-- Dual varieties: 
--  construction of the dual to a hypersurface;
--  example 11.11

R = QQ[x,y,z] -- examples below are for curves in P^2, but all steps work for an arbitrary R
D = (coefficientRing R)[gens R / (x->p_x)]
describe D
RD = (coefficientRing R)[gens R, gens D]
describe RD
use R
d = 3
F = x^d+y^d+z^d
-- F = random(d,R) -- experiment with a random hypersurface: is it smooth? is the dual smooth?
grad = diff (vars R, F) -- partial derivatives
I = ideal sub(F,RD) + ideal(sub(vars D,RD) - sub(grad, RD)) -- I = <F> + <p_x - dF/dx | all vars x in R>
n = numgens R
J = sub(eliminate(take(gens RD,n), I), D) 

-- singular locus computation
m = ideal gens R
sF = ideal F + ideal grad -- defines the singular locus
sFsat = saturate(sF, m) -- if this == 1 then V(F) is smooth (projective variety)

mD = ideal gens D
sJ = J + ideal jacobian J
sJsat = saturate(sJ, mD) -- if this == 1 then the dual of V(F) is smooth 
decompose sJsat