--how to use derivations to determine whether a given form is a power of a linear form (alternative method without computing euqation for the Veronese variety)
--fact: for F(x_0,...,x_n) a degree d form F=L^d for some linear form L iff dim (F)^\perp=n
--i.e. iff the perp ideal of the form F has dimension n in degree 1
--i.e. iff the space of differential linear operators killing F has dimension n
Use S::=Q[x,y,z];
F:=(x+y+z)^5;
G:=x*(x+y+z)^4;
--to check whether F is a power we compute the ideal of derivation killing F
I:=PerpIdealOfForm(F);
--computing the Hilbert function we determine the degree 1 piece
Hilbert(S/I);
J:=PerpIdealOfForm(G);
Hilbert(S/J);
--so G is not a power but a form essentially involving two variables!
--Ex 1a the form (x+y)^3+z^3 is called a binary form as it is an element of the subring k[x+y,z] inside k[x,y,z]. How can we decide whether a given cubic form F(x,y,z) is a binary form?
--Ex 2a for quadratic forms there is a nice description of the variety of quadratic forms which are binary, can you find this description using the Veronese variety?
--Ex 2a a cylinder is R^3 is a surface constructed as the locus of parallel lines touching a fixed curve. Given the implicit equation of a surface how can de decide whether this is a cylinder or not?