# RationalMap ! -- calculates every possible thing

## Synopsis

• Operator: !
• Usage:
phi!
• Inputs:
• phi,
• Outputs:
• , 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 0.0756251 seconds 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 0.0657549 seconds 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

The command phi(*) does more or less the same thing but it uses probabilistic methods and treats them as deterministic (the user should never use this).

 i11 : phi = rationalMap rationalMap map specialQuadraticTransformation(8,ZZ/33331); o11 : RationalMap (quadratic rational map from PP^8 to PP^11) i12 : describe phi o12 = rational map defined by forms of degree 2 source variety: PP^8 target variety: PP^11 image: complete intersection of type (2,2,2) in PP^11 dominance: false birationality: false degree of map: 1 projective degrees: {1, 2, 4, 8, 16, 23, 23, 16, 8} number of minimal representatives: 1 dimension base locus: 3 degree base locus: 9 coefficient ring: ZZ/33331 i13 : time phi(*) ; -- used 7.324e-6 seconds o13 : RationalMap (quadratic rational map from PP^8 to PP^11) i14 : describe phi o14 = rational map defined by forms of degree 2 source variety: PP^8 target variety: PP^11 image: complete intersection of type (2,2,2) in PP^11 dominance: false birationality: false degree of map: 1 projective degrees: {1, 2, 4, 8, 16, 23, 23, 16, 8} number of minimal representatives: 1 dimension base locus: 3 degree base locus: 9 coefficient ring: ZZ/33331