Solving polynomial equations

Question

I really love this website because you really exchange a lot of useful information. Let me tell you about my problem, I have these following equations: $$ T_1=SZ^2+GX+C $$ $$ T_2=SY^2+GY+C $$ $$ T_3=S(Z+a)^2+G(X+a)+C$$ $$ T_4=S(Y+a)^2+G(Y+a)+C$$ $$Z=\frac{T_1-T_2}{T_4-T_2}*a+y$$ $$X=SZ^2+GZ+C$$ These are the known parameters: $T_1,T_2,T_3,T_4,Y,a,Z$ and these are the unknown parameters:$X,S,G,C$ ... I want to solve these 6 equations to find the X value without using (S,G,C). Thanks for helping the science community.

Update: I used GAP to solve this problem and i took the code from someone else in this website (sorry i forget his name) and this is his code after modifying it a little bit:

R := PolynomialRing( Rationals,["t1","t2","t3","t4","a","s","g","c","y","x","z"]);
AssignGeneratorVariables(R);
i1:=s*z^2+g*x+c-t1;
i2:=s*y^2+g*y+c-t2;
i3:=s*(z+a)^2+g*(x+a)+c-t3;
i4:=s*(y+a)^2+g*(y+a)+c-t4;
i5 := s*z^2+g*z+c-x;
I := Ideal(R, [i1,i2,i3,i4]);
ord := EliminationOrdering([s,g,c]);
B:=GroebnerBasis(I,ord);

GAP still in progress for more than an hour and i don't know if it will work or not.

Answer 1

The Gröbner basis functionality in GAP is implemented mainly for illustration purposes and is by far not as strong as dedicated systems. In particular elimination orderings perform far slower. If you try instead a lexicographic ordering, the calculation is much quicker:

gap> ReducedGroebnerBasis(I,MonomialLexOrdering([x,c,g,s,y,z,a,t1,t2,t3,t4]));
[ a*s*y-a*s*z-1/2*t1+1/2*t2+1/2*t3-1/2*t4, 
t1*a*s+2*t1*s*y-t2*a*s-2*t2*s*z-t3*a*s-2  ....

Now there is one polynomial that does not involve s,c,g. (We scale to make it look nicer):

gap> p:=Filtered(last,x->x=Value(x,[s,c,g],[0,0,0]));;
gap> 2*p[1];
t1*a*y+t1*a*x-2*t1*a*z-t1*y^2+2*t1*y*x-t1*z^2-t2*a*y-t2*a*x+2*t2*a*z-t2*y^2+2*t2*y*z-2*t2*x*z+t2*z^2+t3*a*y-\
t3*a*x+t3*y^2-2*t3*y*x+t3*z^2-t4*a*y+t4*a*x+t4*y^2-2*t4*y*z+2*t4*x*z-t4*z^2

This is linear in x, so get the coefficients and solve:

gap> co:=PolynomialCoefficientsOfPolynomial(2*p[1],x);;

and thus -co[1]/co[2] is the solution:

$x=\frac{-T_1ay+2T_1az+T_1y^{2}+T_1z^{2}+T_2ay-2T_2az+T_2y^{2}-2T_2yz-T_2z^{2}-T_3ay-T_3y^{2}-T_3z^{2}+T_4ay-T_4y^{2}+2T_4yz+T_4z^{2}}{T_1a+2T_1y-T_2a-2T_2z-T_3a-2T_3y+T_4a+2T_4z}$