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# isSameMap -- whether two rational maps to between projective varieties are really the same

## Synopsis

• Usage:
b = isSameMap(f1, f2)
phi == psi
b = isSameMap(phi, psi)
• Inputs:
• f1, , a map of rings corresponding to a rational map between projective varieties
• f2, , a map of rings corresponding to a rational map between projective varieties
• phi, an instance of the type RationalMapping, a map between projective varieties
• psi, an instance of the type RationalMapping, a rational map between projective varieties
• Outputs:
• b, , true if the rational maps are the same, false otherwise.

## Description

Checks whether two rational maps between projective varieties are really the same (that is, agree on a dense open set).

 i1 : R=QQ[x,y,z]; i2 : S=QQ[a,b,c]; i3 : f1=map(R, S, {y*z,x*z,x*y}); o3 : RingMap R <--- S i4 : f2=map(R, S, {x*y*z,x^2*z,x^2*y}); o4 : RingMap R <--- S i5 : isSameMap(f1,f2) o5 = true

The Cremona transformation is not the identity, but its square is.

 i6 : R = ZZ/7[x,y,z] o6 = R o6 : PolynomialRing i7 : phi = rationalMapping(R, R, {y*z,x*z,x*y}) o7 = Proj R - - - > Proj R {y*z, x*z, x*y} o7 : RationalMapping i8 : ident = rationalMapping(R, R, {x,y,z}) o8 = Proj R - - - > Proj R {x, y, z} o8 : RationalMapping i9 : phi == ident o9 = false i10 : phi^2 == ident o10 = true

## Ways to use isSameMap :

• "isSameMap(RationalMapping,RationalMapping)"
• "isSameMap(RingMap,RingMap)"

## For the programmer

The object isSameMap is .