A.5.1 G_a -Invariants

We work in characteristic 0 and use the Lie algebra generated by one vectorfield of the form .

LIB "ainvar.lib"; int n=5; int i; ring s=32003,(x(1..n)),wp(1,2,3,4,5); // definition of the vectorfield m=sum m[i,1]*d/dx(i) matrix m[n][1]; for (i=1;i<=n-1;i=i+1) { m[i+1,1]=x(i); } // computation of the ring of invariants ideal in=invariantRing(m,x(2),x(1),0); in; //invariant ring is generated by 5 invariants ==> in[1]=x(1) ==> in[2]=x(2)^2-2*x(1)*x(3) ==> in[3]=x(3)^2-2*x(2)*x(4)+2*x(1)*x(5) ==> in[4]=x(2)^3-3*x(1)*x(2)*x(3)+3*x(1)^2*x(4) ==> in[5]=x(3)^3-3*x(2)*x(3)*x(4)-15997*x(1)*x(4)^2+3*x(2)^2*x(5)-6*x(1)*x(3)\ *x(5) ring q=32003,(x,y,z,u,v,w),dp; matrix m[6][1]; m[2,1]=x; m[3,1]=y; m[5,1]=u; m[6,1]=v; // the vectorfield is: xd/dy+yd/dz+ud/dv+vd/dw ideal in=invariantRing(m,y,x,0); in; //invariant ring is generated by 6 invariants ==> in[1]=x ==> in[2]=u ==> in[3]=v2-2uw ==> in[4]=zu-yv+xw ==> in[5]=yu-xv ==> in[6]=y2-2xz