flatteningRelations -- Compute the flattening relations in this special case

Synopsis

Usage:
J=flatteningRelation(R,A,I)
Inputs:
- R, a polynomial ring, the homogeneous coordinate ring of P^n-1
- A, a polynomial ring, the base ring of the universal apolar unfolding
- I, an ideal, the ideal generated by the unfolding equations in AR=A**R
Outputs:
- J, an ideal, the ideal in the base A, generated by the flattening relations

Description

Flatness means that the syzygies among the generators I ** A/mA in the special fiber lift to syzgies of I over A/J**R where J is the ideal of flattening relations. These can be computed with the Buchberger-Hauser test.

i1 : p=101,n=5

o1 = (101, 5)

o1 : Sequence

i2 : (R,A,I)=unfoldingEquations(p,n)

                    2    2                                           
o2 = (R, A, ideal (x  - x a         - x x a         - x x a         -
                    1    5 (1, 1),5    2 5 (1, 1),2    3 5 (1, 1),3  
     ------------------------------------------------------------------------
                            2                                           
     x x a        , x x  - x a         - x x a         - x x a         -
      4 5 (1, 1),4   1 2    5 (1, 2),5    2 5 (1, 2),2    3 5 (1, 2),3  
     ------------------------------------------------------------------------
                            2                                           
     x x a        , x x  - x a         - x x a         - x x a         -
      4 5 (1, 2),4   1 3    5 (1, 3),5    2 5 (1, 3),2    3 5 (1, 3),3  
     ------------------------------------------------------------------------
                            2                                           
     x x a        , x x  - x a         - x x a         - x x a         -
      4 5 (1, 3),4   1 4    5 (1, 4),5    2 5 (1, 4),2    3 5 (1, 4),3  
     ------------------------------------------------------------------------
                     2           2                                           
     x x a        , x  - x x  - x a         - x x a         - x x a         -
      4 5 (1, 4),4   2    1 5    5 (2, 2),5    2 5 (2, 2),2    3 5 (2, 2),3  
     ------------------------------------------------------------------------
                            2                                           
     x x a        , x x  - x a         - x x a         - x x a         -
      4 5 (2, 2),4   2 3    5 (2, 3),5    2 5 (2, 3),2    3 5 (2, 3),3  
     ------------------------------------------------------------------------
                            2                                           
     x x a        , x x  - x a         - x x a         - x x a         -
      4 5 (2, 3),4   2 4    5 (2, 4),5    2 5 (2, 4),2    3 5 (2, 4),3  
     ------------------------------------------------------------------------
                     2           2                                           
     x x a        , x  - x x  - x a         - x x a         - x x a         -
      4 5 (2, 4),4   3    1 5    5 (3, 3),5    2 5 (3, 3),2    3 5 (3, 3),3  
     ------------------------------------------------------------------------
                            2                                           
     x x a        , x x  - x a         - x x a         - x x a         -
      4 5 (3, 3),4   3 4    5 (3, 4),5    2 5 (3, 4),2    3 5 (3, 4),3  
     ------------------------------------------------------------------------
                     2           2
     x x a        , x  - x x  - x a         - x x a         - x x a         -
      4 5 (3, 4),4   4    1 5    5 (4, 4),5    2 5 (4, 4),2    3 5 (4, 4),3  
     ------------------------------------------------------------------------
     x x a        ))
      4 5 (4, 4),4

o2 : Sequence

i3 : time betti(J=flatteningRelations(R,A,I))
     -- used 0.037282 seconds

            0  1
o3 = total: 1 30
         0: 1  8
         1: . 15
         2: .  6
         3: .  1

o3 : BettiTally

Ways to use `flatteningRelations` :

flatteningRelations(PolynomialRing,PolynomialRing,Ideal)

flatteningRelations -- Compute the flattening relations in this special case

Synopsis

Description

Ways to use flatteningRelations :

Ways to use `flatteningRelations` :