### 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

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