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# RationalMap ! -- calculates every possible thing

- Operator: !
- Usage:
    `phi!`
- Inputs:
  - phi, a rational map
- Outputs:
  - a rational map, the same rational map phi

## Description

This method (mainly used for tests) applies almost all the deterministic methods that are available.

    i1 : QQ[x_0..x_5]; phi = rationalMap {x_4^2-x_3*x_5,x_2*x_4-x_1*x_5,x_2*x_3-x_1*x_4,x_2^2-x_0*x_5,x_1*x_2-x_0*x_4,x_1^2-x_0*x_3};

    o2 : RationalMap (quadratic rational map from PP^5 to PP^5)

    i3 : describe phi

    o3 = rational map defined by forms of degree 2
         source variety: PP^5
         target variety: PP^5
         coefficient ring: QQ

    i4 : time phi! ;
     -- used 0.0537026s (cpu); 0.0532482s (thread); 0s (gc)

    o4 : RationalMap (Cremona transformation of PP^5 of type (2,2))

    i5 : describe phi

    o5 = rational map defined by forms of degree 2
         source variety: PP^5
         target variety: PP^5
         dominance: true
         birationality: true (the inverse map is already calculated)
         projective degrees: {1, 2, 4, 4, 2, 1}
         number of minimal representatives: 1
         dimension base locus: 2
         degree base locus: 4
         coefficient ring: QQ

    i6 : QQ[x_0..x_4]; phi = rationalMap {-x_1^2+x_0*x_2,-x_1*x_2+x_0*x_3,-x_2^2+x_1*x_3,-x_1*x_3+x_0*x_4,-x_2*x_3+x_1*x_4,-x_3^2+x_2*x_4};

    o7 : RationalMap (quadratic rational map from PP^4 to PP^5)

    i8 : describe phi

    o8 = rational map defined by forms of degree 2
         source variety: PP^4
         target variety: PP^5
         coefficient ring: QQ

    i9 : time phi! ;
     -- used 0.0359434s (cpu); 0.0355946s (thread); 0s (gc)

    o9 : RationalMap (quadratic rational map from PP^4 to PP^5)

    i10 : describe phi

    o10 = rational map defined by forms of degree 2
          source variety: PP^4
          target variety: PP^5
          image: smooth quadric hypersurface in PP^5
          dominance: false
          birationality: false
          degree of map: 1
          projective degrees: {1, 2, 4, 4, 2}
          number of minimal representatives: 1
          dimension base locus: 1
          degree base locus: 4
          coefficient ring: QQ

## See also

- describe(RationalMap) -- describe a rational map

## Ways to use this method:

- RationalMap ! -- calculates every possible thing

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Cremona/documentation.m2:512:0.