In this file we have:

1. Ideals
2. Wolfram Mathematica code
3. Singular input/output
4. Wolfram Mathematica test

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1. Ideals

I_2=4*b00*a12*b21-b00*a01*b10-2*b1m1*a12*b10+am11*b10^2+2*b1m1*a01*b01-4*am11*b21*b01
,2*b00*a01*b21-2*b00*a10*b10-b1m1*a01*b10+a00*b10^2+4*b1m1*a10*b01-4*a00*b21*b01
,b00*a01^2-4*b00*a10*a12-am11*a01*b10+2*a00*a12*b10+4*am11*a10*b01-2*a00*a01*b01
,b1m1*a01^2-4*b1m1*a10*a12-2*am11*a01*b21+4*a00*a12*b21+2*am11*a10*b10-a00*a01*b10
;


I_3=a01-2*b01,
2*a10-b10
,4*b1m1^2*a12^2*b21+8*am11*b00*a12*b21^2-4*am11*b1m1*a12*b21*b10-4*a00*b00*a12*b21*b10-4*b00^2*a12*b21*b10+2*a00*b1m1*a12*b10^2+6*b1m1*b00*a12*b10^2+am11^2*b21*b10^2-a00*am11*b10^3+am11*b00*b10^3-8*b1m1*b00*a12*b21*b01-4*am11^2*b21^2*b01-8*b1m1^2*a12*b10*b01+4*a00*am11*b21*b10*b01-8*am11*b00*b21*b10*b01+a00^2*b10^2*b01-b00^2*b10^2*b01-4*a00^2*b21*b01^2+8*am11*b1m1*b21*b01^2+8*a00*b00*b21*b01^2+12*b00^2*b21*b01^2-4*a00*b1m1*b10*b01^2-8*b1m1*b00*b10*b01^2+12*b1m1^2*b01^3
,16*a00*b1m1*a12^2*b21+8*am11^2*a12*b21^2-4*b1m1^2*a12^2*b10-8*a00*am11*a12*b21*b10+6*a00^2*a12*b10^2+4*am11*b1m1*a12*b10^2+4*a00*b00*a12*b10^2-2*b00^2*a12*b10^2+3*am11^2*b10^3-16*a00^2*a12*b21*b01-16*am11*b1m1*a12*b21*b01-16*a00*b00*a12*b21*b01-16*a00*b1m1*a12*b10*b01+8*b1m1*b00*a12*b10*b01-16*am11^2*b21*b10*b01-8*a00*am11*b10^2*b01-4*am11*b00*b10^2*b01+8*b1m1^2*a12*b01^2+48*a00*am11*b21*b01^2+16*am11*b00*b21*b01^2-4*a00^2*b10*b01^2+4*b00^2*b10*b01^2+16*a00*b1m1*b01^3-16*b1m1*b00*b01^3
,8*a00^2*b1m1*a12*b21+8*am11*b1m1^2*a12*b21-24*a00*b1m1*b00*a12*b21+4*a00*am11^2*b21^2+4*am11^2*b00*b21^2+8*b1m1^2*b00*a12*b10-4*a00^2*am11*b21*b10-4*am11^2*b1m1*b21*b10-8*a00*am11*b00*b21*b10-4*am11*b00^2*b21*b10+3*a00^3*b10^2+4*a00*am11*b1m1*b10^2-a00^2*b00*b10^2+8*am11*b1m1*b00*b10^2-3*a00*b00^2*b10^2+b00^3*b10^2-8*b1m1^3*a12*b01-8*a00^3*b21*b01+8*a00*am11*b1m1*b21*b01+16*a00^2*b00*b21*b01-8*am11*b1m1*b00*b21*b01+24*a00*b00^2*b21*b01-12*a00^2*b1m1*b10*b01-12*am11*b1m1^2*b10*b01-4*a00*b1m1*b00*b10*b01-8*b1m1*b00^2*b10*b01+20*a00*b1m1^2*b01^2+4*b1m1^2*b00*b01^2
,4*b1m1^3*a12^2-8*a00*am11*b1m1*a12*b21+8*am11*b1m1*b00*a12*b21-4*am11^3*b21^2-4*am11*b1m1^2*a12*b10+12*a00*b1m1*b00*a12*b10-4*b1m1*b00^2*a12*b10+4*a00*am11^2*b21*b10+4*am11^2*b00*b21*b10-3*a00^2*am11*b10^2-3*am11^2*b1m1*b10^2+4*a00*am11*b00*b10^2-am11*b00^2*b10^2-8*a00*b1m1^2*a12*b01-8*b1m1^2*b00*a12*b01+8*a00^2*am11*b21*b01+8*am11^2*b1m1*b21*b01-24*a00*am11*b00*b21*b01+4*a00*am11*b1m1*b10*b01-4*am11*b1m1*b00*b10*b01+4*a00^2*b1m1*b01^2+12*am11*b1m1^2*b01^2-16*a00*b1m1*b00*b01^2+12*b1m1*b00^2*b01^2
,4*a00*b1m1^2*a12^2+4*b1m1^2*b00*a12^2+8*am11^2*b1m1*a12*b21-24*a00*am11*b00*a12*b21+8*am11*b00^2*a12*b21-4*a00*am11*b1m1*a12*b10+12*a00^2*b00*a12*b10+4*am11*b1m1*b00*a12*b10+8*a00*b00^2*a12*b10-4*b00^3*a12*b10-4*am11^3*b21*b10+a00*am11^2*b10^2+5*am11^2*b00*b10^2-8*a00^2*b1m1*a12*b01-8*am11*b1m1^2*a12*b01-16*a00*b1m1*b00*a12*b01-8*b1m1*b00^2*a12*b01+16*a00*am11^2*b21*b01-8*a00^2*am11*b10*b01-12*am11^2*b1m1*b10*b01-4*a00*am11*b00*b10*b01-12*am11*b00^2*b10*b01+4*a00^3*b01^2+32*a00*am11*b1m1*b01^2-12*a00^2*b00*b01^2+16*am11*b1m1*b00*b01^2-4*a00*b00^2*b01^2+12*b00^3*b01^2
,8*am11^2*b1m1*a12*b21^2-32*a00*am11*b00*a12*b21^2+16*a00^2*b00*a12*b21*b10+8*am11*b1m1*b00*a12*b21*b10+16*a00*b00^2*a12*b21*b10-4*am11^3*b21^2*b10-2*a00^2*b1m1*a12*b10^2-8*a00*b1m1*b00*a12*b10^2-6*b1m1*b00^2*a12*b10^2+4*am11^2*b00*b21*b10^2+a00^2*am11*b10^3-am11*b00^2*b10^3-32*a00*b1m1*b00*a12*b21*b01+24*a00*am11^2*b21^2*b01+8*am11^2*b00*b21^2*b01+8*a00*b1m1^2*a12*b10*b01+16*b1m1^2*b00*a12*b10*b01-16*a00^2*am11*b21*b10*b01-16*am11^2*b1m1*b21*b10*b01-8*a00*am11*b00*b21*b10*b01-8*am11*b00^2*b21*b10*b01+2*a00^3*b10^2*b01+4*a00*am11*b1m1*b10^2*b01-2*a00^2*b00*b10^2*b01+8*am11*b1m1*b00*b10^2*b01-2*a00*b00^2*b10^2*b01+2*b00^3*b10^2*b01-8*b1m1^3*a12*b01^2+32*a00*am11*b1m1*b21*b01^2-8*a00^2*b1m1*b10*b01^2-12*am11*b1m1^2*b10*b01^2+8*a00*b1m1*b00*b10*b01^2+8*a00*b1m1^2*b01^3-8*b1m1^2*b00*b01^3
,8*a00*am11*b1m1*a12*b21^2+4*am11^3*b21^3-16*a00*b1m1*b00*a12*b21*b10-4*a00*am11^2*b21^2*b10-4*am11^2*b00*b21^2*b10+2*a00*b1m1^2*a12*b10^2+6*b1m1^2*b00*a12*b10^2+3*a00^2*am11*b21*b10^2+4*am11^2*b1m1*b21*b10^2-4*a00*am11*b00*b21*b10^2+am11*b00^2*b21*b10^2-a00*am11*b1m1*b10^3+am11*b1m1*b00*b10^3+8*a00*b1m1^2*a12*b21*b01-8*a00^2*am11*b21^2*b01-12*am11^2*b1m1*b21^2*b01+24*a00*am11*b00*b21^2*b01-8*b1m1^3*a12*b10*b01-4*am11*b1m1*b00*b21*b10*b01+a00^2*b1m1*b10^2*b01-b1m1*b00^2*b10^2*b01-8*a00^2*b1m1*b21*b01^2-4*am11*b1m1^2*b21*b01^2+24*a00*b1m1*b00*b21*b01^2-4*a00*b1m1^2*b10*b01^2-8*b1m1^2*b00*b10*b01^2+12*b1m1^3*b01^3
,32*a00^2*am11*b00*a12*b21^2+4*am11^4*b21^3-16*a00^3*b00*a12*b21*b10-24*a00*am11*b1m1*b00*a12*b21*b10-16*a00^2*b00^2*a12*b21*b10-4*am11^3*b00*b21^2*b10+2*a00^3*b1m1*a12*b10^2+2*a00*am11*b1m1^2*a12*b10^2+8*a00^2*b1m1*b00*a12*b10^2+6*am11*b1m1^2*b00*a12*b10^2+6*a00*b1m1*b00^2*a12*b10^2+3*a00^2*am11^2*b21*b10^2+4*am11^3*b1m1*b21*b10^2-8*a00*am11^2*b00*b21*b10^2+am11^2*b00^2*b21*b10^2-a00^3*am11*b10^3-a00*am11^2*b1m1*b10^3+am11^2*b1m1*b00*b10^3+a00*am11*b00^2*b10^3+8*a00*am11*b1m1^2*a12*b21*b01-32*am11*b1m1^2*b00*a12*b21*b01+96*a00*b1m1*b00^2*a12*b21*b01-32*a00^2*am11^2*b21^2*b01-12*am11^3*b1m1*b21^2*b01-16*am11^2*b00^2*b21^2*b01-8*a00^2*b1m1^2*a12*b10*b01-8*am11*b1m1^3*a12*b10*b01-16*a00*b1m1^2*b00*a12*b10*b01-32*b1m1^2*b00^2*a12*b10*b01+16*a00^3*am11*b21*b10*b01+16*a00*am11^2*b1m1*b21*b10*b01+24*a00^2*am11*b00*b21*b10*b01+12*am11^2*b1m1*b00*b21*b10*b01+40*a00*am11*b00^2*b21*b10*b01+16*am11*b00^3*b21*b10*b01-2*a00^4*b10^2*b01-3*a00^2*am11*b1m1*b10^2*b01-10*a00^3*b00*b10^2*b01-24*a00*am11*b1m1*b00*b10^2*b01+6*a00^2*b00^2*b10^2*b01-33*am11*b1m1*b00^2*b10^2*b01+10*a00*b00^3*b10^2*b01-4*b00^4*b10^2*b01+8*a00*b1m1^3*a12*b01^2+32*b1m1^3*b00*a12*b01^2-40*a00^2*am11*b1m1*b21*b01^2-4*am11^2*b1m1^2*b21*b01^2+32*a00^3*b00*b21*b01^2-8*a00*am11*b1m1*b00*b21*b01^2-64*a00^2*b00^2*b21*b01^2+32*am11*b1m1*b00^2*b21*b01^2-96*a00*b00^3*b21*b01^2+8*a00^3*b1m1*b10*b01^2+8*a00*am11*b1m1^2*b10*b01^2+40*a00^2*b1m1*b00*b10*b01^2+40*am11*b1m1^2*b00*b10*b01^2+16*a00*b1m1*b00^2*b10*b01^2+32*b1m1*b00^3*b10*b01^2-8*a00^2*b1m1^2*b01^3+12*am11*b1m1^3*b01^3-72*a00*b1m1^2*b00*b01^3-16*b1m1^2*b00^2*b01^3
,4*am11^4*b1m1*b21^3-16*a00*am11^3*b00*b21^3-8*a00*am11*b1m1^2*b00*a12*b21*b10+24*a00^2*am11^2*b00*b21^2*b10-4*am11^3*b1m1*b00*b21^2*b10+24*a00*am11^2*b00^2*b21^2*b10+2*a00^3*b1m1^2*a12*b10^2+2*a00*am11*b1m1^3*a12*b10^2+6*am11*b1m1^3*b00*a12*b10^2-2*a00*b1m1^2*b00^2*a12*b10^2+3*a00^2*am11^2*b1m1*b21*b10^2+4*am11^3*b1m1^2*b21*b10^2-20*a00^3*am11*b00*b21*b10^2-32*a00*am11^2*b1m1*b00*b21*b10^2+am11^2*b1m1*b00^2*b21*b10^2-12*a00*am11*b00^3*b21*b10^2-a00^3*am11*b1m1*b10^3-a00*am11^2*b1m1^2*b10^3+6*a00^4*b00*b10^3+12*a00^2*am11*b1m1*b00*b10^3+am11^2*b1m1^2*b00*b10^3-2*a00^3*b00^2*b10^3+13*a00*am11*b1m1*b00^2*b10^3-6*a00^2*b00^3*b10^3+2*a00*b00^4*b10^3+8*a00*am11*b1m1^3*a12*b21*b01-32*a00^2*am11^2*b1m1*b21^2*b01-12*am11^3*b1m1^2*b21^2*b01+32*a00^3*am11*b00*b21^2*b01+64*a00*am11^2*b1m1*b00*b21^2*b01-96*a00^2*am11*b00^2*b21^2*b01-8*a00^2*b1m1^3*a12*b10*b01-8*am11*b1m1^4*a12*b10*b01+16*a00^3*am11*b1m1*b21*b10*b01+16*a00*am11^2*b1m1^2*b21*b10*b01-16*a00^4*b00*b21*b10*b01+24*a00^2*am11*b1m1*b00*b21*b10*b01-4*am11^2*b1m1^2*b00*b21*b10*b01+32*a00^3*b00^2*b21*b10*b01+8*a00*am11*b1m1*b00^2*b21*b10*b01+48*a00^2*b00^3*b21*b10*b01-2*a00^4*b1m1*b10^2*b01-3*a00^2*am11*b1m1^2*b10^2*b01-26*a00^3*b1m1*b00*b10^2*b01-32*a00*am11*b1m1^2*b00*b10^2*b01-6*a00^2*b1m1*b00^2*b10^2*b01-am11*b1m1^2*b00^2*b10^2*b01-14*a00*b1m1*b00^3*b10^2*b01+8*a00*b1m1^4*a12*b01^2-40*a00^2*am11*b1m1^2*b21*b01^2-4*am11^2*b1m1^3*b21*b01^2+32*a00^3*b1m1*b00*b21*b01^2+40*a00*am11*b1m1^2*b00*b21*b01^2-96*a00^2*b1m1*b00^2*b21*b01^2+8*a00^3*b1m1^2*b10*b01^2+8*a00*am11*b1m1^3*b10*b01^2+48*a00^2*b1m1^2*b00*b10*b01^2-8*am11*b1m1^3*b00*b10*b01^2+40*a00*b1m1^2*b00^2*b10*b01^2-8*a00^2*b1m1^3*b01^3+12*am11*b1m1^4*b01^3-40*a00*b1m1^3*b00*b01^3
;

I31=a01-2*b01,2*a10-b10,2*am11*b21-a00*b10-b00*b10+2*b1m1*b01,2*b1m1*a12+am11*b10-2*a00*b01-2*b00*b01
;

I32=a01-2*b01,2*a10-b10,4*am11*b1m1*a12*b21-16*a00*b00*a12*b21+2*a00*b1m1*a12*b10+6*b1m1*b00*a12*b10-2*am11^2*b21*b10-a00*am11*b10^2+am11*b00*b10^2-4*b1m1^2*a12*b01+12*a00*am11*b21*b01+4*am11*b00*b21*b01-2*a00^2*b10*b01-6*am11*b1m1*b10*b01+4*a00*b00*b10*b01-2*b00^2*b10*b01+4*a00*b1m1*b01^2-4*b1m1*b00*b01^2
,8*a00*b1m1*a12*b21+4*am11^2*b21^2-4*b1m1^2*a12*b10-4*a00*am11*b21*b10-4*am11*b00*b21*b10+3*a00^2*b10^2+6*am11*b1m1*b10^2-4*a00*b00*b10^2+b00^2*b10^2-8*a00^2*b21*b01-16*am11*b1m1*b21*b01+24*a00*b00*b21*b01-8*a00*b1m1*b10*b01-4*b1m1*b00*b10*b01+12*b1m1^2*b01^2
,4*b1m1^2*a12^2+8*am11*b00*a12*b21-8*am11*b1m1*a12*b10+12*a00*b00*a12*b10-4*b00^2*a12*b10+3*am11^2*b10^2-8*a00*b1m1*a12*b01-8*b1m1*b00*a12*b01-8*am11^2*b21*b01-4*a00*am11*b10*b01-8*am11*b00*b10*b01+4*a00^2*b01^2+24*am11*b1m1*b01^2-16*a00*b00*b01^2+12*b00^2*b01^2
,32*a00^2*b00*a12*b21+4*am11^3*b21^2-4*a00^2*b1m1*a12*b10-4*am11*b1m1^2*a12*b10-12*a00*b1m1*b00*a12*b10-4*am11^2*b00*b21*b10+5*a00^2*am11*b10^2+6*am11^2*b1m1*b10^2-6*a00*am11*b00*b10^2+am11*b00^2*b10^2+8*a00*b1m1^2*a12*b01-32*a00^2*am11*b21*b01-16*am11^2*b1m1*b21*b01+16*a00*am11*b00*b21*b01+4*a00^3*b10*b01+4*a00*am11*b1m1*b10*b01-8*a00^2*b00*b10*b01-4*am11*b1m1*b00*b10*b01+4*a00*b00^2*b10*b01-8*a00^2*b1m1*b01^2+12*am11*b1m1^2*b01^2+8*a00*b1m1*b00*b01^2
,4*am11^3*b1m1*b21^2-16*a00*am11^2*b00*b21^2-4*a00^2*b1m1^2*a12*b10-4*am11*b1m1^3*a12*b10+4*a00*b1m1^2*b00*a12*b10+16*a00^2*am11*b00*b21*b10-4*am11^2*b1m1*b00*b21*b10+16*a00*am11*b00^2*b21*b10+5*a00^2*am11*b1m1*b10^2+6*am11^2*b1m1^2*b10^2-12*a00^3*b00*b10^2-30*a00*am11*b1m1*b00*b10^2+16*a00^2*b00^2*b10^2+am11*b1m1*b00^2*b10^2-4*a00*b00^3*b10^2+8*a00*b1m1^3*a12*b01-32*a00^2*am11*b1m1*b21*b01-16*am11^2*b1m1^2*b21*b01+32*a00^3*b00*b21*b01+80*a00*am11*b1m1*b00*b21*b01-96*a00^2*b00^2*b21*b01+4*a00^3*b1m1*b10*b01+4*a00*am11*b1m1^2*b10*b01+24*a00^2*b1m1*b00*b10*b01-4*am11*b1m1^2*b00*b10*b01+20*a00*b1m1*b00^2*b10*b01-8*a00^2*b1m1^2*b01^2+12*am11*b1m1^3*b01^2-40*a00*b1m1^2*b00*b01^2
;

I6=a00,b00,am11,b1m1,a01-2*b01,2*a10-b10;


/////////////////////////////////////////
Wolfram Mathematica code
Next we provide complete computation for q=3. The similar code, just change q and eps, if necessary, works for other cases. 

/////////////////////////////////////////
Wolfram Mathematica code for system h_1=h_2=...=h_16=0. 

Quit;  Clear[rho,sigma,alpha,beta,gamma,delta,eps,q,a00,am11,b1m1,b00,
a10,a01,a12,b21,b10,b01]
q=3;               (*eps=1 if equivariant, eps =-1; if reversible*)
                   (* rho=0; sigma=0; if linear transformation*)
fi=2Pi/q;

B={{alpha,beta},{gamma,delta}};
detB=Det[B];

B1={{delta,-gamma},{-beta,alpha}}; 
                 (*transposed inverse of B since det B=1 *)
F1={{-a00,-am11},{b1m1,b00}} (*parameters of linear part *);

G={{-a10,-a01/2},{-a01/2,-a12},{b21,b10/2},{b10/2,b01}} 
                 (*parameters of quadratic part *);
BB=KroneckerProduct[B,B1];
X1=F1+2KroneckerProduct[{{1,0},{0,1}},{rho,sigma}].G//Simplify;

h12=F1.{rho,sigma}+(KroneckerProduct[{{1,0},{0,1}},{rho,sigma}].G).{rho,sigma}//Simplify;
h3456=X1.B-eps B.X1//Simplify//Flatten;
h714=2G.B-2 eps BB.G//Simplify//Flatten;
h1516={alpha+delta-2Cos[fi],detB-1};
sys={h12,h3456,h714,h1516}//Simplify//Flatten
sys//.eps->1//Simplify//InputForm


Output: polynomials h_1, h_2, ... h_16, which are given as the input to Singular in the case eps=1 and q=3.

 -(a00*rho) - a10*rho^2 - sigma*(am11 + a01*rho + a12*sigma), 
 b1m1*rho + b21*rho^2 + sigma*(b00 + b10*rho + b01*sigma), 
 -(alpha*(a00 + 2*a10*rho + a01*sigma)) - 
  gamma*(am11 + a01*rho + 2*a12*sigma) + 
  eps*(a00*alpha - b1m1*beta + 2*a10*alpha*rho - 2*b21*beta*rho + 
    a01*alpha*sigma - b10*beta*sigma), -(beta*(a00 + 2*a10*rho + a01*sigma)) - 
  delta*(am11 + a01*rho + 2*a12*sigma) + 
  alpha*eps*(am11 + a01*rho + 2*a12*sigma) - 
  beta*eps*(b00 + b10*rho + 2*b01*sigma), gamma*(b00 + b10*rho + 2*b01*sigma) + 
  alpha*(b1m1 + 2*b21*rho + b10*sigma) + 
  eps*(-(b1m1*delta) + a00*gamma - 2*b21*delta*rho + 2*a10*gamma*rho - 
    b10*delta*sigma + a01*gamma*sigma), delta*(b00 + b10*rho + 2*b01*sigma) + 
  beta*(b1m1 + 2*b21*rho + b10*sigma) + 
  eps*(-(b00*delta) + am11*gamma - b10*delta*rho + a01*gamma*rho - 
    2*b01*delta*sigma + 2*a12*gamma*sigma), -2*b21*beta*delta*eps + 
  2*a10*alpha*(-1 + delta*eps) - (a01 + a01*alpha*eps - b10*beta*eps)*gamma, 
 -2*a10*beta + a01*delta*(-1 + alpha*eps) - 
  eps*(b10*beta*delta + 2*a12*alpha*gamma - 2*b01*beta*gamma), 
 -2*a10*alpha*beta*eps - alpha*b10*beta*eps + 2*b21*beta^2*eps + 
  a01*alpha*(-1 + alpha*eps) - 2*a12*gamma, 
 beta*(-2*alpha*b01 + b10*beta)*eps - 2*a12*(delta - alpha^2*eps) - 
  a01*(beta + alpha*beta*eps), 2*alpha*b21 - 2*b21*delta^2*eps + 
  gamma*(b10 + 2*a10*delta*eps + b10*delta*eps - a01*eps*gamma), 
 2*b21*beta + b10*(delta - delta^2*eps) + 
  eps*gamma*(a01*delta + 2*b01*delta - 2*a12*gamma), 
 2*b21*beta*delta*eps + 2*(b01 - a10*beta*eps)*gamma + 
  alpha*(b10 - b10*delta*eps + a01*eps*gamma), -2*b01*delta*(-1 + alpha*eps) + 
  b10*(beta + beta*delta*eps) + (2*a12*alpha - a01*beta)*eps*gamma, 
 1 + alpha + delta, -1 + alpha*delta - beta*gamma;
//////////////////////////////////
Singular input/output

Singular input file, case q=3, eps=1 for computation of Ideal I_3, the 6th elimination ideal
//////////////////////////////////
LIB"primdec.lib";
LIB "elim.lib";

ring r=0,( a00, am11, b1m1, b00, a10, a01, a12, b21, b10, b01,alpha,beta,gamma,delta,rho,sigma,w,w1,w2,eps ),dp;
r; 

option(redSB);

ideal j=-(a00*rho) - a10*rho^2 - sigma*(am11 + a01*rho + a12*sigma), 
 b1m1*rho + b21*rho^2 + sigma*(b00 + b10*rho + b01*sigma), 
 -(b1m1*beta) - am11*gamma - 2*b21*beta*rho - a01*gamma*rho - b10*beta*sigma - 
  2*a12*gamma*sigma, -(beta*(a00 + 2*a10*rho + a01*sigma)) + 
  alpha*(am11 + a01*rho + 2*a12*sigma) - delta*(am11 + a01*rho + 2*a12*sigma) - 
  beta*(b00 + b10*rho + 2*b01*sigma), -(b1m1*delta) + a00*gamma - 
  2*b21*delta*rho + 2*a10*gamma*rho - b10*delta*sigma + a01*gamma*sigma + 
  gamma*(b00 + b10*rho + 2*b01*sigma) + alpha*(b1m1 + 2*b21*rho + b10*sigma), 
 b1m1*beta + am11*gamma + 2*b21*beta*rho + a01*gamma*rho + b10*beta*sigma + 
  2*a12*gamma*sigma, 2*a10*alpha*(-1 + delta) - 2*b21*beta*delta - 
  (a01 + a01*alpha - b10*beta)*gamma, -2*a10*beta + a01*(-1 + alpha)*delta - 
  b10*beta*delta - 2*a12*alpha*gamma + 2*b01*beta*gamma, 
 a01*(-1 + alpha)*alpha - 2*a10*alpha*beta - alpha*b10*beta + 2*b21*beta^2 - 
  2*a12*gamma, beta*(-(a01*(1 + alpha)) - 2*alpha*b01 + b10*beta) + 
  2*a12*(alpha^2 - delta), 2*alpha*b21 - 2*b21*delta^2 + 
  gamma*(b10 + 2*a10*delta + b10*delta - a01*gamma), 
 2*b21*beta - b10*(-1 + delta)*delta + 
  gamma*(a01*delta + 2*b01*delta - 2*a12*gamma), 
 2*b21*beta*delta + 2*(b01 - a10*beta)*gamma + 
  alpha*(b10 - b10*delta + a01*gamma), -2*(-1 + alpha)*b01*delta + 
  b10*beta*(1 + delta) + (2*a12*alpha - a01*beta)*gamma, 1 + alpha + delta, 
 -1 + alpha*delta - beta*gamma;
 
ideal i=eliminate(j,alpha*beta*gamma*delta*rho*sigma);
i;
$;

Singular output file, case q=3, eps=1 for computation of Ideal I_3, the 6th elimination ideal
//////////////////////////////////
                    SINGULAR                                 /
 A Computer Algebra System for Polynomial Computations       /   version 4.0.1
                                                           0<
 by: W. Decker, G.-M. Greuel, G. Pfister, H. Schoenemann     \   Sep 2014
FB Mathematik der Universitaet, D-67653 Kaiserslautern        \
// ** loaded /opt/Singular-4-0-1/bin/../share/singular/LIB/primdec.lib (4.0.0.0,Jun_2013)
// ** loaded /opt/Singular-4-0-1/bin/../share/singular/LIB/ring.lib (4.0.0.0,Jun_2013)
// ** loaded /opt/Singular-4-0-1/bin/../share/singular/LIB/absfact.lib (4.0.0.0,Jun_2013)
// ** loaded /opt/Singular-4-0-1/bin/../share/singular/LIB/triang.lib (4.0.0.0,Jun_2013)
// ** loaded /opt/Singular-4-0-1/bin/../share/singular/LIB/matrix.lib (4.0.0.0,Jun_2013)
// ** loaded /opt/Singular-4-0-1/bin/../share/singular/LIB/nctools.lib (4.0.0.0,Jun_2013)
// ** loaded /opt/Singular-4-0-1/bin/../share/singular/LIB/inout.lib (4.0.0.0,Jun_2013)
// ** loaded /opt/Singular-4-0-1/bin/../share/singular/LIB/random.lib (4.0.0.0,Jun_2013)
// ** loaded /opt/Singular-4-0-1/bin/../share/singular/LIB/poly.lib (4.0.0.0,Jun_2013)
// ** loaded /opt/Singular-4-0-1/bin/../share/singular/LIB/elim.lib (4.0.0.1,Jan_2014)
// ** loaded /opt/Singular-4-0-1/bin/../share/singular/LIB/general.lib (4.0.0.1,Jan_2014)
// ** redefining blowup0 **
// ** redefining blowup0 **
// ** redefining elimRing **
// ** redefining elimRing **
// ** redefining elim **
// ** redefining elim **
// ** redefining elim2 **
// ** redefining elim2 **
// ** redefining elim1 **
// ** redefining elim1 **
// ** redefining nselect **
// ** redefining nselect **
// ** redefining sat **
// ** redefining sat **
// ** redefining select **
// ** redefining select **
// ** redefining select1 **
// ** redefining select1 **
// ** loaded /opt/Singular-4-0-1/bin/../share/singular/LIB/elim.lib (4.0.0.1,Jan_2014)
//   characteristic : 0
//   number of vars : 20
//        block   1 : ordering dp
//                  : names    a00 am11 b1m1 b00 a10 a01 a12 b21 b10 b01 alpha beta gamma delta rho sigma w w1 w2 eps
//        block   2 : ordering C
i[1]=a01-2*b01
i[2]=2*a10-b10
i[3]=4*b1m1^2*a12^2*b21+8*am11*b00*a12*b21^2-4*am11*b1m1*a12*b21*b10-4*a00*b00*a12*b21*b10-4*b00^2*a12*b21*b10+2*a00*b1m1*a12*b10^2+6*b1m1*b00*a12*b10^2+am11^2*b21*b10^2-a00*am11*b10^3+am11*b00*b10^3-8*b1m1*b00*a12*b21*b01-4*am11^2*b21^2*b01-8*b1m1^2*a12*b10*b01+4*a00*am11*b21*b10*b01-8*am11*b00*b21*b10*b01+a00^2*b10^2*b01-b00^2*b10^2*b01-4*a00^2*b21*b01^2+8*am11*b1m1*b21*b01^2+8*a00*b00*b21*b01^2+12*b00^2*b21*b01^2-4*a00*b1m1*b10*b01^2-8*b1m1*b00*b10*b01^2+12*b1m1^2*b01^3
i[4]=16*a00*b1m1*a12^2*b21+8*am11^2*a12*b21^2-4*b1m1^2*a12^2*b10-8*a00*am11*a12*b21*b10+6*a00^2*a12*b10^2+4*am11*b1m1*a12*b10^2+4*a00*b00*a12*b10^2-2*b00^2*a12*b10^2+3*am11^2*b10^3-16*a00^2*a12*b21*b01-16*am11*b1m1*a12*b21*b01-16*a00*b00*a12*b21*b01-16*a00*b1m1*a12*b10*b01+8*b1m1*b00*a12*b10*b01-16*am11^2*b21*b10*b01-8*a00*am11*b10^2*b01-4*am11*b00*b10^2*b01+8*b1m1^2*a12*b01^2+48*a00*am11*b21*b01^2+16*am11*b00*b21*b01^2-4*a00^2*b10*b01^2+4*b00^2*b10*b01^2+16*a00*b1m1*b01^3-16*b1m1*b00*b01^3
i[5]=8*a00^2*b1m1*a12*b21+8*am11*b1m1^2*a12*b21-24*a00*b1m1*b00*a12*b21+4*a00*am11^2*b21^2+4*am11^2*b00*b21^2+8*b1m1^2*b00*a12*b10-4*a00^2*am11*b21*b10-4*am11^2*b1m1*b21*b10-8*a00*am11*b00*b21*b10-4*am11*b00^2*b21*b10+3*a00^3*b10^2+4*a00*am11*b1m1*b10^2-a00^2*b00*b10^2+8*am11*b1m1*b00*b10^2-3*a00*b00^2*b10^2+b00^3*b10^2-8*b1m1^3*a12*b01-8*a00^3*b21*b01+8*a00*am11*b1m1*b21*b01+16*a00^2*b00*b21*b01-8*am11*b1m1*b00*b21*b01+24*a00*b00^2*b21*b01-12*a00^2*b1m1*b10*b01-12*am11*b1m1^2*b10*b01-4*a00*b1m1*b00*b10*b01-8*b1m1*b00^2*b10*b01+20*a00*b1m1^2*b01^2+4*b1m1^2*b00*b01^2
i[6]=4*b1m1^3*a12^2-8*a00*am11*b1m1*a12*b21+8*am11*b1m1*b00*a12*b21-4*am11^3*b21^2-4*am11*b1m1^2*a12*b10+12*a00*b1m1*b00*a12*b10-4*b1m1*b00^2*a12*b10+4*a00*am11^2*b21*b10+4*am11^2*b00*b21*b10-3*a00^2*am11*b10^2-3*am11^2*b1m1*b10^2+4*a00*am11*b00*b10^2-am11*b00^2*b10^2-8*a00*b1m1^2*a12*b01-8*b1m1^2*b00*a12*b01+8*a00^2*am11*b21*b01+8*am11^2*b1m1*b21*b01-24*a00*am11*b00*b21*b01+4*a00*am11*b1m1*b10*b01-4*am11*b1m1*b00*b10*b01+4*a00^2*b1m1*b01^2+12*am11*b1m1^2*b01^2-16*a00*b1m1*b00*b01^2+12*b1m1*b00^2*b01^2
i[7]=4*a00*b1m1^2*a12^2+4*b1m1^2*b00*a12^2+8*am11^2*b1m1*a12*b21-24*a00*am11*b00*a12*b21+8*am11*b00^2*a12*b21-4*a00*am11*b1m1*a12*b10+12*a00^2*b00*a12*b10+4*am11*b1m1*b00*a12*b10+8*a00*b00^2*a12*b10-4*b00^3*a12*b10-4*am11^3*b21*b10+a00*am11^2*b10^2+5*am11^2*b00*b10^2-8*a00^2*b1m1*a12*b01-8*am11*b1m1^2*a12*b01-16*a00*b1m1*b00*a12*b01-8*b1m1*b00^2*a12*b01+16*a00*am11^2*b21*b01-8*a00^2*am11*b10*b01-12*am11^2*b1m1*b10*b01-4*a00*am11*b00*b10*b01-12*am11*b00^2*b10*b01+4*a00^3*b01^2+32*a00*am11*b1m1*b01^2-12*a00^2*b00*b01^2+16*am11*b1m1*b00*b01^2-4*a00*b00^2*b01^2+12*b00^3*b01^2
i[8]=8*am11^2*b1m1*a12*b21^2-32*a00*am11*b00*a12*b21^2+16*a00^2*b00*a12*b21*b10+8*am11*b1m1*b00*a12*b21*b10+16*a00*b00^2*a12*b21*b10-4*am11^3*b21^2*b10-2*a00^2*b1m1*a12*b10^2-8*a00*b1m1*b00*a12*b10^2-6*b1m1*b00^2*a12*b10^2+4*am11^2*b00*b21*b10^2+a00^2*am11*b10^3-am11*b00^2*b10^3-32*a00*b1m1*b00*a12*b21*b01+24*a00*am11^2*b21^2*b01+8*am11^2*b00*b21^2*b01+8*a00*b1m1^2*a12*b10*b01+16*b1m1^2*b00*a12*b10*b01-16*a00^2*am11*b21*b10*b01-16*am11^2*b1m1*b21*b10*b01-8*a00*am11*b00*b21*b10*b01-8*am11*b00^2*b21*b10*b01+2*a00^3*b10^2*b01+4*a00*am11*b1m1*b10^2*b01-2*a00^2*b00*b10^2*b01+8*am11*b1m1*b00*b10^2*b01-2*a00*b00^2*b10^2*b01+2*b00^3*b10^2*b01-8*b1m1^3*a12*b01^2+32*a00*am11*b1m1*b21*b01^2-8*a00^2*b1m1*b10*b01^2-12*am11*b1m1^2*b10*b01^2+8*a00*b1m1*b00*b10*b01^2+8*a00*b1m1^2*b01^3-8*b1m1^2*b00*b01^3
i[9]=8*a00*am11*b1m1*a12*b21^2+4*am11^3*b21^3-16*a00*b1m1*b00*a12*b21*b10-4*a00*am11^2*b21^2*b10-4*am11^2*b00*b21^2*b10+2*a00*b1m1^2*a12*b10^2+6*b1m1^2*b00*a12*b10^2+3*a00^2*am11*b21*b10^2+4*am11^2*b1m1*b21*b10^2-4*a00*am11*b00*b21*b10^2+am11*b00^2*b21*b10^2-a00*am11*b1m1*b10^3+am11*b1m1*b00*b10^3+8*a00*b1m1^2*a12*b21*b01-8*a00^2*am11*b21^2*b01-12*am11^2*b1m1*b21^2*b01+24*a00*am11*b00*b21^2*b01-8*b1m1^3*a12*b10*b01-4*am11*b1m1*b00*b21*b10*b01+a00^2*b1m1*b10^2*b01-b1m1*b00^2*b10^2*b01-8*a00^2*b1m1*b21*b01^2-4*am11*b1m1^2*b21*b01^2+24*a00*b1m1*b00*b21*b01^2-4*a00*b1m1^2*b10*b01^2-8*b1m1^2*b00*b10*b01^2+12*b1m1^3*b01^3
i[10]=32*a00^2*am11*b00*a12*b21^2+4*am11^4*b21^3-16*a00^3*b00*a12*b21*b10-24*a00*am11*b1m1*b00*a12*b21*b10-16*a00^2*b00^2*a12*b21*b10-4*am11^3*b00*b21^2*b10+2*a00^3*b1m1*a12*b10^2+2*a00*am11*b1m1^2*a12*b10^2+8*a00^2*b1m1*b00*a12*b10^2+6*am11*b1m1^2*b00*a12*b10^2+6*a00*b1m1*b00^2*a12*b10^2+3*a00^2*am11^2*b21*b10^2+4*am11^3*b1m1*b21*b10^2-8*a00*am11^2*b00*b21*b10^2+am11^2*b00^2*b21*b10^2-a00^3*am11*b10^3-a00*am11^2*b1m1*b10^3+am11^2*b1m1*b00*b10^3+a00*am11*b00^2*b10^3+8*a00*am11*b1m1^2*a12*b21*b01-32*am11*b1m1^2*b00*a12*b21*b01+96*a00*b1m1*b00^2*a12*b21*b01-32*a00^2*am11^2*b21^2*b01-12*am11^3*b1m1*b21^2*b01-16*am11^2*b00^2*b21^2*b01-8*a00^2*b1m1^2*a12*b10*b01-8*am11*b1m1^3*a12*b10*b01-16*a00*b1m1^2*b00*a12*b10*b01-32*b1m1^2*b00^2*a12*b10*b01+16*a00^3*am11*b21*b10*b01+16*a00*am11^2*b1m1*b21*b10*b01+24*a00^2*am11*b00*b21*b10*b01+12*am11^2*b1m1*b00*b21*b10*b01+40*a00*am11*b00^2*b21*b10*b01+16*am11*b00^3*b21*b10*b01-2*a00^4*b10^2*b01-3*a00^2*am11*b1m1*b10^2*b01-10*a00^3*b00*b10^2*b01-24*a00*am11*b1m1*b00*b10^2*b01+6*a00^2*b00^2*b10^2*b01-33*am11*b1m1*b00^2*b10^2*b01+10*a00*b00^3*b10^2*b01-4*b00^4*b10^2*b01+8*a00*b1m1^3*a12*b01^2+32*b1m1^3*b00*a12*b01^2-40*a00^2*am11*b1m1*b21*b01^2-4*am11^2*b1m1^2*b21*b01^2+32*a00^3*b00*b21*b01^2-8*a00*am11*b1m1*b00*b21*b01^2-64*a00^2*b00^2*b21*b01^2+32*am11*b1m1*b00^2*b21*b01^2-96*a00*b00^3*b21*b01^2+8*a00^3*b1m1*b10*b01^2+8*a00*am11*b1m1^2*b10*b01^2+40*a00^2*b1m1*b00*b10*b01^2+40*am11*b1m1^2*b00*b10*b01^2+16*a00*b1m1*b00^2*b10*b01^2+32*b1m1*b00^3*b10*b01^2-8*a00^2*b1m1^2*b01^3+12*am11*b1m1^3*b01^3-72*a00*b1m1^2*b00*b01^3-16*b1m1^2*b00^2*b01^3
i[11]=4*am11^4*b1m1*b21^3-16*a00*am11^3*b00*b21^3-8*a00*am11*b1m1^2*b00*a12*b21*b10+24*a00^2*am11^2*b00*b21^2*b10-4*am11^3*b1m1*b00*b21^2*b10+24*a00*am11^2*b00^2*b21^2*b10+2*a00^3*b1m1^2*a12*b10^2+2*a00*am11*b1m1^3*a12*b10^2+6*am11*b1m1^3*b00*a12*b10^2-2*a00*b1m1^2*b00^2*a12*b10^2+3*a00^2*am11^2*b1m1*b21*b10^2+4*am11^3*b1m1^2*b21*b10^2-20*a00^3*am11*b00*b21*b10^2-32*a00*am11^2*b1m1*b00*b21*b10^2+am11^2*b1m1*b00^2*b21*b10^2-12*a00*am11*b00^3*b21*b10^2-a00^3*am11*b1m1*b10^3-a00*am11^2*b1m1^2*b10^3+6*a00^4*b00*b10^3+12*a00^2*am11*b1m1*b00*b10^3+am11^2*b1m1^2*b00*b10^3-2*a00^3*b00^2*b10^3+13*a00*am11*b1m1*b00^2*b10^3-6*a00^2*b00^3*b10^3+2*a00*b00^4*b10^3+8*a00*am11*b1m1^3*a12*b21*b01-32*a00^2*am11^2*b1m1*b21^2*b01-12*am11^3*b1m1^2*b21^2*b01+32*a00^3*am11*b00*b21^2*b01+64*a00*am11^2*b1m1*b00*b21^2*b01-96*a00^2*am11*b00^2*b21^2*b01-8*a00^2*b1m1^3*a12*b10*b01-8*am11*b1m1^4*a12*b10*b01+16*a00^3*am11*b1m1*b21*b10*b01+16*a00*am11^2*b1m1^2*b21*b10*b01-16*a00^4*b00*b21*b10*b01+24*a00^2*am11*b1m1*b00*b21*b10*b01-4*am11^2*b1m1^2*b00*b21*b10*b01+32*a00^3*b00^2*b21*b10*b01+8*a00*am11*b1m1*b00^2*b21*b10*b01+48*a00^2*b00^3*b21*b10*b01-2*a00^4*b1m1*b10^2*b01-3*a00^2*am11*b1m1^2*b10^2*b01-26*a00^3*b1m1*b00*b10^2*b01-32*a00*am11*b1m1^2*b00*b10^2*b01-6*a00^2*b1m1*b00^2*b10^2*b01-am11*b1m1^2*b00^2*b10^2*b01-14*a00*b1m1*b00^3*b10^2*b01+8*a00*b1m1^4*a12*b01^2-40*a00^2*am11*b1m1^2*b21*b01^2-4*am11^2*b1m1^3*b21*b01^2+32*a00^3*b1m1*b00*b21*b01^2+40*a00*am11*b1m1^2*b00*b21*b01^2-96*a00^2*b1m1*b00^2*b21*b01^2+8*a00^3*b1m1^2*b10*b01^2+8*a00*am11*b1m1^3*b10*b01^2+48*a00^2*b1m1^2*b00*b10*b01^2-8*am11*b1m1^3*b00*b10*b01^2+40*a00*b1m1^2*b00^2*b10*b01^2-8*a00^2*b1m1^3*b01^3+12*am11*b1m1^4*b01^3-40*a00*b1m1^3*b00*b01^3

$Bye.

/////////////////////////////////
Singular input file for the decomposition if I_3 to minimal associated primes. Input ideal is the ideal i above.

LIB"primdec.lib";
LIB "elim.lib";

ring r=0,( a00, am11, b1m1, b00, a10, a01, a12, b21, b10, b01,alpha,beta,gamma,delta,rho,sigma,w,w1,w2,eps ),dp;
r; 

option(redSB);
 
ideal i= a01-2*b01,
2*a10-b10
,4*b1m1^2*a12^2*b21+8*am11*b00*a12*b21^2-4*am11*b1m1*a12*b21*b10-4*a00*b00*a12*b21*b10-4*b00^2*a12*b21*b10+2*a00*b1m1*a12*b10^2+6*b1m1*b00*a12*b10^2+am11^2*b21*b10^2-a00*am11*b10^3+am11*b00*b10^3-8*b1m1*b00*a12*b21*b01-4*am11^2*b21^2*b01-8*b1m1^2*a12*b10*b01+4*a00*am11*b21*b10*b01-8*am11*b00*b21*b10*b01+a00^2*b10^2*b01-b00^2*b10^2*b01-4*a00^2*b21*b01^2+8*am11*b1m1*b21*b01^2+8*a00*b00*b21*b01^2+12*b00^2*b21*b01^2-4*a00*b1m1*b10*b01^2-8*b1m1*b00*b10*b01^2+12*b1m1^2*b01^3
,16*a00*b1m1*a12^2*b21+8*am11^2*a12*b21^2-4*b1m1^2*a12^2*b10-8*a00*am11*a12*b21*b10+6*a00^2*a12*b10^2+4*am11*b1m1*a12*b10^2+4*a00*b00*a12*b10^2-2*b00^2*a12*b10^2+3*am11^2*b10^3-16*a00^2*a12*b21*b01-16*am11*b1m1*a12*b21*b01-16*a00*b00*a12*b21*b01-16*a00*b1m1*a12*b10*b01+8*b1m1*b00*a12*b10*b01-16*am11^2*b21*b10*b01-8*a00*am11*b10^2*b01-4*am11*b00*b10^2*b01+8*b1m1^2*a12*b01^2+48*a00*am11*b21*b01^2+16*am11*b00*b21*b01^2-4*a00^2*b10*b01^2+4*b00^2*b10*b01^2+16*a00*b1m1*b01^3-16*b1m1*b00*b01^3
,8*a00^2*b1m1*a12*b21+8*am11*b1m1^2*a12*b21-24*a00*b1m1*b00*a12*b21+4*a00*am11^2*b21^2+4*am11^2*b00*b21^2+8*b1m1^2*b00*a12*b10-4*a00^2*am11*b21*b10-4*am11^2*b1m1*b21*b10-8*a00*am11*b00*b21*b10-4*am11*b00^2*b21*b10+3*a00^3*b10^2+4*a00*am11*b1m1*b10^2-a00^2*b00*b10^2+8*am11*b1m1*b00*b10^2-3*a00*b00^2*b10^2+b00^3*b10^2-8*b1m1^3*a12*b01-8*a00^3*b21*b01+8*a00*am11*b1m1*b21*b01+16*a00^2*b00*b21*b01-8*am11*b1m1*b00*b21*b01+24*a00*b00^2*b21*b01-12*a00^2*b1m1*b10*b01-12*am11*b1m1^2*b10*b01-4*a00*b1m1*b00*b10*b01-8*b1m1*b00^2*b10*b01+20*a00*b1m1^2*b01^2+4*b1m1^2*b00*b01^2
,4*b1m1^3*a12^2-8*a00*am11*b1m1*a12*b21+8*am11*b1m1*b00*a12*b21-4*am11^3*b21^2-4*am11*b1m1^2*a12*b10+12*a00*b1m1*b00*a12*b10-4*b1m1*b00^2*a12*b10+4*a00*am11^2*b21*b10+4*am11^2*b00*b21*b10-3*a00^2*am11*b10^2-3*am11^2*b1m1*b10^2+4*a00*am11*b00*b10^2-am11*b00^2*b10^2-8*a00*b1m1^2*a12*b01-8*b1m1^2*b00*a12*b01+8*a00^2*am11*b21*b01+8*am11^2*b1m1*b21*b01-24*a00*am11*b00*b21*b01+4*a00*am11*b1m1*b10*b01-4*am11*b1m1*b00*b10*b01+4*a00^2*b1m1*b01^2+12*am11*b1m1^2*b01^2-16*a00*b1m1*b00*b01^2+12*b1m1*b00^2*b01^2
,4*a00*b1m1^2*a12^2+4*b1m1^2*b00*a12^2+8*am11^2*b1m1*a12*b21-24*a00*am11*b00*a12*b21+8*am11*b00^2*a12*b21-4*a00*am11*b1m1*a12*b10+12*a00^2*b00*a12*b10+4*am11*b1m1*b00*a12*b10+8*a00*b00^2*a12*b10-4*b00^3*a12*b10-4*am11^3*b21*b10+a00*am11^2*b10^2+5*am11^2*b00*b10^2-8*a00^2*b1m1*a12*b01-8*am11*b1m1^2*a12*b01-16*a00*b1m1*b00*a12*b01-8*b1m1*b00^2*a12*b01+16*a00*am11^2*b21*b01-8*a00^2*am11*b10*b01-12*am11^2*b1m1*b10*b01-4*a00*am11*b00*b10*b01-12*am11*b00^2*b10*b01+4*a00^3*b01^2+32*a00*am11*b1m1*b01^2-12*a00^2*b00*b01^2+16*am11*b1m1*b00*b01^2-4*a00*b00^2*b01^2+12*b00^3*b01^2
,8*am11^2*b1m1*a12*b21^2-32*a00*am11*b00*a12*b21^2+16*a00^2*b00*a12*b21*b10+8*am11*b1m1*b00*a12*b21*b10+16*a00*b00^2*a12*b21*b10-4*am11^3*b21^2*b10-2*a00^2*b1m1*a12*b10^2-8*a00*b1m1*b00*a12*b10^2-6*b1m1*b00^2*a12*b10^2+4*am11^2*b00*b21*b10^2+a00^2*am11*b10^3-am11*b00^2*b10^3-32*a00*b1m1*b00*a12*b21*b01+24*a00*am11^2*b21^2*b01+8*am11^2*b00*b21^2*b01+8*a00*b1m1^2*a12*b10*b01+16*b1m1^2*b00*a12*b10*b01-16*a00^2*am11*b21*b10*b01-16*am11^2*b1m1*b21*b10*b01-8*a00*am11*b00*b21*b10*b01-8*am11*b00^2*b21*b10*b01+2*a00^3*b10^2*b01+4*a00*am11*b1m1*b10^2*b01-2*a00^2*b00*b10^2*b01+8*am11*b1m1*b00*b10^2*b01-2*a00*b00^2*b10^2*b01+2*b00^3*b10^2*b01-8*b1m1^3*a12*b01^2+32*a00*am11*b1m1*b21*b01^2-8*a00^2*b1m1*b10*b01^2-12*am11*b1m1^2*b10*b01^2+8*a00*b1m1*b00*b10*b01^2+8*a00*b1m1^2*b01^3-8*b1m1^2*b00*b01^3
,8*a00*am11*b1m1*a12*b21^2+4*am11^3*b21^3-16*a00*b1m1*b00*a12*b21*b10-4*a00*am11^2*b21^2*b10-4*am11^2*b00*b21^2*b10+2*a00*b1m1^2*a12*b10^2+6*b1m1^2*b00*a12*b10^2+3*a00^2*am11*b21*b10^2+4*am11^2*b1m1*b21*b10^2-4*a00*am11*b00*b21*b10^2+am11*b00^2*b21*b10^2-a00*am11*b1m1*b10^3+am11*b1m1*b00*b10^3+8*a00*b1m1^2*a12*b21*b01-8*a00^2*am11*b21^2*b01-12*am11^2*b1m1*b21^2*b01+24*a00*am11*b00*b21^2*b01-8*b1m1^3*a12*b10*b01-4*am11*b1m1*b00*b21*b10*b01+a00^2*b1m1*b10^2*b01-b1m1*b00^2*b10^2*b01-8*a00^2*b1m1*b21*b01^2-4*am11*b1m1^2*b21*b01^2+24*a00*b1m1*b00*b21*b01^2-4*a00*b1m1^2*b10*b01^2-8*b1m1^2*b00*b10*b01^2+12*b1m1^3*b01^3
,32*a00^2*am11*b00*a12*b21^2+4*am11^4*b21^3-16*a00^3*b00*a12*b21*b10-24*a00*am11*b1m1*b00*a12*b21*b10-16*a00^2*b00^2*a12*b21*b10-4*am11^3*b00*b21^2*b10+2*a00^3*b1m1*a12*b10^2+2*a00*am11*b1m1^2*a12*b10^2+8*a00^2*b1m1*b00*a12*b10^2+6*am11*b1m1^2*b00*a12*b10^2+6*a00*b1m1*b00^2*a12*b10^2+3*a00^2*am11^2*b21*b10^2+4*am11^3*b1m1*b21*b10^2-8*a00*am11^2*b00*b21*b10^2+am11^2*b00^2*b21*b10^2-a00^3*am11*b10^3-a00*am11^2*b1m1*b10^3+am11^2*b1m1*b00*b10^3+a00*am11*b00^2*b10^3+8*a00*am11*b1m1^2*a12*b21*b01-32*am11*b1m1^2*b00*a12*b21*b01+96*a00*b1m1*b00^2*a12*b21*b01-32*a00^2*am11^2*b21^2*b01-12*am11^3*b1m1*b21^2*b01-16*am11^2*b00^2*b21^2*b01-8*a00^2*b1m1^2*a12*b10*b01-8*am11*b1m1^3*a12*b10*b01-16*a00*b1m1^2*b00*a12*b10*b01-32*b1m1^2*b00^2*a12*b10*b01+16*a00^3*am11*b21*b10*b01+16*a00*am11^2*b1m1*b21*b10*b01+24*a00^2*am11*b00*b21*b10*b01+12*am11^2*b1m1*b00*b21*b10*b01+40*a00*am11*b00^2*b21*b10*b01+16*am11*b00^3*b21*b10*b01-2*a00^4*b10^2*b01-3*a00^2*am11*b1m1*b10^2*b01-10*a00^3*b00*b10^2*b01-24*a00*am11*b1m1*b00*b10^2*b01+6*a00^2*b00^2*b10^2*b01-33*am11*b1m1*b00^2*b10^2*b01+10*a00*b00^3*b10^2*b01-4*b00^4*b10^2*b01+8*a00*b1m1^3*a12*b01^2+32*b1m1^3*b00*a12*b01^2-40*a00^2*am11*b1m1*b21*b01^2-4*am11^2*b1m1^2*b21*b01^2+32*a00^3*b00*b21*b01^2-8*a00*am11*b1m1*b00*b21*b01^2-64*a00^2*b00^2*b21*b01^2+32*am11*b1m1*b00^2*b21*b01^2-96*a00*b00^3*b21*b01^2+8*a00^3*b1m1*b10*b01^2+8*a00*am11*b1m1^2*b10*b01^2+40*a00^2*b1m1*b00*b10*b01^2+40*am11*b1m1^2*b00*b10*b01^2+16*a00*b1m1*b00^2*b10*b01^2+32*b1m1*b00^3*b10*b01^2-8*a00^2*b1m1^2*b01^3+12*am11*b1m1^3*b01^3-72*a00*b1m1^2*b00*b01^3-16*b1m1^2*b00^2*b01^3
,4*am11^4*b1m1*b21^3-16*a00*am11^3*b00*b21^3-8*a00*am11*b1m1^2*b00*a12*b21*b10+24*a00^2*am11^2*b00*b21^2*b10-4*am11^3*b1m1*b00*b21^2*b10+24*a00*am11^2*b00^2*b21^2*b10+2*a00^3*b1m1^2*a12*b10^2+2*a00*am11*b1m1^3*a12*b10^2+6*am11*b1m1^3*b00*a12*b10^2-2*a00*b1m1^2*b00^2*a12*b10^2+3*a00^2*am11^2*b1m1*b21*b10^2+4*am11^3*b1m1^2*b21*b10^2-20*a00^3*am11*b00*b21*b10^2-32*a00*am11^2*b1m1*b00*b21*b10^2+am11^2*b1m1*b00^2*b21*b10^2-12*a00*am11*b00^3*b21*b10^2-a00^3*am11*b1m1*b10^3-a00*am11^2*b1m1^2*b10^3+6*a00^4*b00*b10^3+12*a00^2*am11*b1m1*b00*b10^3+am11^2*b1m1^2*b00*b10^3-2*a00^3*b00^2*b10^3+13*a00*am11*b1m1*b00^2*b10^3-6*a00^2*b00^3*b10^3+2*a00*b00^4*b10^3+8*a00*am11*b1m1^3*a12*b21*b01-32*a00^2*am11^2*b1m1*b21^2*b01-12*am11^3*b1m1^2*b21^2*b01+32*a00^3*am11*b00*b21^2*b01+64*a00*am11^2*b1m1*b00*b21^2*b01-96*a00^2*am11*b00^2*b21^2*b01-8*a00^2*b1m1^3*a12*b10*b01-8*am11*b1m1^4*a12*b10*b01+16*a00^3*am11*b1m1*b21*b10*b01+16*a00*am11^2*b1m1^2*b21*b10*b01-16*a00^4*b00*b21*b10*b01+24*a00^2*am11*b1m1*b00*b21*b10*b01-4*am11^2*b1m1^2*b00*b21*b10*b01+32*a00^3*b00^2*b21*b10*b01+8*a00*am11*b1m1*b00^2*b21*b10*b01+48*a00^2*b00^3*b21*b10*b01-2*a00^4*b1m1*b10^2*b01-3*a00^2*am11*b1m1^2*b10^2*b01-26*a00^3*b1m1*b00*b10^2*b01-32*a00*am11*b1m1^2*b00*b10^2*b01-6*a00^2*b1m1*b00^2*b10^2*b01-am11*b1m1^2*b00^2*b10^2*b01-14*a00*b1m1*b00^3*b10^2*b01+8*a00*b1m1^4*a12*b01^2-40*a00^2*am11*b1m1^2*b21*b01^2-4*am11^2*b1m1^3*b21*b01^2+32*a00^3*b1m1*b00*b21*b01^2+40*a00*am11*b1m1^2*b00*b21*b01^2-96*a00^2*b1m1*b00^2*b21*b01^2+8*a00^3*b1m1^2*b10*b01^2+8*a00*am11*b1m1^3*b10*b01^2+48*a00^2*b1m1^2*b00*b10*b01^2-8*am11*b1m1^3*b00*b10*b01^2+40*a00*b1m1^2*b00^2*b10*b01^2-8*a00^2*b1m1^3*b01^3+12*am11*b1m1^4*b01^3-40*a00*b1m1^3*b00*b01^3
;

list l=minAssGTZ(i);

list l1;
 
int k;
for(k=1;k<=size(l);k++)
{ l1[k]=groebner(l[k]); print(k); print(dim(l1[k])); print(l1[k]); }

l1;

$;
/////////////////////////////////
Output; The first component with 4 polynomials is the ideal I_{31}, and the second one, having 7 polynomials is the ideal I_{32}. 

                    SINGULAR                                 /
 A Computer Algebra System for Polynomial Computations       /   version 4.0.1
                                                           0<
 by: W. Decker, G.-M. Greuel, G. Pfister, H. Schoenemann     \   Sep 2014
FB Mathematik der Universitaet, D-67653 Kaiserslautern        \
// ** loaded /opt/Singular-4-0-1/bin/../share/singular/LIB/primdec.lib (4.0.0.0,Jun_2013)
// ** loaded /opt/Singular-4-0-1/bin/../share/singular/LIB/ring.lib (4.0.0.0,Jun_2013)
// ** loaded /opt/Singular-4-0-1/bin/../share/singular/LIB/absfact.lib (4.0.0.0,Jun_2013)
// ** loaded /opt/Singular-4-0-1/bin/../share/singular/LIB/triang.lib (4.0.0.0,Jun_2013)
// ** loaded /opt/Singular-4-0-1/bin/../share/singular/LIB/matrix.lib (4.0.0.0,Jun_2013)
// ** loaded /opt/Singular-4-0-1/bin/../share/singular/LIB/nctools.lib (4.0.0.0,Jun_2013)
// ** loaded /opt/Singular-4-0-1/bin/../share/singular/LIB/inout.lib (4.0.0.0,Jun_2013)
// ** loaded /opt/Singular-4-0-1/bin/../share/singular/LIB/random.lib (4.0.0.0,Jun_2013)
// ** loaded /opt/Singular-4-0-1/bin/../share/singular/LIB/poly.lib (4.0.0.0,Jun_2013)
// ** loaded /opt/Singular-4-0-1/bin/../share/singular/LIB/elim.lib (4.0.0.1,Jan_2014)
// ** loaded /opt/Singular-4-0-1/bin/../share/singular/LIB/general.lib (4.0.0.1,Jan_2014)
// ** redefining blowup0 **
// ** redefining blowup0 **
// ** redefining elimRing **
// ** redefining elimRing **
// ** redefining elim **
// ** redefining elim **
// ** redefining elim2 **
// ** redefining elim2 **
// ** redefining elim1 **
// ** redefining elim1 **
// ** redefining nselect **
// ** redefining nselect **
// ** redefining sat **
// ** redefining sat **
// ** redefining select **
// ** redefining select **
// ** redefining select1 **
// ** redefining select1 **
// ** loaded /opt/Singular-4-0-1/bin/../share/singular/LIB/elim.lib (4.0.0.1,Jan_2014)
//   characteristic : 0
//   number of vars : 20
//        block   1 : ordering dp
//                  : names    a00 am11 b1m1 b00 a10 a01 a12 b21 b10 b01 alpha beta gamma delta rho sigma w w1 w2 eps
//        block   2 : ordering C
1
16
a01-2*b01,
2*a10-b10,
2*am11*b21-a00*b10-b00*b10+2*b1m1*b01,
2*b1m1*a12+am11*b10-2*a00*b01-2*b00*b01
2
16
a01-2*b01,
2*a10-b10,
4*am11*b1m1*a12*b21-16*a00*b00*a12*b21+2*a00*b1m1*a12*b10+6*b1m1*b00*a12*b10-2*am11^2*b21*b10-a00*am11*b10^2+am11*b00*b10^2-4*b1m1^2*a12*b01+12*a00*am11*b21*b01+4*am11*b00*b21*b01-2*a00^2*b10*b01-6*am11*b1m1*b10*b01+4*a00*b00*b10*b01-2*b00^2*b10*b01+4*a00*b1m1*b01^2-4*b1m1*b00*b01^2,
8*a00*b1m1*a12*b21+4*am11^2*b21^2-4*b1m1^2*a12*b10-4*a00*am11*b21*b10-4*am11*b00*b21*b10+3*a00^2*b10^2+6*am11*b1m1*b10^2-4*a00*b00*b10^2+b00^2*b10^2-8*a00^2*b21*b01-16*am11*b1m1*b21*b01+24*a00*b00*b21*b01-8*a00*b1m1*b10*b01-4*b1m1*b00*b10*b01+12*b1m1^2*b01^2,
4*b1m1^2*a12^2+8*am11*b00*a12*b21-8*am11*b1m1*a12*b10+12*a00*b00*a12*b10-4*b00^2*a12*b10+3*am11^2*b10^2-8*a00*b1m1*a12*b01-8*b1m1*b00*a12*b01-8*am11^2*b21*b01-4*a00*am11*b10*b01-8*am11*b00*b10*b01+4*a00^2*b01^2+24*am11*b1m1*b01^2-16*a00*b00*b01^2+12*b00^2*b01^2,
32*a00^2*b00*a12*b21+4*am11^3*b21^2-4*a00^2*b1m1*a12*b10-4*am11*b1m1^2*a12*b10-12*a00*b1m1*b00*a12*b10-4*am11^2*b00*b21*b10+5*a00^2*am11*b10^2+6*am11^2*b1m1*b10^2-6*a00*am11*b00*b10^2+am11*b00^2*b10^2+8*a00*b1m1^2*a12*b01-32*a00^2*am11*b21*b01-16*am11^2*b1m1*b21*b01+16*a00*am11*b00*b21*b01+4*a00^3*b10*b01+4*a00*am11*b1m1*b10*b01-8*a00^2*b00*b10*b01-4*am11*b1m1*b00*b10*b01+4*a00*b00^2*b10*b01-8*a00^2*b1m1*b01^2+12*am11*b1m1^2*b01^2+8*a00*b1m1*b00*b01^2,
4*am11^3*b1m1*b21^2-16*a00*am11^2*b00*b21^2-4*a00^2*b1m1^2*a12*b10-4*am11*b1m1^3*a12*b10+4*a00*b1m1^2*b00*a12*b10+16*a00^2*am11*b00*b21*b10-4*am11^2*b1m1*b00*b21*b10+16*a00*am11*b00^2*b21*b10+5*a00^2*am11*b1m1*b10^2+6*am11^2*b1m1^2*b10^2-12*a00^3*b00*b10^2-30*a00*am11*b1m1*b00*b10^2+16*a00^2*b00^2*b10^2+am11*b1m1*b00^2*b10^2-4*a00*b00^3*b10^2+8*a00*b1m1^3*a12*b01-32*a00^2*am11*b1m1*b21*b01-16*am11^2*b1m1^2*b21*b01+32*a00^3*b00*b21*b01+80*a00*am11*b1m1*b00*b21*b01-96*a00^2*b00^2*b21*b01+4*a00^3*b1m1*b10*b01+4*a00*am11*b1m1^2*b10*b01+24*a00^2*b1m1*b00*b10*b01-4*am11*b1m1^2*b00*b10*b01+20*a00*b1m1*b00^2*b10*b01-8*a00^2*b1m1^2*b01^2+12*am11*b1m1^3*b01^2-40*a00*b1m1^2*b00*b01^2
[1]:
   _[1]=a01-2*b01
   _[2]=2*a10-b10
   _[3]=2*am11*b21-a00*b10-b00*b10+2*b1m1*b01
   _[4]=2*b1m1*a12+am11*b10-2*a00*b01-2*b00*b01
[2]:
   _[1]=a01-2*b01
   _[2]=2*a10-b10
   _[3]=4*am11*b1m1*a12*b21-16*a00*b00*a12*b21+2*a00*b1m1*a12*b10+6*b1m1*b00*a12*b10-2*am11^2*b21*b10-a00*am11*b10^2+am11*b00*b10^2-4*b1m1^2*a12*b01+12*a00*am11*b21*b01+4*am11*b00*b21*b01-2*a00^2*b10*b01-6*am11*b1m1*b10*b01+4*a00*b00*b10*b01-2*b00^2*b10*b01+4*a00*b1m1*b01^2-4*b1m1*b00*b01^2
   _[4]=8*a00*b1m1*a12*b21+4*am11^2*b21^2-4*b1m1^2*a12*b10-4*a00*am11*b21*b10-4*am11*b00*b21*b10+3*a00^2*b10^2+6*am11*b1m1*b10^2-4*a00*b00*b10^2+b00^2*b10^2-8*a00^2*b21*b01-16*am11*b1m1*b21*b01+24*a00*b00*b21*b01-8*a00*b1m1*b10*b01-4*b1m1*b00*b10*b01+12*b1m1^2*b01^2
   _[5]=4*b1m1^2*a12^2+8*am11*b00*a12*b21-8*am11*b1m1*a12*b10+12*a00*b00*a12*b10-4*b00^2*a12*b10+3*am11^2*b10^2-8*a00*b1m1*a12*b01-8*b1m1*b00*a12*b01-8*am11^2*b21*b01-4*a00*am11*b10*b01-8*am11*b00*b10*b01+4*a00^2*b01^2+24*am11*b1m1*b01^2-16*a00*b00*b01^2+12*b00^2*b01^2
   _[6]=32*a00^2*b00*a12*b21+4*am11^3*b21^2-4*a00^2*b1m1*a12*b10-4*am11*b1m1^2*a12*b10-12*a00*b1m1*b00*a12*b10-4*am11^2*b00*b21*b10+5*a00^2*am11*b10^2+6*am11^2*b1m1*b10^2-6*a00*am11*b00*b10^2+am11*b00^2*b10^2+8*a00*b1m1^2*a12*b01-32*a00^2*am11*b21*b01-16*am11^2*b1m1*b21*b01+16*a00*am11*b00*b21*b01+4*a00^3*b10*b01+4*a00*am11*b1m1*b10*b01-8*a00^2*b00*b10*b01-4*am11*b1m1*b00*b10*b01+4*a00*b00^2*b10*b01-8*a00^2*b1m1*b01^2+12*am11*b1m1^2*b01^2+8*a00*b1m1*b00*b01^2
   _[7]=4*am11^3*b1m1*b21^2-16*a00*am11^2*b00*b21^2-4*a00^2*b1m1^2*a12*b10-4*am11*b1m1^3*a12*b10+4*a00*b1m1^2*b00*a12*b10+16*a00^2*am11*b00*b21*b10-4*am11^2*b1m1*b00*b21*b10+16*a00*am11*b00^2*b21*b10+5*a00^2*am11*b1m1*b10^2+6*am11^2*b1m1^2*b10^2-12*a00^3*b00*b10^2-30*a00*am11*b1m1*b00*b10^2+16*a00^2*b00^2*b10^2+am11*b1m1*b00^2*b10^2-4*a00*b00^3*b10^2+8*a00*b1m1^3*a12*b01-32*a00^2*am11*b1m1*b21*b01-16*am11^2*b1m1^2*b21*b01+32*a00^3*b00*b21*b01+80*a00*am11*b1m1*b00*b21*b01-96*a00^2*b00^2*b21*b01+4*a00^3*b1m1*b10*b01+4*a00*am11*b1m1^2*b10*b01+24*a00^2*b1m1*b00*b10*b01-4*am11*b1m1^2*b00*b10*b01+20*a00*b1m1*b00^2*b10*b01-8*a00^2*b1m1^2*b01^2+12*am11*b1m1^3*b01^2-40*a00*b1m1^2*b00*b01^2

$Bye.

/////////////////////////////
Wolfram Mathemtica test
/////////////////////////////

I31={a01-2*b01,2*a10-b10,2*am11*b21-a00*b10-b00*b10+2*b1m1*b01,2*b1m1*a12+am11*b10-2*a00*b01-2*b00*b01};
I32={a01-2*b01,2*a10-b10,4*am11*b1m1*a12*b21-16*a00*b00*a12*b21+2*a00*b1m1*a12*b10+6*b1m1*b00*a12*b10-2*am11^2*b21*b10-a00*am11*b10^2+am11*b00*b10^2-4*b1m1^2*a12*b01+12*a00*am11*b21*b01+4*am11*b00*b21*b01-2*a00^2*b10*b01-6*am11*b1m1*b10*b01+4*a00*b00*b10*b01-2*b00^2*b10*b01+4*a00*b1m1*b01^2-4*b1m1*b00*b01^2,8*a00*b1m1*a12*b21+4*am11^2*b21^2-4*b1m1^2*a12*b10-4*a00*am11*b21*b10-4*am11*b00*b21*b10+3*a00^2*b10^2+6*am11*b1m1*b10^2-4*a00*b00*b10^2+b00^2*b10^2-8*a00^2*b21*b01-16*am11*b1m1*b21*b01+24*a00*b00*b21*b01-8*a00*b1m1*b10*b01-4*b1m1*b00*b10*b01+12*b1m1^2*b01^2,4*b1m1^2*a12^2+8*am11*b00*a12*b21-8*am11*b1m1*a12*b10+12*a00*b00*a12*b10-4*b00^2*a12*b10+3*am11^2*b10^2-8*a00*b1m1*a12*b01-8*b1m1*b00*a12*b01-8*am11^2*b21*b01-4*a00*am11*b10*b01-8*am11*b00*b10*b01+4*a00^2*b01^2+24*am11*b1m1*b01^2-16*a00*b00*b01^2+12*b00^2*b01^2,32*a00^2*b00*a12*b21+4*am11^3*b21^2-4*a00^2*b1m1*a12*b10-4*am11*b1m1^2*a12*b10-12*a00*b1m1*b00*a12*b10-4*am11^2*b00*b21*b10+5*a00^2*am11*b10^2+6*am11^2*b1m1*b10^2-6*a00*am11*b00*b10^2+am11*b00^2*b10^2+8*a00*b1m1^2*a12*b01-32*a00^2*am11*b21*b01-16*am11^2*b1m1*b21*b01+16*a00*am11*b00*b21*b01+4*a00^3*b10*b01+4*a00*am11*b1m1*b10*b01-8*a00^2*b00*b10*b01-4*am11*b1m1*b00*b10*b01+4*a00*b00^2*b10*b01-8*a00^2*b1m1*b01^2+12*am11*b1m1^2*b01^2+8*a00*b1m1*b00*b01^2,4*am11^3*b1m1*b21^2-16*a00*am11^2*b00*b21^2-4*a00^2*b1m1^2*a12*b10-4*am11*b1m1^3*a12*b10+4*a00*b1m1^2*b00*a12*b10+16*a00^2*am11*b00*b21*b10-4*am11^2*b1m1*b00*b21*b10+16*a00*am11*b00^2*b21*b10+5*a00^2*am11*b1m1*b10^2+6*am11^2*b1m1^2*b10^2-12*a00^3*b00*b10^2-30*a00*am11*b1m1*b00*b10^2+16*a00^2*b00^2*b10^2+am11*b1m1*b00^2*b10^2-4*a00*b00^3*b10^2+8*a00*b1m1^3*a12*b01-32*a00^2*am11*b1m1*b21*b01-16*am11^2*b1m1^2*b21*b01+32*a00^3*b00*b21*b01+80*a00*am11*b1m1*b00*b21*b01-96*a00^2*b00^2*b21*b01+4*a00^3*b1m1*b10*b01+4*a00*am11*b1m1^2*b10*b01+24*a00^2*b1m1*b00*b10*b01-4*am11*b1m1^2*b00*b10*b01+20*a00*b1m1*b00^2*b10*b01-8*a00^2*b1m1^2*b01^2+12*am11*b1m1^3*b01^2-40*a00*b1m1^2*b00*b01^2};

(*System (4.1)*)
I31//.{a00->1, am11->0,b1m1->0,b00->-1,
       a10->2,a01->-2,a12->-1,
       b21->-2,b10->4,b01->-1}//Simplify
{0,0,0,0}

(*System (4.2)*)
I31//.{a00->1, am11->0,b1m1->0,b00->-1,
       a10->0,a01->2,a12->0,
       b21->-1,b10->0,b01->1}//Simplify
{0,0,0,0}

Z3test=a12==-a10&&b00==-a00&&b01==a01/2&&b10==2 a10&&b1m1==am11&&b21==-(a01/2);

Z3test//.{a00->1, am11->0,b1m1->0,b00->-1,a10->0,a01->2,a12->0,b21->-1,b10->0,b01->1}//Simplify
True

(*System (4.3)*)
I6={a00,b00,am11,b1m1,a01-2*b01,2*a10-b10};
I6 //. {a00 -> 0, am11 -> 0, b1m1 -> 0, b00 -> 0, 
        a10 -> -1,  a01 -> -1, a12 -> 2, 
        b21 -> 2, b10 -> -2, b01 -> -1/2}//Simplify
{0, 0, 0, 0, 0, 0}


(*System (4.4)*)
Z6test = 
  a00 == 0 && a12 == -a10 && am11 == 0 && b00 == 0 && b01 == a01/2 && 
   b10 == 2 a10 && b1m1 == 0 && b21 == -(a01/2);
Z6test //. {a00 -> 0, am11 -> 0, b1m1 -> 0, b00 -> 0, 
   a10 -> -3 Sqrt[23]/32, a01 -> -23/16, a12 -> 3 Sqrt[23]/32, 
   b21 -> 23/32, b10 -> -3 Sqrt[23]/16, b01 -> -(23/32)} // Simplify

True

(*System (4.5)*)
I32//.{a00->1, am11->0,b1m1->0,b00->1,a10->1,a01->-4,a12->0,b21->0,b10->2,b01->-2}//Simplify

{0,0,0,0,0,0,0}

(*System (4.6)*)
Z3test//.{a00->0,am11->-1/Sqrt[3],b1m1->-1/Sqrt[3],b00->0,
a10->-Sqrt[3],a01->0,a12->Sqrt[3],b21->0,b10->-2Sqrt[3],
b01->0}//Simplify
True

(*System (4.7)*)

I2 = {4*b00*a12*b21 - b00*a01*b10 - 2*b1m1*a12*b10 + 
    am11*b10^2 + 2*b1m1*a01*b01 - 4*am11*b21*b01, 
   2*b00*a01*b21 - 2*b00*a10*b10 - b1m1*a01*b10 + a00*b10^2 + 
    4*b1m1*a10*b01 - 4*a00*b21*b01, 
   b00*a01^2 - 4*b00*a10*a12 - am11*a01*b10 + 2*a00*a12*b10 + 
    4*am11*a10*b01 - 2*a00*a01*b01, 
   b1m1*a01^2 - 4*b1m1*a10*a12 - 2*am11*a01*b21 + 4*a00*a12*b21 + 
    2*am11*a10*b10 - a00*a01*b10};
I2 //. {a00 -> 0, am11 -> 1, b1m1 -> 1, b00 -> 0,
   a10 -> -2, a01 -> 1, a12 -> -25/8, b21 -> 4, b10 -> -8, 
   b01 -> 1} // Simplify

{0, 0, 0, 0}

(*System (4.8)*)

I2rtest = a00 == 0 && am11 == 0 && b1m1 == 0 && b00 == 0;  

(* The system 4.8. clearly satisfies this criteria. *)