Primary&Secondary_Invariants

2698 days ago by J.Wisniewski

# the following calculations are done in SINGULAR with SAGE working as an interface # Singular help at http://www.singular.uni-kl.de/Manual/latest/sing_1387.htm#SEC1463 # we load the singular library singular.lib('finvar.lib') # declare the ring R=singular.ring(0,'(x,y)','dp') # and the matrix transformation A=singular.matrix(2,2,'0,1,-1,0') print 'The matrix of the linear transformation \n', A invA=singular.invariant_ring(A) print '\n primary invariants \n', invA[1] print '\n secondary invariants \n', invA[3] 
       
The matrix of the linear transformation 
0, 1,
-1,0 

 primary invariants 
x^2+y^2,x^2*y^2

 secondary invariants 
x^3*y-x*y^3
# the same for another matrix B=singular.matrix(2,2,'0,-1,1,-1') print 'The matrix of the linear transformation \n', B invB=singular.invariant_ring(B) print '\n primary invariants \n', invB[1] print '\n secondary invariants \n', invB[3] 
       
The matrix of the linear transformation 
0,-1,
1,-1 

 primary invariants 
x^2-x*y+y^2,x^2*y-x*y^2

 secondary invariants 
x^3-3*x*y^2+y^3
# now we try binary dihedral group BD_{12} over finite field R=singular.ring(7,'(x,y)','dp') A=singular.matrix(2,2,'0,1,-1,0') C = singular.matrix(2,2,'3,0,0,5') print 'Matrix of order 6 \n', C print 'and matrix of order 4\n', A inv=singular.invariant_ring(A,C) print '\n primary invariants \n', inv[1] print '\n secondary invariants \n', inv[3] 
       
Matrix of order 6 
3,0,
0,-2
and matrix of order 4
0, 1,
-1,0 

 primary invariants 
x^2*y^2,x^6+y^6

 secondary invariants 
x^7*y-x*y^7