# Singular Book 1.8.6 -- Zariski closure of the image

We compute an implicit equation for the surface defined parametrically by the map $f : A^2 \rightarrow{} A^3, (u,v) \mapsto{} (uv,uv^2,u^2)$.
 i1 : A = QQ[u,v,x,y,z]; i2 : I = ideal "x-uv,y-uv2,z-u2" 2 2 o2 = ideal (- u*v + x, - u*v + y, - u + z) o2 : Ideal of A i3 : eliminate(I,{u,v}) 4 2 o3 = ideal(x - y z) o3 : Ideal of A
This ideal defines the closure of the map $f$, the Whitney umbrella.

Alternatively, we could take the coimage of the ring homomorphism g corresponding to f.

 i4 : g = map(QQ[u,v],QQ[x,y,z],{x => u*v, y => u*v^2, z => u^2}) 2 2 o4 = map (QQ[u..v], QQ[x..z], {u*v, u*v , u }) o4 : RingMap QQ[u..v] <--- QQ[x..z] i5 : coimage g QQ[x..z] o5 = -------- 4 2 x - y z o5 : QuotientRing