Given $f : X \to Y \subset P^N$, this returns the defining ideal of $f(x) \subseteq P^N$. It should be noted for inputs that all rings are quotients of polynomial rings, and all ideals and ring maps are of these. In particular, this function returns an ideal defining a subset of the the ambient projective space of the image. In the following example we consider the image of $P^1$ inside $P^1 \times P^1$.
i1 : S = QQ[x,y,z,w]; |
i2 : b = ideal(x*y-z*w); o2 : Ideal of S |
i3 : R = QQ[u,v]; |
i4 : a = ideal(sub(0,R)); o4 : Ideal of R |
i5 : f = matrix {{u,0,v,0}}; 1 4 o5 : Matrix R <--- R |
i6 : idealOfImageOfMap(a,b,f) o6 = ideal (w, y) o6 : Ideal of S |
This function frequently just calls ker from Macaulay2. However, if the target of the ring map is a polynomial ring, then it first tries to verify if the ring map is injective. This is done by computing the rank of an appropriate jacobian matrix.
The object idealOfImageOfMap is a method function with options.