

<p>We define the 6-dimensional affine space A6 with coordianates y, yb= \bar{y}, z, zb=\bar{z}, alpha et beta.</p>
<p>The aim is to compute the radical of the ideal generated by the equations in Theorem 1.2 (3)</p>

{{{id=81|
A6.<y,yb,z,zb,alpha,beta> = AffineSpace(6,QQ); A6;
///
Affine Space of dimension 6 over Rational Field
}}}

{{{id=121|
S = A6.coordinate_ring(); S;
///
Multivariate Polynomial Ring in y, yb, z, zb, alpha, beta over Rational Field
}}}

<p>Here are the equations of Theorem 1.2 (3)</p>

{{{id=122|
P1 = y*yb-(alpha+1)*(beta+1); 
P2 = y^3+yb^3 - (alpha^2*beta+alpha*beta^2+6*alpha*beta+3*alpha+3*beta+2);
P3 = z^3 +zb^3 -(alpha^4*beta^2+10*alpha^2*beta+9*alpha^2-2*alpha^3-2);
P4 = z*zb-(2*alpha^2*beta+alpha^2+1);
P5 = y*z+yb*zb-(alpha^2*beta+3*alpha*beta+3*alpha+1);
P6 = yb^2*z + y^2*zb - (1 + 2*alpha + 2*beta + 4*alpha*beta + alpha^2 *(2 + 4*beta + beta^2) );
P7 = yb*z^2 +  y*zb^2 - (-1 + alpha^3*beta*(1 + beta) + alpha^2*(3 + 4*beta) + alpha*(3 + 5*beta));
///
}}}

{{{id=124|
I0 = ideal(P1,P2,P3,P4,P5,P6,P7); I0;
///
Ideal (y*yb - alpha*beta - alpha - beta - 1, y^3 + yb^3 - alpha^2*beta - alpha*beta^2 - 6*alpha*beta - 3*alpha - 3*beta - 2, -alpha^4*beta^2 + z^3 + zb^3 + 2*alpha^3 - 10*alpha^2*beta - 9*alpha^2 + 2, -2*alpha^2*beta + z*zb - alpha^2 - 1, -alpha^2*beta + y*z + yb*zb - 3*alpha*beta - 3*alpha - 1, -alpha^2*beta^2 + yb^2*z + y^2*zb - 4*alpha^2*beta - 2*alpha^2 - 4*alpha*beta - 2*alpha - 2*beta - 1, -alpha^3*beta^2 - alpha^3*beta + yb*z^2 + y*zb^2 - 4*alpha^2*beta - 3*alpha^2 - 5*alpha*beta - 3*alpha + 1) of Multivariate Polynomial Ring in y, yb, z, zb, alpha, beta over Rational Field
}}}

<p>The ideal I0 generated by the seven polynomials is not reduced. We have I0 \neq rad(I0).</p>

{{{id=169|
I0 == I0.radical();
///
False
}}}

{{{id=125|
radI0 = I0.radical();
///
}}}

<p>Here are the 18 generators of the ideal rad(I0):</p>

{{{id=170|
for i in radI0.gens():
    print '  ',i;
///
   2*y*z + 2*yb*zb - z*zb + alpha^2 - 6*alpha*beta - 6*alpha - 1
   y*yb - alpha*beta - alpha - beta - 1
   2*alpha^2*beta - z*zb + alpha^2 + 1
   yb*alpha*beta + y^2 - y*zb + yb*alpha - z*beta - z
   y*alpha*beta + yb^2 - yb*z + y*alpha - zb*beta - zb
   y*zb*alpha - yb*alpha^2 - zb^2 + 3*yb*alpha - z*alpha - z
   yb*z*alpha - y*alpha^2 - z^2 + 3*y*alpha - zb*alpha - zb
   yb^2*alpha - yb*z + y*alpha - zb*alpha + y - zb
   y^2*alpha - y*zb + yb*alpha - z*alpha + yb - z
   2*yb*zb^2 - z*zb^2 + 2*y*alpha^2 + zb*alpha^2 - 2*zb*alpha*beta - 4*yb*z + 4*z^2 - 8*y*alpha - 2*zb*alpha + 6*y - zb
   yb*z*zb - yb*alpha^2 - 2*z*alpha*beta + 2*y*zb - 2*zb^2 + 4*yb*alpha - 2*z*alpha - 3*yb
   yb^2*zb + 3*y^2 - 2*y*zb - zb^2 + 2*yb*alpha - 2*z*beta - 2*yb - 2*z
   4*yb^2*z + 4*y^2*zb - 2*z*zb*beta - 7*z*zb - alpha^2 - 16*alpha*beta - 8*alpha - 6*beta + 3
   yb^3 + y^2*zb - yb*zb*beta + alpha*beta^2 - yb*zb - z*zb - 2*alpha*beta - alpha - 2*beta
   2*y^3 - 2*y^2*zb + 2*yb*zb*beta - 4*alpha*beta^2 + 2*yb*zb + z*zb + alpha^2 - 8*alpha*beta - 4*alpha - 2*beta - 3
   2*z*zb*alpha*beta - 4*yb*z^2 - 4*y*zb^2 + z*zb*alpha - alpha^3 + 8*z*zb + 4*alpha^2 + 18*alpha*beta + 11*alpha - 12
   z^2*zb^2 - 2*z*zb*alpha^2 + alpha^4 - 4*z^3 - 4*zb^3 - 8*alpha^3 + 18*z*zb + 18*alpha^2 - 27
   4*y^2*zb^2 - 2*z*zb^2*beta - 3*z*zb^2 - zb*alpha^2 + 8*z^2*beta + 36*yb^2 - 32*yb*z + 4*z^2 + 16*y*alpha - 30*zb*beta - 29*zb
}}}