# bettiBounds -- Compute the Betti bounds

## Synopsis

• Usage:
bettiBounds(L)
bettiBounds(c)
• Inputs:
• L, a list, with the Betti bounds in codimension c
• c, an integer, positive
• Outputs:
• B, a list, with the Betti bounds in codimension c+1

## Description

Computes the Betti bounds for complexes which are obtained by c-1 iterated stellar subdivisions from the boundary complex of a simplex on q+1 vertices.

The procedure can also be applied recursively to lists starting with the Betti numbers L={1,1} of a hypersurface.

The i-th entry of B corresponds to the bound of the i-th Betti number. For example, B#1 is the bound on the number of generators. The first and the last entry of B is always 1. The number of entries of B is c+1.

 ```i1 : bettiBounds(1) o1 = {1, 1} o1 : List``` ```i2 : bettiBounds(2) o2 = {1, 2, 1} o2 : List``` ```i3 : bettiBounds(3) o3 = {1, 5, 5, 1} o3 : List``` ```i4 : bettiBounds(4) o4 = {1, 11, 20, 11, 1} o4 : List``` ```i5 : bettiBounds(5) o5 = {1, 23, 62, 62, 23, 1} o5 : List```

Alternatively one can also do:

 ```i6 : L={1,1} o6 = {1, 1} o6 : List``` ```i7 : L=bettiBounds(L) o7 = {1, 2, 1} o7 : List``` ```i8 : L=bettiBounds(L) o8 = {1, 5, 5, 1} o8 : List``` ```i9 : L=bettiBounds(L) o9 = {1, 11, 20, 11, 1} o9 : List``` ```i10 : L=bettiBounds(L) o10 = {1, 23, 62, 62, 23, 1} o10 : List```