champion -- Compute simplicial complexes with maximal total Betti numbers

Synopsis

• Usage:
champion(c)
• Inputs:
• c, an integer, codimension, positive
• Outputs:

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```