recurrence relation for a system of simultaneous equations
David Stahl
I have one rational solution. I am trying to find a recurrence
relation that will allow me to generate additional rational
solutions. The equations are:
A6x0^2+A5x1^2+A4x2^2+A3x0x1+A2x0x2+A1x1x2+A0=0 (a)
B6y0^2+B5y1^2+B4y2^2+B3y0y1+B2y0y2+B1y1y2+B0=0 (b)
C8x0y0+C7x0y1+C6x0y2+C5x1y0+C4x1y1+C3x1y2+C2x2y0+C1x2y1+C0x2y2=0 (c)
I can find a recurrence relation for the equations separately. If a
solution to (a) is x0=d, x1=e, and x2=f then additional solutions can
be found. First define the x elements as:
x0=d+t
x1=e+mt
x2=f+nt
Substitute these values into (a) to get a homogeneous equation in t.
Then solve for t:
t=(N2m+N1n+N0)/(A5m^2+A1mn+A4n^2+A3m+A2n+A6)
N2= -2A5e-A3d-A1f
N1= -A1e-A2d-2A4f
N0= -A3e-2A6d-A2f
Since t is rational in m and n we simply pick rational values for m
and n to generate rational solutions for x0, x1, and x2. Similar
relations can be made for (b) and (c). My problem is that I do not
know how to find a recurrence relation for the simultaneous system.
Any guidance would be appreciated. Thank you.
David
David Joyner
You might be interested in looking at Martin Rubey's GUESS package
in Axiom, a bit of which is described in
http://wiki.sagemath.org/Axiom_as_an_OSCAS
I think it is included in Bill Page's SAGE package axiom4sage-0.3.
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