degree(MultirationalMap,Option) -- degree of a multi-rational map using a probabilistic approach

Synopsis

• Function: degree

• Usage:

  degree(Phi,Strategy=>"random point")

  degree(Phi,Strategy=>"0-th projective degree")

• Inputs:
  • Phi, a multi-rational map
  • Strategy, an option
• Outputs:
  • an integer, the degree of Phi. So this value is 1 if and only if (with high probability) the map is birational onto its image.

Description

i1 : R = ZZ/33331[x_0..x_4];

i2 : Phi = (last graph multirationalMap rationalMap transpose jacobian(-x_2^3+2*x_1*x_2*x_3-x_0*x_3^2-x_1^2*x_4+x_0*x_2*x_4))||projectiveVariety ideal(random(2,R));

o2 : MultirationalMap (rational map from threefold in PP^4 x PP^4 to hypersurface in PP^4)

i3 : ? Phi

o3 = multi-rational map consisting of one single rational map
     source variety: threefold in PP^4 x PP^4 cut out by 13 hypersurfaces of
     target variety: hypersurface in PP^4 defined by a form of degree 2
     ------------------------------------------------------------------------
     multi-degrees (0,2)^1 (1,1)^3 (2,1)^8 (4,0)^1

i4 : time degree(Phi,Strategy=>"random point")
 -- used 4.59667s (cpu); 2.80722s (thread); 0s (gc)

o4 = 2

i5 : time degree(Phi,Strategy=>"0-th projective degree")
 -- used 0.37107s (cpu); 0.25456s (thread); 0s (gc)

o5 = 2

i6 : time degree Phi
 -- used 0.290951s (cpu); 0.293579s (thread); 0s (gc)

o6 = 2

Note, as in the example above, that calculation times may vary depending on the strategy used.

See also

• degree(MultirationalMap) -- degree of a multi-rational map
• degreeMap(RationalMap) -- degree of a rational map
• multidegree(ZZ,MultirationalMap) -- i-th projective degree of a multi-rational map using a probabilistic approach
• point(MultiprojectiveVariety) -- pick a random rational point on a multi-projective variety
• MultirationalMap ^* -- inverse image via a multi-rational map