ehrhartRing -- Ehrhart ring

Synopsis

Usage:
ehrhartRing(I)
Inputs:
- an ideal, the leading monomials of the elements of the ideal are considered as generators of a lattice polytope
Outputs:
- a list, a list containing one or two ideals

Description

The exponent vectors of the leading monomials of the elements of I are considered as generators of a lattice polytope. The function returns a list of ideals:

(i) If the last ring variable is not used by the monomials, it is treated as the auxiliary variable of the Ehrhart ring. The function returns two ideals, the first containing the monomials representing the lattice points of the polytope, the second containing the generators of the Ehrhart ring.

(ii) If the last ring variable is used by the monomials, the function returns only one ideal, namely the monomials representing the lattice points of the polytope.

i1 : R=ZZ/37[x,y,t];

i2 : I=ideal(x^3, x^2*y, y^3, x*y^2);

o2 : Ideal of R

i3 : ehrhartRing(I)

              3   2      2   3           3    2        2    3
o3 = {ideal (x , x y, x*y , y ), ideal (x t, x y*t, x*y t, y t)}

o3 : List

i4 : J=I+ideal(x*y^2*t^7);

o4 : Ideal of R

i5 : ehrhartRing(J)

                2 7     2 6     2 5     2 4   2   3     2 3   2   2     2 2 
o5 = {ideal (x*y t , x*y t , x*y t , x*y t , x y*t , x*y t , x y*t , x*y t ,
     ------------------------------------------------------------------------
      2        2    3   2      2   3
     x y*t, x*y t, x , x y, x*y , y )}

o5 : List

Ways to use `ehrhartRing` :

ehrhartRing(Ideal)

ehrhartRing -- Ehrhart ring

Synopsis

Description

Ways to use ehrhartRing :

Ways to use `ehrhartRing` :