry a bit since I know there are at least four solutions. The fact that there are three also makes me think this might be the method that Maxima is using behind the scenes since it found 3 solutions. And for the record, my fourth solution (x1 and x2 both equal to sqrt(5) - 2) does satisfy the Grobner bases. On Jul 16, 10:46 am, David Joyner <wdjoy...@gmail.com> wrote: > Does this help any? > > sage: R = PolynomialRing(QQ, 2, 'x1,x2', order='lp') > sage: x1,x2 = R.gens() > sage: f1 = 1/2*((x1^2 + 2*x1 - 4)*x2^2 + 2*(x1^2 + x1)*x2 + x1^2) > sage: f2 = 1/2*((x1^2 + 2*x1 + 1)*x2^2 + 2*(x1^2 + x1)*x2 - 4*x1^2) > sage: I = (f1,f2)*R; I > Ideal (1/2*x1^2*x2^2 + x1^2*x2 + 1/2*x1^2 + x1*x2^2 + x1*x2 - 2*x2^2, > 1/2*x1^2*x2^2 + x1^2*x2 - 2*x1^2 + x1*x2^2+ x1*x2 + 1/2*x2^2) of > Multivariate Polynomial Ring in x1, x2 over Rational Field > sage: B = I.groebner_basis(); B > [x1^2 - x2^2, x1*x2 - 1/8*x2^6 - 3/8*x2^5 + 9/8*x2^4 + 15/8*x2^3 - > 3/2*x2^2, x2^7 + 4*x2^6 - 6*x2^5 - 20*x2^4 + 5*x2^3] > > Readhttp://www.sagemath.org/doc/constructions/polynomials.html#gr-bner-bases > for more examples. --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-support@googlegroups.com To unsubscribe from this group, send email to sage-support-unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-support URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
