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# isIsomorphism(RationalMap) -- whether a birational map is an isomorphism

- Function: isIsomorphism
- Usage:
    `isIsomorphism phi`
- Inputs:
  - phi, a rational map
- Outputs:
  - a Boolean value, whether phi is an isomorphism

## Description

This method computes the inverse rational map using inverse.

    i1 : P1 := QQ[a,b]; P4 := QQ[x,y,z,w];

    i3 : phi = rationalMap({a^4,a^3*b,a^2*b^2,a*b^3,b^4},Dominant=>true)

    o3 = -- rational map --
         source: Proj(QQ[a, b])
         target: subvariety of Proj(QQ[t , t , t , t , t ]) defined by
                                        0   1   2   3   4
                 {
                   2
                  t  - t t ,
                   3    2 4

                  t t  - t t ,
                   2 3    1 4

                  t t  - t t ,
                   1 3    0 4

                   2
                  t  - t t ,
                   2    0 4

                  t t  - t t ,
                   1 2    0 3

                   2
                  t  - t t
                   1    0 2
                 }
         defining forms: {
                           4
                          a ,

                           3
                          a b,

                           2 2
                          a b ,

                             3
                          a*b ,

                           4
                          b
                         }

    o3 : RationalMap (dominant rational map from PP^1 to curve in PP^4)

    i4 : isIsomorphism phi

    o4 = true

## See also

- ideal(RationalMap) -- base locus of a rational map
- isBirational -- whether a rational map is birational
- isMorphism -- whether a rational map is a morphism

## Ways to use this method:

- isIsomorphism(RationalMap) -- whether a birational map is an isomorphism

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Cremona/documentation.m2:1147:0.