It looks as if there are infinitely many solutions. If you clear the
denominators, you get polynomials, with which you can compute a
lexicographical Gröbner basis of 15 elements (given below). The dimension
of the corresponding ideal is 4, which is admittedly the dimension of the
set of complex solutions, but I can find quite a few real solutions, as
well: for example, {r1=r2=s1=s2=1,t1=-4,t2=6,t3=-4,t4=1}. To obtain those,
I first converted the result of the Gröbner basis back to SR. It's actually
pretty quick once you get the hang of it. As long as you keep t4=/=0, this
method should work.
There's probably a better way, though.
john perry
Gröbner basis:
{
r1*r2*t1 + 2*r1*s2 + 2*r2*s1,
r1*r2*t2 - r1 - r2 - 4*s1*s2,
r1*s2*t2 + 1/2*r1*t1 - r2*s2^2*t3 - 8*s1*s2^3*t3 - 4*s1*s2^2*t2 + s1 +
8*s2^5*t4 + 4*s2^4*t3 - 2*s2^3*t2 - s2^2*t1 - s2,
r1*t3 + r2*t3 + 4*s1*s2*t3 + 2*s1*t2 + 2*s2*t2,
r1*t4 + r2*t4 - 4*s2^2*t4 - 2*s2*t3 - t2,
r2^2*t3 + 2*r2*s2^2*t3 + 2*r2*s2*t2 - r2*t1 + 16*s2^5*t4 + 8*s2^4*t3 +
4*s2^3*t2 + 2*s2^2*t1 - 4*s2,
r2^2*t4 - r2*s2*t3 - r2*t2 - 8*s2^4*t4 - 4*s2^3*t3 - 2*s2^2*t2 - s2*t1 + 1,
r2*s1*t2 + r2*s2^2*t3 + 1/2*r2*t1 + 8*s1*s2^3*t3 + 4*s1*s2^2*t2 +
2*s1*s2*t1 - s1 - 8*s2^5*t4 - 4*s2^4*t3 + 2*s2^3*t2 + s2^2*t1 + s2,
r2*s1*t3 - r2*s2*t3 - 4*s1*s2^2*t3 - 2*s1*s2*t2 - s1*t1 - 2*s2^2*t2 -
s2*t1,
r2*s2*t3^2 - 2*r2*t1*t4 + r2*t2*t3 + 32*s2^5*t4^2 + 40*s2^4*t3*t4 +
16*s2^3*t2*t4 + 12*s2^3*t3^2 + 4*s2^2*t1*t4 + 10*s2^2*t2*t3 + 3*s2*t1*t3 +
2*s2*t2^2 - 8*s2*t4 + t1*t2 - 2*t3,
r2*s2*t4 + 1/4*r2*t3 - 2*s2^3*t4 - s2^2*t3 - 1/2*s2*t2 - 1/4*t1,
r2*t1*t4^2 - 1/2*r2*t2*t3*t4 + 1/8*r2*t3^3 - 16*s2^5*t4^3 -
20*s2^4*t3*t4^2 - 8*s2^3*t2*t4^2 - 7*s2^3*t3^2*t4 - 2*s2^2*t1*t4^2 -
5*s2^2*t2*t3*t4 - 1/2*s2^2*t3^3 - 3/2*s2*t1*t3*t4 - s2*t2^2*t4 -
1/4*s2*t2*t3^2 + 4*s2*t4^2 - 1/2*t1*t2*t4 - 1/8*t1*t3^2 + t3*t4,
s1*s2^3*t3^2 + s1*s2^2*t2*t3 + 1/4*s1*s2*t1*t3 + 1/4*s1*s2*t2^2 +
1/8*s1*t1*t2 - 1/8*s1*t3 - 4*s2^6*t4^2 - 6*s2^5*t3*t4 - 2*s2^4*t2*t4 -
2*s2^4*t3^2 - s2^3*t2*t3 + s2^2*t4 + 1/8*s2*t1*t2 + 3/8*s2*t3 + 1/16*t1^2,
s1*t4 + s2*t4 + 1/2*t3,
s2^6*t4^3 + 3/2*s2^5*t3*t4^2 + 1/2*s2^4*t2*t4^2 + 3/4*s2^4*t3^2*t4 +
1/2*s2^3*t2*t3*t4 + 1/8*s2^3*t3^3 + 1/16*s2^2*t1*t3*t4 + 1/16*s2^2*t2^2*t4
+ 1/8*s2^2*t2*t3^2 - 1/4*s2^2*t4^2 + 1/32*s2*t1*t3^2 + 1/32*s2*t2^2*t3 -
1/8*s2*t3*t4 - 1/64*t1^2*t4 + 1/64*t1*t2*t3 - 1/64*t3^2
}
On Tuesday, March 18, 2014 6:38:28 AM UTC-5, Urs Hackstein wrote:
>
> Dear all,
>
> I am dealing with the following system of nonlinear equations
> [R_1*S_2+S_1*R_2==-(1/2)*(T_1/T_4),S_1+S_2==-(1/2)*(T_3/T_4),R_1+R_2+4*S_1*S_2==(T_2/T_4),R_1*R_2==1/T_4]
>
>
> in the real variables R_1,R_2,S_1 and S_2. What is the best way to
> solve this system using Sage, if there is any?
> I tried to use solve, but we didn't receive any result within 26 hours.
> Thanks a lot in advance!
>
> Best regards,
>
> Urs Hackstein
>
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