John,
> As far as I know you cannot check associativity in this naive way.
>
> For a start, nowhere in your code do you use the equation of the
> curve. If that is (say) y^2=x^3+a*x+b, then your equation will only
> be correct modulo the relations y1^2=x1^3+a*x1+b and so on.
this is correct, but the first statement is not, since Maple can do it
(sorry for that :-)
|\^/| Maple 10 (IBM INTEL LINUX)
._|\| |/|_. Copyright (c) Maplesoft, a division of Waterloo Maple Inc. 2005
\ MAPLE / All rights reserved. Maple is a trademark of
<____ ____> Waterloo Maple Inc.
| Type ? for help.
> lambda12 := (y1 - y2)/(x1 - x2):
> x4 := (lambda12*lambda12 - x1 - x2):
> nu12 := (y1 - lambda12*x1):
> y4 := (-lambda12*x4 - nu12):
> lambda23 := ((y2 - y3)/(x2 - x3)):
> x5 := (lambda23*lambda23 - x2 - x3):
> nu23 := (y2 - lambda23*x2):
> y5 := (-lambda23*x5 - nu23):
> s1 :=(x1 - x5)*(x1 - x5)*((y3 - y4)*(y\
> 3 - y4) - (x3 + x4)*(x3 - x4)*(x3 - x4)):
> s2 :=(x3 - x4)*(x3 - x4)*((y1 - y5)*(y\
> 1 - y5) - (x1 + x5)*(x1 - x5)*(x1 - x5)):
> numer(s1-s2):
> simplify(%, {y1^2=x1^3+a*x1+b,y2^2=x2^3+a*x2+b,y3^2=x3^3+a*x3+b});
0
Note the simplify(..., {eqs}) command, which computes the normal form of a
polynomial with respect to a set of polynomial equations. This is quite useful
for the user who is not aware of Gröbner bases (or the aware-user who prefers
a simple command). Does a similar command exist in SAGE?
Paul Zimmermann
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