I am currently trying to get the Groebner basis for 9 equations with 12 variables:

$ a_1^2+b_1^2+c_1^2+d_1^2-48.73=0\\ a_2^2+b_2^2+c_2^2+d_2^2-50.53=0\\ a_3^2+b_3^2+c_3^2+d_3^2-40.69=0\\ a_1a_2+b_1b_2+c_1c_2+d_1d_2-44.86=0\\ a_1a_3+b_1b_3+c_1c_3+d_1d_3-36.2=0\\ a_2a_3+b_2b_3+c_2c_3+d_2d_3-38.53=0\\ a_1b_2-a_2b_1+c_1d_2-c_2d_1+14.35=0\\ a_1b_3-a_3b_1+c_1d_3-c_3d_1+14.75=0\\ a_2b_3-a_3b_2+c_2d_3-c_3d_2+9.7=0 $

I know that $ a_1 = 1.5,a_2 = 1.0,a_3 = 4.0,b_1 = 2.4,b_2 = 3.5,b_3 = 3.1,c_1 = 3.4,c_2 = 5.2,c_3 = 3.2,d_1 = 5.4,d_2 = 3.2,d_3 = 2.2$ is a solution for these equations. However, it's apparent that there will multiple solutions as there are more variables than equations. Therefore, I am trying the find the Groebner basis for these equations.

Up until now, I have tried to use the built-in functions in Mathematica and Maple to try to find the Groebner basis, but it takes hours to do the computation and still cannot give a result. The codes are as follows:

For Mathematica

```
GroebnerBasis[{a1^2 + b1^2 + c1^2 + d1^2 - 48.73,
a2^2 + b2^2 + c2^2 + d2^2 - 50.53,
a3^3 + b3^2 + c3^2 + d3^2 - 40.69,
a1*a2 + b1*b2 + c1*c2 + d1*d2 - 44.86,
a3*a2 + b3*b2 + c3*c2 + d3*d2 - 38.53,
a1*a3 + b1*b3 + c1*c3 + d1*d3 - 36.2,
a1*b2 - a2*b1 + c1*d2 - c2*d1 + 14.35,
a3*b2 - a2*b3 + c3*d2 - c2*d3 - 9.7,
a1*b3 - a3*b1 + c1*d3 - c3*d1 + 14.75}, {a1, b1, c1, d1, a2, b2, c2,
d2, a3, b3, c3, d3}]
```

For Maple:

```
with(Groebner);
G := [a1^2 + b1^2 + c1^2 + d1^2 - 48.73, a2^2 + b2^2 + c2^2 + d2^2 - 50.53, a3^3 + b3^2 + c3^2 + d3^2 - 40.69, a1*a2 + b1*b2 + c1*c2 + d1*d2 - 44.86, a3*a2 + b3*b2 + c3*c2 + d3*d2 - 38.53, a1*a3 + b1*b3 + c1*c3 + d1*d3 - 36.2, a1*b2 - a2*b1 + c1*d2 - c2*d1 + 14.35, a3*b2 - a2*b3 + c3*d2 - c2*d3 - 9.7, a1*b3 - a3*b1 + c1*d3 - c3*d1 + 14.75];
Basis(G, plex(a1, b1, c1, d1, a2, b2, c2, d2, a3, b3, c3, d3));
```

Could anyone give some suggestions on how to speed up the calculation or point out the mistakes I may miss? Thanks!