Dear Andrzej,
> I don't know if elegant Paul Zimmermann's MAPLE solution
> has counterpart in SAGE (I'm affraid it hasn't) but his remark on
> the Groebner bases (SINGULAR, MACAULAY2 interf?) may be essentiall.
> Does the discussion mean that I have to use the closed source
> software?
> Still waiting for reply.
I was able to do the job with SAGE, but I have to confess it was not as easy
as in Maple (however I am still more fluent in Maple):
----------------------------------------------------------------------
| SAGE Version 2.9.3, Release Date: 2008-01-05 |
| Type notebook() for the GUI, and license() for information. |
----------------------------------------------------------------------
sage: var('x1,y1,x2,y2,x3,y3,a,b')
sage: eq1 = y1^2 -(x1^3+a*x1+b)
sage: eq2 = y2^2 -(x2^3+a*x2+b)
sage: eq3 = y3^2 -(x3^3+a*x3+b)
sage: lambda12 = (y1 - y2)/(x1 - x2)
sage: x4 = (lambda12*lambda12 - x1 - x2)
sage: nu12 = (y1 - lambda12*x1)
sage: y4 = (-lambda12*x4 - nu12)
sage: lambda23 = ((y2 - y3)/(x2 - x3))
sage: x5 = (lambda23*lambda23 - x2 - x3)
sage: nu23 = (y2 - lambda23*x2)
sage: y5 = (-lambda23*x5 - nu23)
sage: s1 =(x1 - x5)*(x1 - x5)*((y3 - y4)*(y3-y4) - (x3+x4)*(x3-x4)*(x3-x4))
sage: s2 =(x3 - x4)*(x3 - x4)*((y1 - y5)*(y1-y5) - (x1+x5)*(x1-x5)*(x1-x5))
sage: n12 = numerator(factor(s1-s2))
sage: R = singular.ring(0, '(a,b,x1,x2,x3,y1,y2,y3)')
sage: I = singular.ideal([repr(eq1), repr(eq2), repr(eq3)])
sage: I2 = I.groebner()
sage: singular.reduce(repr(n12), I2)
0
In particular:
(1) is there a better way to normalize a rational expression that calling
factor? Apparently numerator alone does not do the job.
(2) can we get rid of the repr() calls?
Paul Zimmermann
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