// Counting the number of terms of general formula of //
// the discriminant up to degree 12                   //
// Last modified: February 7, 2014                    //

var:=12;
P<a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12>:=PolynomialRing(Rationals(),var);
V:=[1,a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12];

// No need to change below //

SDIV:=function(f,W,k)
	X,Y:=CoefficientsAndMonomials(f);
	U:=#X;
	I:=[i : i in [1..U] | Degree(Y[i],W) eq 0];
	if #I eq 0 then
		return f/(k*W);
	else
		xx:=f-(&+[X[j]*Y[j] : j in I]);
		return xx/(k*W);
		end if;
	end function;

S:=V[2]^2-4*V[3];
for N in [3..var] do
	T:=V[N]^2*S;
	S:=T;
	for K in [1..N-1] do
		U:=(-N)*Derivative(P!T,V[2]);
		for J in [2..N-1] do
			U:=U+(J-N-1)*V[J]*Derivative(P!T,V[J+1]);
			end for;
		T:=SDIV(P!U,P!V[N],K);
		S:=S+T*V[N+1]^K;
		end for;
	"(deg,#monomial)=(",N,",",#CoefficientsAndMonomials(P!S),")";
	end for;