Use R ::= QQ[x,y,z,w];
Homogenized(w, x^3y);
x^3  yw^2

Homogenized(w, [x^3y, x^4z]);
[x^3  yw^2, x^4  zw^3]

I := Ideal(x^3y, x^4z);
 same as Homogenized5(w, I); Homogenized([w], I);
Homogenized(w, I);  don't just get the homogenizations of
 the generators of I
Ideal(x^3  yw^2, xy + zw, x^2z  y^2w, y^3  xz^2)

Homogenized(w,[[I,yz^2],zy^4]);
[[Ideal(x^3  yw^2, xy + zw, x^2z  y^2w, y^3  xz^2), z^2 + yw], y^4 + zw^3]

