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Divisor :: reflexify(..., Strategy => ...)

reflexify(..., Strategy => ...) -- specify a strategy for the reflexification

Synopsis

Description

There are two strategies for computing a reflexification (at least if the module embeds as an ideal).

IdealStrategy. In the case that R is a domain, and our module is isomorphic to an ideal I, then one can compute the reflexification by computing colons.

ModuleStrategy. This computes the reflexification simply by computing Hom twice.

ModuleStrategy is the default strategy for modules, IdealStrategy is the default Strategy for Ideals. In our experience, IdealStrategy is faster on average. Note that calling ModuleStrategy for Ideals or IdealStrategy for modules creates overhead which can slow things down substantially (since we must embed various modules as ideals).

i1 : R = ZZ/13[x,y,z]/ideal(x^3 + y^3-z^11*x*y);
i2 : I = ideal(x-4*y, z);

o2 : Ideal of R
i3 : J = I^20;

o3 : Ideal of R
i4 : M = J*R^1;
i5 : J1 = time reflexify( J, Strategy=>IdealStrategy )
     -- used 0.0573534 seconds

             2            2     9       9   11
o5 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)

o5 : Ideal of R
i6 : J2 = time reflexify( J, Strategy=>ModuleStrategy )
     -- used 5.28877 seconds

             2            2     9       9      11
o6 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , - 4z   - 4x + 3y)

o6 : Ideal of R
i7 : J1 == J2

o7 = true
i8 : time reflexify( M, Strategy=>IdealStrategy );
     -- used 5.27135 seconds
i9 : time reflexify( M, Strategy=>ModuleStrategy );
     -- used 0.475142 seconds

However, sometimes ModuleStrategy is faster, especially for Monomial ideals.

i10 : R = QQ[x,y,u,v]/ideal(x*y-u*v);
i11 : I = ideal(x,u);

o11 : Ideal of R
i12 : J = I^20;

o12 : Ideal of R
i13 : M = I^20*R^1;
i14 : time reflexify( J, Strategy=>IdealStrategy )
     -- used 0.0489745 seconds

              20     19   2 18   3 17   4 16   5 15   6 14   7 13   8 12 
o14 = ideal (u  , x*u  , x u  , x u  , x u  , x u  , x u  , x u  , x u  ,
      -----------------------------------------------------------------------
       9 11   10 10   11 9   12 8   13 7   14 6   15 5   16 4   17 3   18 2 
      x u  , x  u  , x  u , x  u , x  u , x  u , x  u , x  u , x  u , x  u ,
      -----------------------------------------------------------------------
       19    20
      x  u, x  )

o14 : Ideal of R
i15 : time reflexify( J, Strategy=>ModuleStrategy )
     -- used 0.0126358 seconds

              20     19   2 18   3 17   4 16   5 15   6 14   7 13   8 12 
o15 = ideal (u  , x*u  , x u  , x u  , x u  , x u  , x u  , x u  , x u  ,
      -----------------------------------------------------------------------
       9 11   10 10   11 9   12 8   13 7   14 6   15 5   16 4   17 3   18 2 
      x u  , x  u  , x  u , x  u , x  u , x  u , x  u , x  u , x  u , x  u ,
      -----------------------------------------------------------------------
       19    20
      x  u, x  )

o15 : Ideal of R
i16 : time reflexify( M, Strategy=>IdealStrategy );
     -- used 0.157007 seconds
i17 : time reflexify( M, Strategy=>ModuleStrategy );
     -- used 0.00504966 seconds

Further information

See also