twodimToricIrrationalPoincare -- Produces the example of a two-dimensional toric ideal whose Poincar\'e series is irrational.

Synopsis

Usage:

I = twodimToricIrrationalPoincare ()
Outputs:
- I, an ideal,

Description

Returns the example from Roos and Sturmfels' "A toric ring with irrational Poincar\'e-Betti series" (1998) of a toric ideal whose quotient ring has an irrational Poincar\'e series. It is the toric ideal of the numerical subsemigroup of $\mathbb{N}^2$ generated by ${(36,0), (33,3), (30,6), (28,8), (26,10), (25,11), (24,12), (18,18), (0,36)}.$

i1 : I = onedimToricIrrationalPoincare

             2          2                       2                       3  
o1 = ideal (w  - w w , w  - w w , w w  - w w , w  - w w , w w  - w w , w  -
             2    1 6   3    2 4   3 4    1 7   4    2 5   4 5    2 6   1  
     ------------------------------------------------------------------------
            2                       2      2   2            2      2        
     w w , w  - w w , w w  - w w , w w  - w , w w  - w w , w w  - w , w w w 
      2 6   5    4 6   5 6    3 7   1 2    6   1 3    5 7   1 6    7   2 3 5
     ------------------------------------------------------------------------
                                   2      2     2              2            3
     - w w w , w w w  - w w w , w w  - w w , w w  - w w w , w w  - w w w , w 
        1 4 7   2 3 6    1 5 7   2 6    1 7   3 6    2 5 7   4 6    3 5 7   6
     ------------------------------------------------------------------------
          2
     - w w )
        2 7

o1 : Ideal of QQ[w ..w ]
                  1   7

For the programmer

The object twodimToricIrrationalPoincare is an ideal.