% The examples from the paper 
% "On Factorized Groebner Bases"
% H.-G. Graebe | Univ. Leipzig

% =========== Example G_1 =================

A1:=-2*L1**2+L1*L2+2*L1*L3-L2**2-7*L5+21*L6;
A2:=7*L7-2*L1*L4+3/7*L1**3;
B1:=L1*(5*L1-3*L2+L3);
B2:=L1*(2*L6-4*L4);
B3:=L1*L7/2;
P1:=L1*(L4-L5/2+L6);
P2:=(2/7*L1**2-L4)*(-10*L1+5*L2-L3);
P3:=(2/7*L1**2-L4)*(3*L4-L5+L6);
P4:=A1*(-3*L1+2*L2)+21*A2;
P5:=A1*(2*L4-2*L5)+A2*(-45*L1+15*L2-3*L3);
P6:=2*A1*L7+A2*(12*L4-3*L5+2*L6);
P7:=B1*(2*L2-L1)+7*B2;
P8:=B1*L3+7*B2;
P9:=B1*(-2*L4-2*L5)+B2*(2*L2-8*L1)+84*B3;
P10:=B1*(8/3*L5+6*L6)+B2*(11*L1-17/3*L2+5/3*L3)-168*B3;
P11:=15*B1*L7+B2*(5*L4-2*L5)+B3*(-120*L1+30*L2-6*L3);
P12:=-3*B1*L7+B2*(-L4/2+L5/4-L6/2)+B3*(24*L1-6*L2);
P13:=3*B2*L7+B3*(40*L4-8*L5+4*L6);

polys:={P1,P2,P3,P4,P5,P6,P7,P8,P9,P10,P11,P12,P13};
vars:={L7,L6,L5,L4,L3,L2,L1};
% clear a1,a2,b1,b2,b3;

% =========== Example G_6 =================

vars:={t,x,y,z};
polys:={
-2*z^3*t+(3*z^2*t-2*z^2-6*z*y*t+6*z*y+6*y^2*t-6*y^2)*x-z*t*x^2,
18*z^3*t^2-9*z^3*t-18*z^2*y*t^2+18*z^2*y*t+18*z*y^2*t^2-18*z*y^2*t+
        (-27*z^2*t^2+24*z^2*t-5*z^2+63*z*y*t^2-78*z*y*t+15*z*y-63*y^2*t^2+
        78*y^2*t-15*y^2)*x+9*z*t^2*x^2,
-8*z^4*t+(6*z^3*t-6*z^3-12*z^2*y*t+12*z^2*y+12*z*y^2*t-12*z*y^2)*x+
        (5*z^2*t-4*z^2-18*z*y*t+18*z*y+18*y^2*t-18*y^2)*x^2-3*z*t*x^3,
(3t-5)*z^2*y-15*(t-1)*z*y^2+10*(t-1)*y^3+(z*y+3*y^2*t-3*y^2)*x-y*t*x^2};

% =========== Example A_5 =================

Vars:={V,W,X,Y,Z};

m:={V + W + X + Y + Z,
 V*W + V*Z + W*X + X*Y + Y*Z,
 V*W*X + V*W*Z + V*Y*Z + W*X*Y + X*Y*Z,
 V*W*X*Y + V*W*X*Z + V*W*Y*Z + V*X*Y*Z + W*X*Y*Z,
 V*W*X*Y*Z - 1};

% =========== Example K_4 =================

vars:={u0,u1,u2,u3,u4,u5};

m:={u0+2u1+2u2+2u3-1,
 +u0^2+2u1^2+2u2^2+2u3^2-u0,
 +2*u0*u1+2*u1*u2+2*u2*u3-u1,
 +2*u0*u2+u1^2+2*u1*u3-u2};
 
% =========== Example G_7 =================

vars:={x,y,z,t,u,v,w};
m:={
x*(t-1/2u+v),
(2/7x^2-t)*(-10x+5y-z),
(2/7x^2-t)*(3t-u+v),
(-2x^2+x*y+2x*z-y^2-7u+21v)*(-3x+2y)+21*(7w-2x*t+3/7x^3),
(-2x^2+x*y+2x*z-y^2-7u+21v)*(2t-2u)+(7w-2x*t+3/7x^3)*(-45x+15y-3z),
2*(-2x^2+x*y+2x*z-y^2-7u+21v)*w+(7w-2x*t+3/7x^3)*(12t-3u+2v),
(x*(5x-3y+z))*(2y-x)+7*(x*(2v-4t)),
(x*(5x-3y+z))*(-2t-2u)+(x*(2v-4t))*(2y-8x)+42x*w,
(x*(5x-3y+z))*(8/3u+6v)+(x*(2v-4t))*(11x-17/3y+5/3z)-84x*w,
15w*(x*(5x-3y+z))+(x*(2v-4t))*(5t-2u)+1/2x*w*(-120x+30y-6z),
-3*(x+5x-3y+z)*w+(x*(2v-4t))*(-1/2t+1/4u-1/2v)+1/2x*w*(24x-6y),
3*(x*(2v-4t))*w+1/2x*w*(40t-8u+4v)};

% =========== Example Go =================

vars:= {a0,a1,a2,a3,a4,a5,b0,b1,b2,b3,b4,b5,c0,c1,c2,c3,c4,c5};
m:={a4*b4,
a5*b1+a1*b5+a4*b3+a3*b4,
a2*b2,a5*b5,
(a0+a1+a4)*b2+a2*(b0+b1+b4)+c2,
(a0+a1+a4)*(b0+b1+b4)+(a3+a5)*b2+a2*(b3+b5)+c0+c1+c4,
(a3+a5)*(b0+b1+b4)+(b3+b5)*(a0+a1+a4)+c3+c5-a1^2,
(a3+a5)*(b3+b5),
a5*(b3+b5)+b5*(a3+a5),
b5*(a0+a1+2*a4)+a5*(b0+b1+2*b4)+a3*b4+a4*b3+c5,
a4*(b0+b1+2*b4)+a2*b5+a5*b2+(a0+a1)*b4+c4,
a2*b4+a4*b2,
a4*b5+a5*b4,
2*a3*b3+a3*b5+a5*b3,
c3+b3*(a0+2*a1+a4)+a3*(b0+2*b1+b4)+a1*b5+a5*b1,
c1+(a0+2*a1+a4)*b1+a2*b3+a3*b2+a1*(b0+b4),
a2*b1+a1*b2,
a5*b3+a3*b5,
a1*b4+a4*b1};

% =========== Examples S_k =================

operator x;

vars:=for k:=1:8 collect x(k);

s5:={
-1*x(1)^2-1*x(1)-2*x(2)*x(5)-2*x(3)*x(4),
-2*x(1)*x(2)-1*x(2)-2*x(3)*x(5)-1*x(4)^2,
-2*x(1)*x(3)-1*x(2)^2-1*x(3)-2*x(4)*x(5),
-2*x(1)*x(4)-2*x(2)*x(3)-1*x(4)-1*x(5)^2,
-2*x(1)*x(5)-2*x(2)*x(4)-1*x(3)^2-1*x(5)};

s6:={
-1*x(1)^2-1*x(1)-2*x(2)*x(6)-2*x(3)*x(5)-1*x(4)^2,
-2*x(1)*x(2)-1*x(2)-2*x(3)*x(6)-2*x(4)*x(5),
-2*x(1)*x(3)-1*x(2)^2-1*x(3)-2*x(4)*x(6)-1*x(5)^2,
-2*x(1)*x(4)-2*x(2)*x(3)-1*x(4)-2*x(5)*x(6),
-2*x(1)*x(5)-2*x(2)*x(4)-1*x(3)^2-1*x(5)-1*x(6)^2,
-2*x(1)*x(6)-2*x(2)*x(5)-2*x(3)*x(4)-1*x(6)};


s7:={
-1*x(1)^2-2*x(4)*x(5)-2*x(3)*x(6)-2*x(2)*x(7)-1*x(1),
-2*x(1)*x(2)-1*x(5)^2-2*x(4)*x(6)-2*x(3)*x(7)-1*x(2),
-1*x(2)^2-2*x(1)*x(3)-2*x(5)*x(6)-2*x(4)*x(7)-1*x(3),
-2*x(2)*x(3)-2*x(1)*x(4)-1*x(6)^2-2*x(5)*x(7)-1*x(4),
-1*x(3)^2-2*x(2)*x(4)-2*x(1)*x(5)-2*x(6)*x(7)-1*x(5),
-2*x(3)*x(4)-2*x(2)*x(5)-2*x(1)*x(6)-1*x(7)^2-1*x(6),
-1*x(4)^2-2*x(3)*x(5)-2*x(2)*x(6)-2*x(1)*x(7)-1*x(7)};


s8:={
-1*x(1)^2-1*x(1)-2*x(2)*x(8)-2*x(3)*x(7)-2*x(4)*x(6)-1*x(5)^2,
-2*x(1)*x(2)-1*x(2)-2*x(3)*x(8)-2*x(4)*x(7)-2*x(5)*x(6),
-2*x(1)*x(3)-1*x(2)^2-1*x(3)-2*x(4)*x(8)-2*x(5)*x(7)-1*x(6)^2,
-2*x(1)*x(4)-2*x(2)*x(3)-1*x(4)-2*x(5)*x(8)-2*x(6)*x(7),
-2*x(1)*x(5)-2*x(2)*x(4)-1*x(3)^2-1*x(5)-2*x(6)*x(8)-1*x(7)^2,
-2*x(1)*x(6)-2*x(2)*x(5)-2*x(3)*x(4)-1*x(6)-2*x(7)*x(8),
-2*x(1)*x(7)-2*x(2)*x(6)-2*x(3)*x(5)-1*x(4)^2-1*x(7)-1*x(8)^2,
-2*x(1)*x(8)-2*x(2)*x(7)-2*x(3)*x(6)-2*x(4)*x(5)-1*x(8)};