-- Implicitization

--Find implicit equations for the curve t->(t^4, t^3, t^2)

R=ZZ/101[t,x,y,z, MonomialOrder=>Lex]
graph = ideal(x-t^4, y-t^3, z-t^2)
GB = gens gb graph
eqs = selectInSubring(1,GB)


--Find the equations of the tangent developable to the twisted cubic

R=ZZ/31991[t,u,x,y,z, MonomialOrder=>Lex]
twistedCubic = matrix {{t, t^2, t^3}} 
tangVector = diff(t, twistedCubic)
develop = twistedCubic + u*tangVector
graph = ideal (submatrix(vars R,{0},{2,3,4})-develop)
GB = gens gb graph
eqs = selectInSubring(2,GB)


--Whitney umbrella
R=ZZ/31991[u,v,x,y,z, MonomialOrder=>Lex]
graph = ideal (x-u*v, y-v, z-u^2)
GB = gens gb graph
eqs = selectInSubring(2,GB)