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:
• 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 3.81845 seconds o4 = 2 i5 : time degree(Phi,Strategy=>"0-th projective degree") -- used 0.936889 seconds o5 = 2 i6 : time degree Phi -- used 0.606725 seconds o6 = 2

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