sourceInversionFactor -- computes the common factor among the components of the composition of the inverse map and the original map

Synopsis

• Usage:

  s = sourceInversionFactor(g)

• Inputs:
  • g, a ring map, a birational map $f : X \to Y$.
• Optional inputs:
  • Verbosity => an integer, default value 1, if 0 then silence the function, if 1 then generate informative output which can be used to adjust strategies, if > 1 then generate a detailed description of the execution
  • CheckBirational => a Boolean value, default value false, whether to check birationality (if it is not birational, and CheckBirational is set to true, then an error will be thrown)
  • Strategy => a symbol, default value HybridStrategy, choose the strategy to use: HybridStrategy, SimisStrategy, or ReesStrategy
  • HybridLimit => an integer, default value 15, within HybridStrategy, the option HybridLimit controls how often SimisStrategy and ReesStrategy are used, larger numbers means SimisStrategy will be executed longer
  • MinorsLimit => an integer, default value 0, how many submatrices of a variant of the Jacobian dual matrix to consider before switching to a different strategy
  • AssumeDominant => a Boolean value, default value false, whether to assume a rational map of schemes is dominant, if set to true it can speed up computation
  • QuickRank => a Boolean value, default value true, whether to compute rank via the package FastMinors
• Outputs:
  • s, a ring element, an element of the coordinate ring of $X$ .

Description

i1 : R = ZZ/7[x,y,z];

i2 : S = ZZ/7[a,b,c];

i3 : h = map(R, S, {y*z, x*z, x*y});

o3 : RingMap R <-- S

i4 : sourceInversionFactor h
sourceInversionFactor: About to find the image of the map.  If you know the image, 
        you may want to use the AssumeDominant option if this is slow.
inverseOfMapSimis: About to find the image of the map.  If you know the image, 
        you may want to use the AssumeDominant option if this is slow.
inverseOfMapSimis: About to check rank, if this is very slow, you may want to try turning QuickRank=>false.
inverseOfMapSimis: rank found, we computed enough of the Groebner basis.

o4 = x*y*z

o4 : R

i5 : S=QQ[a,b,c,d];

i6 : g=(b^2-a*c)*c*d;

i7 : phi=map(S,S,transpose jacobian ideal g);

o7 : RingMap S <-- S

i8 : sourceInversionFactor(phi, Verbosity=>0)

      2 4 2      5 2
o8 = b c d  - a*c d

o8 : S

Caveat

See also

• HybridStrategy -- A strategy for determining whether a map is birational and computing its inverse
• SimisStrategy -- a strategy for determining whether a map is birational and computing its inverse
• ReesStrategy -- a strategy for determining whether a map is birational and computing its inverse