(* The output of this procedure is:
If f = x[1]x[2]^3 + 5 x[1]x[2]x[3]^2 + 7x[2]^2 and V = {x[1],x[2],x[3]} then
Monomials[f,V]={{{1, 3, 0}, 1}, {{1, 1, 2}, 5}, {{0, 2, 0}, 7}}.
Notice that the terms are given in Lex order and the coefficient is the second term in each pair. *)

Monomials[f_, V_] :=
  Block[{M}, InputForm[M = Internal`DistributedTermsList[f, V]];
    M[[1]]]

(* The leading monomial in Lex of f withrespect to the variables V in Lex is *) 

LeadMon[f_,V_] := Monomials[f,V][[1,1]]

(* The leading coefficient relative to Lex is *)
 
LeadCoeff[f_,V_] := Monomials[f,V][[1,2]]

(* Given a set of polynomials L in the variables V this outputs the set of lead monomials 
realtive to Lex *)
MkLeads[L_,V_] := Block[{M,i,j,f},
        p = Length[L];
        M = {};
        For[i=1,i<=p,i++,
           f = Monomials[L[[i]],V];
           M = Append[M,f[[1,1]]]];
        M]


(* Once one has a Groebner basis relative to Lex and one has applied MkLeads to get L this procedure
gives the dimension *)

Dim[L_,n_] := Block[{j,s,d,u,TMP,TMP1,Z,z},
				If[Length[L]<=0, Return[n]];
				If[n==1,If[MemberQ[L,{0}],Return[-1]];Return[0]];
				Z = Array[z,n];
				Z = 0 Z;
				If[MemberQ[L,Z], Return[-1]];
				s = Length[L];
				TMP = {};
				TMP1 = {};
				For[j=1,j<=s,j++, If[L[[j,1]]==0, TMP = Append[TMP,Delete[L[[j]],1]]];
				TMP1 = Append[TMP1,Delete[L[[j]],1]]];
				d = Dim[TMP,n-1];
				u = Dim[TMP1,n-1] + 1;
				Max[d,u]]

(* This outputs the hilbert series same steps as for dimension but the output is unreadable  use Hilb instead.*)							
hilb[L_,n_] := Block[{m,p,i,j,t,Z,TMP},
        If[Length[L]<=0,Return[1/(1-q)^n]];
        Z=Array[0 # &,n];
        If[MemberQ[L,Z],Return[0]];
        p = Length[L];
        If[n==1,m = L[[1,1]];
        For[i=2,i<=p,i++,If[L[[i,1]]<m,m = L[[i,1]]]];
        Return[Sum[q^i,{i,0,m-1}]]];
        m = L[[1,1]];
        For[i=2,i<=p,i++,If[L[[i,1]]>m,m = L[[i,1]]]];
        t=0;
        For[i=0,i<m,i++,TMP={};
                For[j=1,j<=p,j++,If[L[[j,1]]<=i,
			TMP=Union[TMP,{Delete[L[[j]],1]}]]];
        t= t + (q^i)*hilb[TMP,n-1];
        ];
        TMP={};
        For[j=1,j<=p,j++,TMP=Union[TMP,{Delete[L[[j]],1]}]];
        t= t + q^m/(1-q)*hilb[TMP,n-1]]
        
Hilb[L_,n_] := Simplify[hilb[L,n]]