|
Applications: Algebraic Dependency In Quantum Algebra
ring r=(0,Q),(a,b,c,d,x,y,z),dp; We denote Q=q1/2
minpoly=Q^4+Q^2+1; And set Q6=1
matrix A[7][7]; matrix B[7][7];
int u,j;
for(u=1;u<=7;u++)
{ for(j=u;j<=7;j++) {A[u,j]=1;} }
A[5,6]=Q2; A[5,7]=1/Q2; A[6,7]=Q2;
B[5,6]=-Q*z; B[5,7]=1/Q*y; B[6,7]=-Q*x;
system("PLURAL",A,B);
poly Cq=Q^4*x2+y2+Q^4*z2+Q*(1-Q^4)*x*y*z;
poly C1=1/3*(x^3+x);
poly C2=1/3*(y^3+y);
poly C3=1/3*(z^3+z);
ideal I=a-Cq,b-C1,c-C2,d-C3;
I=std(I);
I;
|
==>
|
...
I[5]=a3+(81Q3+162Q)*bcd+(-Q2)*a2-9*b2-9*c2-9*d2
...
I[26]=bcdxy+(1/81Q3-1/81Q)*abx2y+(1/729Q3-4/729Q)*ax3y+(1/81Q3-1/81Q)*acxy2+
(1/81Q3+2/81Q)*aby3+(-2/729Q3-1/729Q)*axy3+(2/243Q2+4/243)*a2x2z+(-5/81Q3-4/81Q)*adxyz+
(-1/81Q2+1/81)*a2y2z+(1/9Q3+2/9Q)*abcz2+(7/81Q3+8/81Q)*acxz2+(-5/81Q3-7/81Q)*abyz2+
(22/729Q3+20/729Q)*axyz2+(2/243Q2+1/81)*a2z3+(-2/81Q3-1/81Q)*a2cx+(7/81Q2+4/81)*adx2+
(-2/81Q3-1/81Q)*a2by+(-13/729Q3-11/729Q)*a2xy+(1/27Q3+2/27Q)*b2xy+(1/27Q3+2/27Q)*c2xy+
(1/27Q3+2/27Q)*d2xy+(1/81Q2-5/81)*ady2+(-1/243Q2-1/243)*a3z+(4/81Q2+4/81)*abxz+
(-14/729Q2+8/729)*ax2z+(4/81Q2-2/81)*acyz+(-2/243Q2)*ay2z+(1/27Q2)*adz2-2/243*az3+
(1/27Q3+2/27Q)*abc+(5/81Q3+1/81Q)*acx+(-1/81Q3-1/27Q)*aby+(-1/729Q3-5/729Q)*axy+(1/243Q2+5/243)*a2z
|
So, the needed polynomial is
I[5] =
a3+81q1/2(q+2)bcd-qa2
-9(b2+c2+d2).
|