ToricDivisor == ToricDivisor -- equality of toric divisors

Synopsis

Operator: ==
Usage:

D1 == D2
Inputs:
- D1, a toric divisor,
- D2, a toric divisor,
Outputs:
- a Boolean value, that is true if the underlying varieties are equal and lists of coefficients are equal

Description

Two torus-invariant Weil divisors are equal when their underlying normal toric varieties are equal and, for each irreducible torus-invariant divisor, the corresponding coefficients are equal.

i1 : X = normalToricVariety(id_(ZZ^3) | -id_(ZZ^3));

i2 : D1 = toricDivisor({2,-7,3,0,7,5,8,-8}, X)

o2 = 2*X  - 7*X  + 3*X  + 7*X  + 5*X  + 8*X  - 8*X
        0      1      2      4      5      6      7

o2 : ToricDivisor on X

i3 : D2 = 2 * X_0 - 7 * X_1 + 3 * X_2 + 7 * X_4 + 5 * X_5 + 8 * X_6 - 8 * X_7

o3 = 2*X  - 7*X  + 3*X  + 7*X  + 5*X  + 8*X  - 8*X
        0      1      2      4      5      6      7

o3 : ToricDivisor on X

i4 : D1 == D2

o4 = true

i5 : D1 == - D2

o5 = false

i6 : assert (D1 == D2 and D2 == D1 and D1 =!= - D2)

Since the group of torus-equivariant Weil divisors form an abelian group, it also makes sense to compare a toric divisor with the zero integer (which we identify with the toric divisor whose coefficients are equal to zero).

i7 : D1 == 0

o7 = false

i8 : 0*D1 == 0

o8 = true

i9 : assert (D1 =!= 0 and 0*D1 == 0 and 0 == 0*D2)

Ways to use this method: