Maxima fail to solve system of equations
(%i1) s:[b*c+a^2-4/5=0,b*d+a*b-1/2=0,c*d+a*c-1/5=0,d^2+b*c-1/2=0];
(%o1) [b*c+a^2-4/5=0,b*d+a*b-1/2=0,c*d+a*c-1/5=0,d^2+b*c-1/2=0]
(%i2) solve(s);
(%o2) []
(%i3) to_poly_solve(s, [a,b,c,d]);
to_poly_solve: to_poly_solver.mac is obsolete; I'm loading
to_poly_solve.mac instead.
Loading maxima-grobner $Revision: 1.6 $ $Date: 2009-06-02 07:49:49 $
(%o3) %union()
How solve this system? I think that one way is
(%i4) eliminate(s,[d]);
(%o4) [2*(4*b^3*c+(4*a^2-2)*b^2-4*a*b+1),5*c-2*b,5*b*c+5*a^2-4]
(%i5) eliminate(%,[c]);
(%o5) [-2*b*(6*b^2-20*a*b+5),-2*(8*b^4+(20*a^2-10)*b^2-20*a*b+5)]
(%i6) eliminate(%,[b]);
(%o6) [-1024000*(1225*a^4-1310*a^2+289)]
(%i7) factor(%);
(%o7) [-1024000*(35*a^2-50*a+17)*(35*a^2+50*a+17)]
(%i8) as:solve(%);
(%o8)
[a=-(sqrt(30)+25)/35,a=(sqrt(30)-25)/35,a=-(sqrt(30)-25)/35,a=(sqrt(30)+25)/35]
But
(%i9) solve(1225*a^4-1310*a^2+289);
(%o9)
[a=-sqrt(10*sqrt(30)+131)/(7*sqrt(5)),a=sqrt(10*sqrt(30)+131)/(7*sqrt(5)),a=-sqrt(131-10*sqrt(30))/(7*sqrt(5)),a=sqrt(131-10*sqrt(30))/(7*sqrt(5))]
best
Aleksas D