# mapOntoImage -- Given a map of rings, correspoing to X mapping to Y, this returns the map of rings corresponding to X mapping to f(X).

## Synopsis

• Usage:
h = mapOntoImage(f)
h = mapOntoImage(a,b,l)
h = mapOntoImage(R,S,l)
• Inputs:
• a, an ideal, defining equations for X
• b, an ideal, defining equations for Y
• l, , projective rational map given by polynomial represenatives of the same degree
• f, , the ring map corresponding to $f : X \to Y$
• R, a ring, coordinate ring for X
• S, a ring, coordinate ring for Y
• Optional inputs:
• QuickRank => ..., default value true, An option for computing how rank is computed
• Outputs:
• h, , the map of rings corresponding to $f : X \to f(X)$.

## Description

This function is really simple, given $S \to R$, this just returns $S/kernel \to R$.

 i1 : R = QQ[x,y]; i2 : S = QQ[a,b,c]; i3 : f = map(R, S, {x^2, x*y, y^2}); o3 : RingMap R <--- S i4 : mapOntoImage(f) S 2 2 o4 = map (R, --------, {x , x*y, y }) 2 b - a*c S o4 : RingMap R <--- -------- 2 b - a*c i5 : mapOntoImage(R,S,{x^2,x*y,y^2}) S 2 2 o5 = map (R, --------, {x , x*y, y }) 2 b - a*c S o5 : RingMap R <--- -------- 2 b - a*c

## Ways to use mapOntoImage :

• "mapOntoImage(Ideal,Ideal,BasicList)"
• "mapOntoImage(Ring,Ring,BasicList)"
• "mapOntoImage(RingMap)"

## For the programmer

The object mapOntoImage is .