gap> g:=Group(g.1,g.2);
Group([ [ [ 0, -1 ], [ 1, 0 ] ], [ [ 1, 1 ], [ 0, 1 ] ] ])
gap> h:=Group(HomogPolyAction(g,2)); # own fct;
Group([ [ [ 0, 0, 1 ], [ 0, -1, 0 ], [ 1, 0, 0 ] ], 
  [ [ 1, 0, 0 ], [ 2, 1, 0 ], [ 1, 1, 1 ] ] ])
gap> HomogeneousInvariants(h,1);
[  ]
gap> HomogeneousInvariants(h,2);
[ x_1*x_3-1/4*x_2^2 ]
gap> invar:=last*4;
[ 4*x_1*x_3-x_2^2 ]
gap> HomogeneousInvariants(h,3);
[  ]
gap> HomogeneousInvariants(h,4);
[ x_1^2*x_3^2-1/2*x_1*x_2^2*x_3+1/16*x_2^4 ]
gap> 16*last[1]-invar[1]^2;
0

gap> g:=Image(SLNZFP(3)); #own fct
<matrix group with 6 generators>
gap> AllHomogeneousPolynomials(vars{[1..3]},3);
[ [ x_3^3, 1 ], [ x_2*x_3^2, 3 ], [ x_2^2*x_3, 3 ], [ x_2^3, 1 ], 
  [ x_1*x_3^2, 3 ], [ x_1*x_2*x_3, 6 ], [ x_1*x_2^2, 3 ], [ x_1^2*x_3, 3 ], 
  [ x_1^2*x_2, 3 ], [ x_1^3, 1 ] ]

gap> h:=Group(HomogPolyAction(g,3));
<matrix group with 6 generators>

gap> Display(h.1);
[ [  1,  0,  0,  0,  0,  0,  0,  0,  0,  0 ],
  [  0,  1,  0,  0,  0,  0,  0,  0,  0,  0 ],
  [  0,  0,  1,  0,  0,  0,  0,  0,  0,  0 ],
  [  0,  0,  0,  1,  0,  0,  0,  0,  0,  0 ],
  [  0,  1,  0,  0,  1,  0,  0,  0,  0,  0 ],
  [  0,  0,  2,  0,  0,  1,  0,  0,  0,  0 ],
  [  0,  0,  0,  3,  0,  0,  1,  0,  0,  0 ],
  [  0,  0,  1,  0,  0,  1,  0,  1,  0,  0 ],
  [  0,  0,  0,  3,  0,  0,  2,  0,  1,  0 ],
  [  0,  0,  0,  1,  0,  0,  1,  0,  1,  1 ] ]
gap> HomogeneousInvariants(h,1);
[  ]
gap> HomogeneousInvariants(h,2);
[  ]
gap> HomogeneousInvariants(h,3);
[  ]
gap> HomogeneousInvariants(h,4);
[ x_1*x_3*x_7*x_10-1/3*x_1*x_3*x_9^2-3/2*x_1*x_4*x_6*x_10+x_1*x_4*x_8*x_9+1/6\
*x_1*x_6*x_7*x_9-1/3*x_1*x_7^2*x_8-1/3*x_2^2*x_7*x_10+1/9*x_2^2*x_9^2+1/6*x_2\
*x_3*x_6*x_10-1/9*x_2*x_3*x_8*x_9+x_2*x_4*x_5*x_10-1/3*x_2*x_4*x_8^2-1/9*x_2*\
x_5*x_7*x_9-1/18*x_2*x_6^2*x_9+1/6*x_2*x_6*x_7*x_8-1/3*x_3^2*x_5*x_10+1/9*x_3\
^2*x_8^2+1/6*x_3*x_5*x_6*x_9-1/9*x_3*x_5*x_7*x_8-1/18*x_3*x_6^2*x_8-1/3*x_4*x\
_5^2*x_9+1/6*x_4*x_5*x_6*x_8+1/9*x_5^2*x_7^2-1/18*x_5*x_6^2*x_7+1/144*x_6^4 ]
gap> invar:=last*144;;
gap> HomogeneousInvariants(h,5);
[  ]
gap> HomogeneousInvariants(h,6);
[ x_1^2*x_4^2*x_10^2-2/3*x_1^2*x_4*x_7*x_9*x_10+4/27*x_1^2*x_4*x_9^3+4/27*x_1\
^2*x_7^3*x_10-1/27*x_1^2*x_7^2*x_9^2-2/3*x_1*x_2*x_3*x_4*x_10^2+2/9*x_1*x_2*x\
_3*x_7*x_9*x_10-4/81*x_1*x_2*x_3*x_9^3+2/9*x_1*x_2*x_4*x_6*x_9*x_10+2/9*x_1*x\
_2*x_4*x_7*x_8*x_10-4/27*x_1*x_2*x_4*x_8*x_9^2-4/27*x_1*x_2*x_6*x_7^2*x_10+2/\
81*x_1*x_2*x_6*x_7*x_9^2+2/81*x_1*x_2*x_7^2*x_8*x_9+4/27*x_1*x_3^3*x_10^2-4/2\
7*x_1*x_3^2*x_6*x_9*x_10-4/27*x_1*x_3^2*x_7*x_8*x_10+8/81*x_1*x_3^2*x_8*x_9^2\
+2/9*x_1*x_3*x_4*x_5*x_9*x_10+2/9*x_1*x_3*x_4*x_6*x_8*x_10-4/27*x_1*x_3*x_4*x\
_8^2*x_9-4/27*x_1*x_3*x_5*x_7^2*x_10+2/81*x_1*x_3*x_5*x_7*x_9^2+1/9*x_1*x_3*x\
_6^2*x_7*x_10+1/81*x_1*x_3*x_6^2*x_9^2-10/81*x_1*x_3*x_6*x_7*x_8*x_9+8/81*x_1\
*x_3*x_7^2*x_8^2-2/3*x_1*x_4^2*x_5*x_8*x_10+4/27*x_1*x_4^2*x_8^3+2/9*x_1*x_4*\
x_5*x_6*x_7*x_10-4/27*x_1*x_4*x_5*x_6*x_9^2+2/9*x_1*x_4*x_5*x_7*x_8*x_9-5/54*\
x_1*x_4*x_6^3*x_10+1/9*x_1*x_4*x_6^2*x_8*x_9-4/27*x_1*x_4*x_6*x_7*x_8^2+2/81*\
x_1*x_5*x_6*x_7^2*x_9-4/81*x_1*x_5*x_7^3*x_8-1/162*x_1*x_6^3*x_7*x_9+1/81*x_1\
*x_6^2*x_7^2*x_8+4/27*x_2^3*x_4*x_10^2-4/81*x_2^3*x_7*x_9*x_10+8/729*x_2^3*x_\
9^3-1/27*x_2^2*x_3^2*x_10^2+2/81*x_2^2*x_3*x_6*x_9*x_10+2/81*x_2^2*x_3*x_7*x_\
8*x_10-4/243*x_2^2*x_3*x_8*x_9^2-4/27*x_2^2*x_4*x_5*x_9*x_10-4/27*x_2^2*x_4*x\
_6*x_8*x_10+8/81*x_2^2*x_4*x_8^2*x_9+8/81*x_2^2*x_5*x_7^2*x_10-4/243*x_2^2*x_\
5*x_7*x_9^2+1/81*x_2^2*x_6^2*x_7*x_10-2/243*x_2^2*x_6^2*x_9^2+2/81*x_2^2*x_6*\
x_7*x_8*x_9-1/27*x_2^2*x_7^2*x_8^2+2/81*x_2*x_3^2*x_5*x_9*x_10+2/81*x_2*x_3^2\
*x_6*x_8*x_10-4/243*x_2*x_3^2*x_8^2*x_9+2/9*x_2*x_3*x_4*x_5*x_8*x_10-4/81*x_2\
*x_3*x_4*x_8^3-10/81*x_2*x_3*x_5*x_6*x_7*x_10+2/81*x_2*x_3*x_5*x_6*x_9^2-2/24\
3*x_2*x_3*x_5*x_7*x_8*x_9-1/162*x_2*x_3*x_6^3*x_10-1/243*x_2*x_3*x_6^2*x_8*x_\
9+2/81*x_2*x_3*x_6*x_7*x_8^2-4/27*x_2*x_4*x_5^2*x_7*x_10+8/81*x_2*x_4*x_5^2*x\
_9^2+1/9*x_2*x_4*x_5*x_6^2*x_10-10/81*x_2*x_4*x_5*x_6*x_8*x_9+2/81*x_2*x_4*x_\
5*x_7*x_8^2+1/81*x_2*x_4*x_6^2*x_8^2-4/243*x_2*x_5^2*x_7^2*x_9-1/243*x_2*x_5*\
x_6^2*x_7*x_9+2/81*x_2*x_5*x_6*x_7^2*x_8+1/486*x_2*x_6^4*x_9-1/162*x_2*x_6^3*\
x_7*x_8-4/81*x_3^3*x_5*x_8*x_10+8/729*x_3^3*x_8^3+8/81*x_3^2*x_5^2*x_7*x_10-1\
/27*x_3^2*x_5^2*x_9^2+1/81*x_3^2*x_5*x_6^2*x_10+2/81*x_3^2*x_5*x_6*x_8*x_9-4/\
243*x_3^2*x_5*x_7*x_8^2-2/243*x_3^2*x_6^2*x_8^2-4/27*x_3*x_4*x_5^2*x_6*x_10+2\
/81*x_3*x_4*x_5^2*x_8*x_9+2/81*x_3*x_4*x_5*x_6*x_8^2+2/81*x_3*x_5^2*x_6*x_7*x\
_9-4/243*x_3*x_5^2*x_7^2*x_8-1/162*x_3*x_5*x_6^3*x_9-1/243*x_3*x_5*x_6^2*x_7*\
x_8+1/486*x_3*x_6^4*x_8+4/27*x_4^2*x_5^3*x_10-1/27*x_4^2*x_5^2*x_8^2-4/81*x_4\
*x_5^3*x_7*x_9+1/81*x_4*x_5^2*x_6^2*x_9+2/81*x_4*x_5^2*x_6*x_7*x_8-1/162*x_4*\
x_5*x_6^3*x_8+8/729*x_5^3*x_7^3-2/243*x_5^2*x_6^2*x_7^2+1/486*x_5*x_6^4*x_7-1\
/5832*x_6^6 ]
gap> Add(invar,-last[1]*5832);

gap> delta:=invar[1]^3-invar[2]^2;
-34012224*x_1^4*x_4^4*x_10^4+45349632*x_1^4*x_4^3*x_7*x_9*x_10^3-10077696*x_1\
^4*x_4^3*x_9^3*x_10^2-10077696*x_1^4*x_4^2*x_7^3*x_10^3-12597120*x_1^4*x_4^2*\
[...]

gap> Length(ExtRepPolynomialRatFun(delta)); # 2040 terms
4080

###

AllHomogeneousPolynomials:=function(vars,deg)
local n,p,i,l;
  n:=Length(vars);
  p:=OrderedPartitions(deg+n,n)-1;
  l:=[];
  for i in p do
    Add(l,[Product(List([1..n],x->vars[x]^i[x])),
      Factorial(n)/Product(i,x->Factorial(x))]);
  od;
  return l;

end;

HomogeneousInvariants:=function(G,deg)
local n,vars,b,be,eqs,g,imgs,o,a,i,j,e,p,m;
  n:=Length(One(G));
  vars:=IndeterminatesOfPolynomialRing(
    PolynomialRing(DefaultFieldOfMatrixGroup(G),n));
  b:=AllHomogeneousPolynomials(vars,deg);
  b:=List(b,x->x[1]);
  be:=List(b,x->ExtRepPolynomialRatFun(x)[1]);

  eqs:=[];
  for g in GeneratorsOfGroup(G) do
    #images
    imgs:=List([1..n],i->Sum([1..n],j->g[i][j]*vars[j]));
    m:=[];
    for i in b do
      e:=ListWithIdenticalEntries(Length(b),0);
      e[Position(b,i)]:=-1;
      a:=Value(i,vars,imgs);
      a:=ExtRepPolynomialRatFun(a);
      for j in [1,3..Length(a)-1] do
        p:=Position(be,a[j]);
        e[p]:=e[p]+a[j+1];
      od;
      Add(m,e);
    od;
    Append(eqs,TransposedMat(m));
  od;
  e:=[];
  for a in NullspaceMat(TransposedMat(eqs)) do
    i:=Sum([1..Length(b)],j->b[j]*a[j]);
    #Print(a," ",i,"\n");
    Add(e,i);
  od;
  return e;
end;

HomogPolyAction:=function(G,deg)
local n,vars,b,be,l,m,imgs,i,e,a,j,p,g,d;
  n:=Length(One(G));
  vars:=IndeterminatesOfPolynomialRing(
    PolynomialRing(DefaultFieldOfMatrixGroup(G),n));
  b:=AllHomogeneousPolynomials(vars,deg);
  d:=List(b,x->x[2]);
  d:=DiagonalMat(d)^-1;
  b:=List(b,x->x[1]);
  be:=List(b,x->ExtRepPolynomialRatFun(x)[1]);
  l:=[];
  for g in GeneratorsOfGroup(G) do
    m:=[];
    imgs:=List([1..n],i->Sum([1..n],j->g[i][j]*vars[j]));
    for i in b do
      e:=ListWithIdenticalEntries(Length(b),0);
      a:=Value(i,vars,imgs);
      a:=ExtRepPolynomialRatFun(a);
      for j in [1,3..Length(a)-1] do
        p:=Position(be,a[j]);
        e[p]:=e[p]+a[j+1];
      od;
      Add(m,e);
    od;
    Add(l,m);
  od;
  return List(l,x->x^d);
end;