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]
Read http://www.sagemath.org/doc/constructions/polynomials.html#gr-bner-bases for more examples. On Wed, Jul 15, 2009 at 11:19 PM, Doug<mcke...@gmail.com> wrote: > > I'm sorry for the deluge of questions, but I hope you will bear with > me as I come up to speed. This one is more substantive and I think > related to my first question yesterday about simplify() not > simplifying enough. Now, I have two equations with two unknowns I'm > trying to solve and I'm getting 3 solutions. Two are correct, but the > third yields a division by zero error if I try to plug it back into > the equations. And the real problem is that the solution I care about > is missing! These particular two equations are symmetric, so I could > (and have) solved it by subsituting x1 for x2 and solving just one > equation with one unknown. That works and that's how I know the > missing solution. But I want to solve slightly different sets of > equations that are _not_ symmetric and the fact that sage/Maxima fails > me here makes me pessimistic that my results for the asymmetric > versions are correct. > > Here's my code: > > var('x1, x2') > > eqn1 = (1/2*((x1^2 + 2*x1 - 4)*x2^2 + 2*(x1^2 + x1)*x2 + x1^2) > / ((x1^2 + 2*x1 + 1)*x2^2 + 2*(x1^2 + x1)*x2 + x1^2) == > 0) > eqn2 = (1/2*((x1^2 + 2*x1 + 1)*x2^2 + 2*(x1^2 + x1)*x2 - 4*x1^2) > / ((x1^2 + 2*x1 + 1)*x2^2 + 2*(x1^2 + x1)*x2 + x1^2) == > 0) > > print "eqn1: ", eqn1 > print "eqn2: ", eqn1 > > sols = solve([eqn1, eqn2], x1, x2, solution_dict=True) > > print "number of sols: ", len(sols) > print "sols: ", sols > > foo = eqn1.lhs().subs(x1=sqrt(5) - 2, x2=sqrt(5) - 2) > print "What about sqrt(5)-2 in eqn1?", foo > bar = eqn2.lhs().subs(x1=sqrt(5) - 2, x2=sqrt(5) - 2) > print "What about sqrt(5)-2 in eqn2?", bar > > print "But these both simplify to zero after a simplify_full: ", > foo.simplify_full() > print "But these both simplify to zero after a simplify_full: ", > bar.simplify_full() > > print "The third solution I do get yields a divide by zero error, but > I'm willing to forgive this:" > eqn1.lhs().subs(sols[2]) > > *** And here's what I get when I run it: *** > > sage: load problem.sage > eqn1: 1/2*((x1^2 + 2*x1 - 4)*x2^2 + 2*(x1^2 + x1)*x2 + x1^2)/((x1^2 + > 2*x1 + 1)*x2^2 + 2*(x1^2 + x1)*x2 + x1^2) == 0 > eqn2: 1/2*((x1^2 + 2*x1 - 4)*x2^2 + 2*(x1^2 + x1)*x2 + x1^2)/((x1^2 + > 2*x1 + 1)*x2^2 + 2*(x1^2 + x1)*x2 + x1^2) == 0 > number of sols: 3 > sols: [{x2: -sqrt(5), x1: -(sqrt(5) - 10)/(2*sqrt(5) - 1)}, {x2: sqrt > (5), x1: -(sqrt(5) + 10)/(2*sqrt(5) + 1)}, {x2: 0, x1: 0}] > What about sqrt(5)-2 in eqn1? 1/2*((sqrt(5) - 2)^2*((sqrt(5) - 2)^2 + > 2*sqrt(5) - 8) + (sqrt(5) - 2)^2 + 2*(sqrt(5) - 2)*((sqrt(5) - 2)^2 + > sqrt(5) - 2))/((sqrt(5) - 2)^2*((sqrt(5) - 2)^2 + 2*sqrt(5) - 3) + > (sqrt(5) - 2)^2 + 2*(sqrt(5) - 2)*((sqrt(5) - 2)^2 + sqrt(5) - 2)) > What about sqrt(5)-2 in eqn2? 1/2*((sqrt(5) - 2)^2*((sqrt(5) - 2)^2 + > 2*sqrt(5) - 3) - 4*(sqrt(5) - 2)^2 + 2*(sqrt(5) - 2)*((sqrt(5) - 2)^2 > + sqrt(5) - 2))/((sqrt(5) - 2)^2*((sqrt(5) - 2)^2 + 2*sqrt(5) - 3) + > (sqrt(5) - 2)^2 + 2*(sqrt(5) - 2)*((sqrt(5) - 2)^2 + sqrt(5) - 2)) > But these both simplify to zero after a simplify_full: 0 > But these both simplify to zero after a simplify_full: 0 > The third solution I do get yields a divide by zero error, but I'm > willing to forgive this: > --------------------------------------------------------------------------- > RuntimeError Traceback (most recent call > last) > ... > RuntimeError: power::eval(): division by zero > > As usual, any help would be greatly appreciated! > > Thank you, > > Doug > > > > --~--~---------~--~----~------------~-------~--~----~ 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 -~----------~----~----~----~------~----~------~--~---
