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##
#A toric.gd toric library David Joyner
##
## this file contains declarations for toric
##
##
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##
#F InDualCone(<v>,<L>)
##
## returns true if v belongs to the dual of the code generated by
## the vectors in L
##
DeclareGlobalFunction("InDualCone");
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##
#F DualSemigroupGenerators(<L>,<F>)
##
## Input: L is a list of integral n-vectors
## generating a cone sigma
## Output: the generators of S_sigma,
##
## Idea: let M be the max of the abs values of the
## coords of the L[i]'s, for each vector v in
## [1..M]^n, test if v in sigma^*. If so,
## add to list of possible generators.
## Once this for loop is finished, one can check
## this list for redundant generators. The
## trick below is to simply omit those elements
## which are of the form d1+d2, where d1 and d2
## are "small" elements in the integral dual cone.
##
DeclareGlobalFunction("DualSemigroupGenerators");
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##
#F EmbeddingAffineToricVariety(<L>)
##
## Input: L is a list generating a cone
## (as in dual_semigp_gens),
## Output: the toroidal embedding of X=affine_toric_variety(L)
## (given as a list of multinomials)
##
DeclareGlobalFunction("EmbeddingAffineToricVariety");
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##
#F DivisorPolytope(<D>, <Rays>)
##
## Input: D is the list of coeffs for the Weil divisor D
## Rays is the list of vectors in the rays for the fan Delta
## (Delta is determined by a list of generators L
## but these are not explicitly needed here)
## Output: the linear expressions in the affine coordinates of the
## space of the cone which must be >0 for a point to be in the
## desired polytope
##
DeclareGlobalFunction("DivisorPolytope");
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##
#F PolytopeLatticePoints(<A>, <Perps>)
##
## Input: Perps=[v1,...,vk] is the list of "inward normal"
## vectors perpendicular to the walls of the polytope P,
## A=[a1,...,ak] is the amount wall is shifted from the
## origin (ai>=0)
## Eg, x=0, x=a, y=0, y=b has Perps=[[1,0],[-1,0],[0,1],[0,-1]]
## and A=[0,a,0,b]
## Output: the list of points in P \cap L^\perp
##
DeclareGlobalFunction("PolytopeLatticePoints");
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#F DivisorPolytopeLatticePoints(<D>, <Cones>, <Rays_ordered> )
##
## Input: Cones is the fan
## Rays_ordered is the *ordered* list of rays for Cones
## D is the list of coeffs for the Weil divisor D
## Output: the list of points in P_D \cap L^\perp
## which paramterize the elements in L(D)
##
## Caution: The rays do not determine the cones in a fan.
##
DeclareGlobalFunction("DivisorPolytopeLatticePoints");
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#F RiemannRochBasis(<D>, <Cones>, <Rays> )
##
## Input: Cones is the fan
## Rays is the *ordered* list of rays for Cones
## D is the list of coeffs for the Weil divisor D
##
## Output: computes a basis (a list of monomials)
## for the RR space of the divisor represented by D
##
##
DeclareGlobalFunction("RiemannRochBasis");
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#F Faces( <Rays> )
##
## Input: Rays is a list of rays for the fan Delta
##
## Output: all the normals to the faces (hyperplanes of the cone)
##
##
DeclareGlobalFunction("Faces");
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#F InsideCone(<v>, <Rays> )
##
## Input: Rays is a list of rays for the fan Delta
##
## Output: true if v is in the cone represented by Rays
## (a cone is a list of vectors generating it),
## false otherwise
##
## Idea: look at all dim-subsets of Rays and test if
## there is a positive solns vector all the normals
## to the faces (hyperplanes of the cone)
##
##
DeclareGlobalFunction("InsideCone");
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#F NumberOfConesOfFan( <Delta>, <k> )
##
## Input: Delta is the fan of cones
## k is the dimension of the cones counted
## Output: number of k-diml cones in the fan
## (each hyperplane of one cone should not form the face
## of any other cone in the fan)
##
## Idea: the fan Delta is represented as a set of max cones.
## for each max cone, look at the k-diml faces
## obtained by taking n choose k subsets of the
## rays describing the cone.
## **Certain** of these k-subsets yield the
## desired cones
##
DeclareGlobalFunction("NumberOfConesOfFan");
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#F ConesOfFan( <Delta>, <k> )
##
## Input: Delta is the fan of cones
## k is the dimension of the cones taken
## Output: the set of k-diml cones in the fan
## (each hyperplane of one cone should not form the face
## of any other cone in the fan, ie,
## only takes those subcones which are *not* interior)
##
##
DeclareGlobalFunction("ConesOfFan");
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##
#F ToricStar( <cone>, <Cones> )
##
## Input: Cones is the fan of cones
## <cone> is a cone of <Cones>
## Output: the star of the cone in a fan
## ie, the cones tau which have <cone> as a face
##
##
DeclareGlobalFunction("ToricStar");
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#F BettiNumberToric( <Cones>, <k> )
##
## Input: Cones is the fan of cones
## k is an integer
## Output: the k-th betti number of the toric variety
## X(Delta), where Delta is a fan defined by its maximal
## cones, Cones
##
##
DeclareGlobalFunction("BettiNumberToric");
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#F EulerCharacteristic( <Cones> )
##
## Input: Cones is the fan of cones
## Output: Euler char of X(Delta), where
## Delta is a fan defined by its maximal cones, Cones
##
## Note: X(Delta) **must be non-singular**
##
## was DeclareGlobalFunction("EulerCharacteristic");
DeclareAttribute( "EulerCharacteristic", IsList );
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#F CardinalityOfToricVariety( <Cones>, <q> )
##
## Input: Cones is the fan of cones
## Output: the number of elements of X(Delta) over GF(q), where
## Delta is a fan defined by its maximal cones, Cones,
## and where q is a prime power
##
## Note: X(Delta) **must be non-singular***
##
DeclareGlobalFunction("CardinalityOfToricVariety");
