On Jan 11, 3:09 pm, Paul Zimmermann <[EMAIL PROTECTED]> wrote:
> 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
Here is a more idiomatic way to do this computation in Sage. We work
in the fraction field of a multivariate polynomial ring; this means
that our polynomial arithmetic is handled by libSingular instead of by
maxima, and that we can get the numerator directly with "numerator",
since fraction field elements are always normalized. Also, we use
Sage's wrapper of ideals and Groebner bases (which I believe is
implemented with libSingular), rather than calling Singular.
(Avoiding the call to "factor(s1-s2)" means that this version is much
faster.)
sage: R.<x1,y1,x2,y2,x3,y3,a,b> = QQ[]
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(s1-s2)
sage: I = ideal([eq1,eq2,eq3])
sage: I.reduce(n12)
0
> Paul Zimmermann
Carl Witty
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