Find isomorphisms between fans¶
- exception sage.geometry.fan_isomorphism.FanNotIsomorphicError[source]¶
Bases:
ExceptionException to return if there is no fan isomorphism
- sage.geometry.fan_isomorphism.fan_2d_cyclically_ordered_rays(fan)[source]¶
Return the rays of a 2-dimensional
fanin cyclic order.INPUT:
fan– a 2-dimensional fan
OUTPUT:
A
PointCollectioncontaining the rays in one particular cyclic order.EXAMPLES:
sage: rays = ((1, 1), (-1, -1), (-1, 1), (1, -1)) sage: cones = [(0,2), (2,1), (1,3), (3,0)] sage: fan = Fan(cones, rays) sage: fan.rays() N( 1, 1), N(-1, -1), N(-1, 1), N( 1, -1) in 2-d lattice N sage: from sage.geometry.fan_isomorphism import fan_2d_cyclically_ordered_rays sage: fan_2d_cyclically_ordered_rays(fan) N(-1, -1), N(-1, 1), N( 1, 1), N( 1, -1) in 2-d lattice N
>>> from sage.all import * >>> rays = ((Integer(1), Integer(1)), (-Integer(1), -Integer(1)), (-Integer(1), Integer(1)), (Integer(1), -Integer(1))) >>> cones = [(Integer(0),Integer(2)), (Integer(2),Integer(1)), (Integer(1),Integer(3)), (Integer(3),Integer(0))] >>> fan = Fan(cones, rays) >>> fan.rays() N( 1, 1), N(-1, -1), N(-1, 1), N( 1, -1) in 2-d lattice N >>> from sage.geometry.fan_isomorphism import fan_2d_cyclically_ordered_rays >>> fan_2d_cyclically_ordered_rays(fan) N(-1, -1), N(-1, 1), N( 1, 1), N( 1, -1) in 2-d lattice N
rays = ((1, 1), (-1, -1), (-1, 1), (1, -1)) cones = [(0,2), (2,1), (1,3), (3,0)] fan = Fan(cones, rays) fan.rays() from sage.geometry.fan_isomorphism import fan_2d_cyclically_ordered_rays fan_2d_cyclically_ordered_rays(fan)
- sage.geometry.fan_isomorphism.fan_2d_echelon_form(fan)[source]¶
Return echelon form of a cyclically ordered ray matrix.
INPUT:
fan– a fan
OUTPUT:
A matrix. The echelon form of the rays in one particular cyclic order.
EXAMPLES:
sage: fan = toric_varieties.P2().fan() # needs palp sage.graphs sage: from sage.geometry.fan_isomorphism import fan_2d_echelon_form sage: fan_2d_echelon_form(fan) # needs palp sage.graphs [ 1 0 -1] [ 0 1 -1]
>>> from sage.all import * >>> fan = toric_varieties.P2().fan() # needs palp sage.graphs >>> from sage.geometry.fan_isomorphism import fan_2d_echelon_form >>> fan_2d_echelon_form(fan) # needs palp sage.graphs [ 1 0 -1] [ 0 1 -1]
fan = toric_varieties.P2().fan() # needs palp sage.graphs from sage.geometry.fan_isomorphism import fan_2d_echelon_form fan_2d_echelon_form(fan) # needs palp sage.graphs
- sage.geometry.fan_isomorphism.fan_2d_echelon_forms(fan)[source]¶
Return echelon forms of all cyclically ordered ray matrices.
Note that the echelon form of the ordered ray matrices are unique up to different cyclic orderings.
INPUT:
fan– a fan
OUTPUT:
A set of matrices. The set of all echelon forms for all different cyclic orderings.
EXAMPLES:
sage: fan = toric_varieties.P2().fan() # needs palp sage.graphs sage: from sage.geometry.fan_isomorphism import fan_2d_echelon_forms sage: fan_2d_echelon_forms(fan) # needs palp sage.graphs frozenset({[ 1 0 -1] [ 0 1 -1]}) sage: fan = toric_varieties.dP7().fan() # needs palp sage.graphs sage: sorted(fan_2d_echelon_forms(fan)) # needs palp sage.graphs [ [ 1 0 -1 -1 0] [ 1 0 -1 -1 0] [ 1 0 -1 -1 1] [ 1 0 -1 0 1] [ 0 1 0 -1 -1], [ 0 1 1 0 -1], [ 0 1 1 0 -1], [ 0 1 0 -1 -1], [ 1 0 -1 0 1] [ 0 1 1 -1 -1] ]
>>> from sage.all import * >>> fan = toric_varieties.P2().fan() # needs palp sage.graphs >>> from sage.geometry.fan_isomorphism import fan_2d_echelon_forms >>> fan_2d_echelon_forms(fan) # needs palp sage.graphs frozenset({[ 1 0 -1] [ 0 1 -1]}) >>> fan = toric_varieties.dP7().fan() # needs palp sage.graphs >>> sorted(fan_2d_echelon_forms(fan)) # needs palp sage.graphs [ [ 1 0 -1 -1 0] [ 1 0 -1 -1 0] [ 1 0 -1 -1 1] [ 1 0 -1 0 1] [ 0 1 0 -1 -1], [ 0 1 1 0 -1], [ 0 1 1 0 -1], [ 0 1 0 -1 -1], <BLANKLINE> [ 1 0 -1 0 1] [ 0 1 1 -1 -1] ]
fan = toric_varieties.P2().fan() # needs palp sage.graphs from sage.geometry.fan_isomorphism import fan_2d_echelon_forms fan_2d_echelon_forms(fan) # needs palp sage.graphs fan = toric_varieties.dP7().fan() # needs palp sage.graphs sorted(fan_2d_echelon_forms(fan)) # needs palp sage.graphs
- sage.geometry.fan_isomorphism.fan_isomorphic_necessary_conditions(fan1, fan2)[source]¶
Check necessary (but not sufficient) conditions for the fans to be isomorphic.
INPUT:
fan1,fan2– two fans
OUTPUT: boolean;
Falseif the two fans cannot be isomorphic.Trueif the two fans may be isomorphic.EXAMPLES:
sage: fan1 = toric_varieties.P2().fan() # needs palp sage.graphs sage: fan2 = toric_varieties.dP8().fan() # needs palp sage.graphs sage: from sage.geometry.fan_isomorphism import fan_isomorphic_necessary_conditions sage: fan_isomorphic_necessary_conditions(fan1, fan2) # needs palp sage.graphs False
>>> from sage.all import * >>> fan1 = toric_varieties.P2().fan() # needs palp sage.graphs >>> fan2 = toric_varieties.dP8().fan() # needs palp sage.graphs >>> from sage.geometry.fan_isomorphism import fan_isomorphic_necessary_conditions >>> fan_isomorphic_necessary_conditions(fan1, fan2) # needs palp sage.graphs False
fan1 = toric_varieties.P2().fan() # needs palp sage.graphs fan2 = toric_varieties.dP8().fan() # needs palp sage.graphs from sage.geometry.fan_isomorphism import fan_isomorphic_necessary_conditions fan_isomorphic_necessary_conditions(fan1, fan2) # needs palp sage.graphs
- sage.geometry.fan_isomorphism.fan_isomorphism_generator(fan1, fan2)[source]¶
Iterate over the isomorphisms from
fan1tofan2.ALGORITHM:
The
sage.geometry.fan.Fan.vertex_graph()of the two fans is compared. For each graph isomorphism, we attempt to lift it to an actual isomorphism of fans.INPUT:
fan1,fan2– two fans
OUTPUT:
Yields the fan isomorphisms as matrices acting from the right on rays.
EXAMPLES:
sage: fan = toric_varieties.P2().fan() # needs palp sage.graphs sage: from sage.geometry.fan_isomorphism import fan_isomorphism_generator sage: sorted(fan_isomorphism_generator(fan, fan)) # needs palp sage.graphs [ [-1 -1] [-1 -1] [ 0 1] [0 1] [ 1 0] [1 0] [ 0 1], [ 1 0], [-1 -1], [1 0], [-1 -1], [0 1] ] sage: m1 = matrix([(1, 0), (0, -5), (-3, 4)]) sage: m2 = matrix([(3, 0), (1, 0), (-2, 1)]) sage: m1.elementary_divisors() == m2.elementary_divisors() == [1,1,0] True sage: fan1 = Fan([Cone([m1*vector([23, 14]), m1*vector([ 3,100])]), ....: Cone([m1*vector([-1,-14]), m1*vector([-100, -5])])]) sage: fan2 = Fan([Cone([m2*vector([23, 14]), m2*vector([ 3,100])]), ....: Cone([m2*vector([-1,-14]), m2*vector([-100, -5])])]) sage: sorted(fan_isomorphism_generator(fan1, fan2)) # needs sage.graphs [ [-12 1 -5] [ -4 0 -1] [ -5 0 -1] ] sage: m0 = identity_matrix(ZZ, 2) sage: m1 = matrix([(1, 0), (0, -5), (-3, 4)]) sage: m2 = matrix([(3, 0), (1, 0), (-2, 1)]) sage: m1.elementary_divisors() == m2.elementary_divisors() == [1,1,0] True sage: fan0 = Fan([Cone([m0*vector([1,0]), m0*vector([1,1])]), ....: Cone([m0*vector([1,1]), m0*vector([0,1])])]) sage: fan1 = Fan([Cone([m1*vector([1,0]), m1*vector([1,1])]), ....: Cone([m1*vector([1,1]), m1*vector([0,1])])]) sage: fan2 = Fan([Cone([m2*vector([1,0]), m2*vector([1,1])]), ....: Cone([m2*vector([1,1]), m2*vector([0,1])])]) sage: sorted(fan_isomorphism_generator(fan0, fan0)) # needs sage.graphs [ [0 1] [1 0] [1 0], [0 1] ] sage: sorted(fan_isomorphism_generator(fan1, fan1)) # needs sage.graphs [ [ -3 -20 28] [1 0 0] [ -1 -4 7] [0 1 0] [ -1 -5 8], [0 0 1] ] sage: sorted(fan_isomorphism_generator(fan1, fan2)) # needs sage.graphs [ [-24 -3 7] [-12 1 -5] [ -7 -1 2] [ -4 0 -1] [ -8 -1 2], [ -5 0 -1] ] sage: sorted(fan_isomorphism_generator(fan2, fan1)) # needs sage.graphs [ [ 0 1 -1] [ 0 1 -1] [ 1 -13 8] [ 2 -8 1] [ 0 -5 4], [ 1 0 -3] ]
>>> from sage.all import * >>> fan = toric_varieties.P2().fan() # needs palp sage.graphs >>> from sage.geometry.fan_isomorphism import fan_isomorphism_generator >>> sorted(fan_isomorphism_generator(fan, fan)) # needs palp sage.graphs [ [-1 -1] [-1 -1] [ 0 1] [0 1] [ 1 0] [1 0] [ 0 1], [ 1 0], [-1 -1], [1 0], [-1 -1], [0 1] ] >>> m1 = matrix([(Integer(1), Integer(0)), (Integer(0), -Integer(5)), (-Integer(3), Integer(4))]) >>> m2 = matrix([(Integer(3), Integer(0)), (Integer(1), Integer(0)), (-Integer(2), Integer(1))]) >>> m1.elementary_divisors() == m2.elementary_divisors() == [Integer(1),Integer(1),Integer(0)] True >>> fan1 = Fan([Cone([m1*vector([Integer(23), Integer(14)]), m1*vector([ Integer(3),Integer(100)])]), ... Cone([m1*vector([-Integer(1),-Integer(14)]), m1*vector([-Integer(100), -Integer(5)])])]) >>> fan2 = Fan([Cone([m2*vector([Integer(23), Integer(14)]), m2*vector([ Integer(3),Integer(100)])]), ... Cone([m2*vector([-Integer(1),-Integer(14)]), m2*vector([-Integer(100), -Integer(5)])])]) >>> sorted(fan_isomorphism_generator(fan1, fan2)) # needs sage.graphs [ [-12 1 -5] [ -4 0 -1] [ -5 0 -1] ] >>> m0 = identity_matrix(ZZ, Integer(2)) >>> m1 = matrix([(Integer(1), Integer(0)), (Integer(0), -Integer(5)), (-Integer(3), Integer(4))]) >>> m2 = matrix([(Integer(3), Integer(0)), (Integer(1), Integer(0)), (-Integer(2), Integer(1))]) >>> m1.elementary_divisors() == m2.elementary_divisors() == [Integer(1),Integer(1),Integer(0)] True >>> fan0 = Fan([Cone([m0*vector([Integer(1),Integer(0)]), m0*vector([Integer(1),Integer(1)])]), ... Cone([m0*vector([Integer(1),Integer(1)]), m0*vector([Integer(0),Integer(1)])])]) >>> fan1 = Fan([Cone([m1*vector([Integer(1),Integer(0)]), m1*vector([Integer(1),Integer(1)])]), ... Cone([m1*vector([Integer(1),Integer(1)]), m1*vector([Integer(0),Integer(1)])])]) >>> fan2 = Fan([Cone([m2*vector([Integer(1),Integer(0)]), m2*vector([Integer(1),Integer(1)])]), ... Cone([m2*vector([Integer(1),Integer(1)]), m2*vector([Integer(0),Integer(1)])])]) >>> sorted(fan_isomorphism_generator(fan0, fan0)) # needs sage.graphs [ [0 1] [1 0] [1 0], [0 1] ] >>> sorted(fan_isomorphism_generator(fan1, fan1)) # needs sage.graphs [ [ -3 -20 28] [1 0 0] [ -1 -4 7] [0 1 0] [ -1 -5 8], [0 0 1] ] >>> sorted(fan_isomorphism_generator(fan1, fan2)) # needs sage.graphs [ [-24 -3 7] [-12 1 -5] [ -7 -1 2] [ -4 0 -1] [ -8 -1 2], [ -5 0 -1] ] >>> sorted(fan_isomorphism_generator(fan2, fan1)) # needs sage.graphs [ [ 0 1 -1] [ 0 1 -1] [ 1 -13 8] [ 2 -8 1] [ 0 -5 4], [ 1 0 -3] ]
fan = toric_varieties.P2().fan() # needs palp sage.graphs from sage.geometry.fan_isomorphism import fan_isomorphism_generator sorted(fan_isomorphism_generator(fan, fan)) # needs palp sage.graphs m1 = matrix([(1, 0), (0, -5), (-3, 4)]) m2 = matrix([(3, 0), (1, 0), (-2, 1)]) m1.elementary_divisors() == m2.elementary_divisors() == [1,1,0] fan1 = Fan([Cone([m1*vector([23, 14]), m1*vector([ 3,100])]), Cone([m1*vector([-1,-14]), m1*vector([-100, -5])])]) fan2 = Fan([Cone([m2*vector([23, 14]), m2*vector([ 3,100])]), Cone([m2*vector([-1,-14]), m2*vector([-100, -5])])]) sorted(fan_isomorphism_generator(fan1, fan2)) # needs sage.graphs m0 = identity_matrix(ZZ, 2) m1 = matrix([(1, 0), (0, -5), (-3, 4)]) m2 = matrix([(3, 0), (1, 0), (-2, 1)]) m1.elementary_divisors() == m2.elementary_divisors() == [1,1,0] fan0 = Fan([Cone([m0*vector([1,0]), m0*vector([1,1])]), Cone([m0*vector([1,1]), m0*vector([0,1])])]) fan1 = Fan([Cone([m1*vector([1,0]), m1*vector([1,1])]), Cone([m1*vector([1,1]), m1*vector([0,1])])]) fan2 = Fan([Cone([m2*vector([1,0]), m2*vector([1,1])]), Cone([m2*vector([1,1]), m2*vector([0,1])])]) sorted(fan_isomorphism_generator(fan0, fan0)) # needs sage.graphs sorted(fan_isomorphism_generator(fan1, fan1)) # needs sage.graphs sorted(fan_isomorphism_generator(fan1, fan2)) # needs sage.graphs sorted(fan_isomorphism_generator(fan2, fan1)) # needs sage.graphs
- sage.geometry.fan_isomorphism.find_isomorphism(fan1, fan2, check=False)[source]¶
Find an isomorphism of the two fans.
INPUT:
fan1,fan2– two fanscheck– boolean (default:False); passed to the fan morphism constructor, seeFanMorphism()
OUTPUT:
A fan isomorphism. If the fans are not isomorphic, a
FanNotIsomorphicErroris raised.EXAMPLES:
sage: rays = ((1, 1), (0, 1), (-1, -1), (3, 1)) sage: cones = [(0,1), (1,2), (2,3), (3,0)] sage: fan1 = Fan(cones, rays) sage: m = matrix([[-2,3],[1,-1]]) sage: m.det() == -1 True sage: fan2 = Fan(cones, [vector(r)*m for r in rays]) sage: from sage.geometry.fan_isomorphism import find_isomorphism sage: find_isomorphism(fan1, fan2, check=True) # needs sage.graphs Fan morphism defined by the matrix [-2 3] [ 1 -1] Domain fan: Rational polyhedral fan in 2-d lattice N Codomain fan: Rational polyhedral fan in 2-d lattice N sage: find_isomorphism(fan1, toric_varieties.P2().fan()) # needs palp sage.graphs Traceback (most recent call last): ... FanNotIsomorphicError sage: fan1 = Fan(cones=[[1,3,4,5],[0,1,2,3],[2,3,4],[0,1,5]], ....: rays=[(-1,-1,0),(-1,-1,3),(-1,1,-1),(-1,3,-1),(0,2,-1),(1,-1,1)]) sage: fan2 = Fan(cones=[[0,2,3,5],[0,1,4,5],[0,1,2],[3,4,5]], ....: rays=[(-1,-1,-1),(-1,-1,0),(-1,1,-1),(0,2,-1),(1,-1,1),(3,-1,-1)]) sage: fan1.is_isomorphic(fan2) # needs sage.graphs True
>>> from sage.all import * >>> rays = ((Integer(1), Integer(1)), (Integer(0), Integer(1)), (-Integer(1), -Integer(1)), (Integer(3), Integer(1))) >>> cones = [(Integer(0),Integer(1)), (Integer(1),Integer(2)), (Integer(2),Integer(3)), (Integer(3),Integer(0))] >>> fan1 = Fan(cones, rays) >>> m = matrix([[-Integer(2),Integer(3)],[Integer(1),-Integer(1)]]) >>> m.det() == -Integer(1) True >>> fan2 = Fan(cones, [vector(r)*m for r in rays]) >>> from sage.geometry.fan_isomorphism import find_isomorphism >>> find_isomorphism(fan1, fan2, check=True) # needs sage.graphs Fan morphism defined by the matrix [-2 3] [ 1 -1] Domain fan: Rational polyhedral fan in 2-d lattice N Codomain fan: Rational polyhedral fan in 2-d lattice N >>> find_isomorphism(fan1, toric_varieties.P2().fan()) # needs palp sage.graphs Traceback (most recent call last): ... FanNotIsomorphicError >>> fan1 = Fan(cones=[[Integer(1),Integer(3),Integer(4),Integer(5)],[Integer(0),Integer(1),Integer(2),Integer(3)],[Integer(2),Integer(3),Integer(4)],[Integer(0),Integer(1),Integer(5)]], ... rays=[(-Integer(1),-Integer(1),Integer(0)),(-Integer(1),-Integer(1),Integer(3)),(-Integer(1),Integer(1),-Integer(1)),(-Integer(1),Integer(3),-Integer(1)),(Integer(0),Integer(2),-Integer(1)),(Integer(1),-Integer(1),Integer(1))]) >>> fan2 = Fan(cones=[[Integer(0),Integer(2),Integer(3),Integer(5)],[Integer(0),Integer(1),Integer(4),Integer(5)],[Integer(0),Integer(1),Integer(2)],[Integer(3),Integer(4),Integer(5)]], ... rays=[(-Integer(1),-Integer(1),-Integer(1)),(-Integer(1),-Integer(1),Integer(0)),(-Integer(1),Integer(1),-Integer(1)),(Integer(0),Integer(2),-Integer(1)),(Integer(1),-Integer(1),Integer(1)),(Integer(3),-Integer(1),-Integer(1))]) >>> fan1.is_isomorphic(fan2) # needs sage.graphs True
rays = ((1, 1), (0, 1), (-1, -1), (3, 1)) cones = [(0,1), (1,2), (2,3), (3,0)] fan1 = Fan(cones, rays) m = matrix([[-2,3],[1,-1]]) m.det() == -1 fan2 = Fan(cones, [vector(r)*m for r in rays]) from sage.geometry.fan_isomorphism import find_isomorphism find_isomorphism(fan1, fan2, check=True) # needs sage.graphs find_isomorphism(fan1, toric_varieties.P2().fan()) # needs palp sage.graphs fan1 = Fan(cones=[[1,3,4,5],[0,1,2,3],[2,3,4],[0,1,5]], rays=[(-1,-1,0),(-1,-1,3),(-1,1,-1),(-1,3,-1),(0,2,-1),(1,-1,1)]) fan2 = Fan(cones=[[0,2,3,5],[0,1,4,5],[0,1,2],[3,4,5]], rays=[(-1,-1,-1),(-1,-1,0),(-1,1,-1),(0,2,-1),(1,-1,1),(3,-1,-1)]) fan1.is_isomorphic(fan2) # needs sage.graphs