next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
BettiBounds :: champion

champion -- Compute simplicial complexes with maximal total Betti numbers

Synopsis

Description

Computes a simplicial complex of dimension q on q+c variables with maximal total Betti numbers bounds among all complexes obtained by a sequence of c-1 stellar subdivisions from the boundary complex of any simplex.

Given c the function chooses q appropriately.

This implements the construction given in Section 5 of

[6] J. Boehm, S. Papadakis: Bounds on the Betti numbers of successive stellar subdivisions of a simplex, http://arxiv.org/abs/1212.4358

i1 : champion 3

o1 = | x_3x_4x_5x_7 x_2x_4x_5x_7 x_1x_3x_5x_7 x_1x_2x_5x_7 x_1x_3x_4x_7 x_1x_2x_4x_7 x_2x_3x_5x_6 x_1x_3x_5x_6 x_1x_2x_5x_6 x_2x_3x_4x_6 x_1x_3x_4x_6 x_1x_2x_4x_6 x_2x_3x_4x_5 |

o1 : SimplicialComplex
i2 : ideal champion 3

o2 = ideal (x x x , x x x , x x x , x x x , x x )
             1 2 3   1 4 5   4 5 6   2 3 7   6 7

o2 : Ideal of QQ[x , x , x , x , x , x , x ]
                  1   2   3   4   5   6   7
i3 : betti res ideal champion 1

            0 1
o3 = total: 1 1
         0: 1 .
         1: . .
         2: . 1

o3 : BettiTally
i4 : betti res ideal champion 2

            0 1 2
o4 = total: 1 2 1
         0: 1 . .
         1: . 2 .
         2: . . 1

o4 : BettiTally
i5 : betti res ideal champion 3

            0 1 2 3
o5 = total: 1 5 5 1
         0: 1 . . .
         1: . 1 . .
         2: . 4 4 .
         3: . . 1 .
         4: . . . 1

o5 : BettiTally
i6 : betti res ideal champion 4

            0  1  2  3 4
o6 = total: 1 11 20 11 1
         0: 1  .  .  . .
         1: .  .  .  . .
         2: .  1  .  . .
         3: .  9  9  1 .
         4: .  .  2  . .
         5: .  1  9  9 .
         6: .  .  .  1 .
         7: .  .  .  . .
         8: .  .  .  . 1

o6 : BettiTally
i7 : betti res ideal champion 5

            0  1  2  3  4 5
o7 = total: 1 23 62 62 23 1
         0: 1  .  .  .  . .
         1: .  .  .  .  . .
         2: .  .  .  .  . .
         3: .  1  .  .  . .
         4: . 18 22  7  1 .
         5: .  .  2  .  . .
         6: .  3 17 14  . .
         7: .  . 14 17  3 .
         8: .  .  .  2  . .
         9: .  1  7 22 18 .
        10: .  .  .  .  1 .
        11: .  .  .  .  . .
        12: .  .  .  .  . .
        13: .  .  .  .  . 1

o7 : BettiTally

See also

Ways to use champion :

  • champion(ZZ)