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A.5.1 G_a -Invariants

We work in characteristic 0 and use the Lie algebra generated by one vectorfield of the form $\sum x_i \partial /\partial x_{i+1}$.

  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)
  // 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)\
  ring q=32003,(x,y,z,u,v,w),dp;
  matrix m[6][1];
  // 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