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-- 0. Service routines
R = QQ[x,y,z]
W = makeWA R
-- fill an m-by-n matrix with elements of W
-- that have monomials up to degree d
randomMatrix = (m,n,d,W) -> (
RW := (coefficientRing W) (monoid [gens W]);
toW := map(W,RW,gens W);
matrix apply(m, i->apply(n,j->(
sum(d+1, e->toW random(e,RW))
)))
)
M = randomMatrix(2,3,1,W)
M = randomMatrix(3,2,1,W)
Dprune M == prune M
-- 1. Basic invariants
-- GKZ of the twisted quartic
A = matrix{{1,1,1,1},{0,1,3,4}}
b = {1,2}
I = gkz(A,b,Vars=>Local)
Ddim I -- check it's holonomic
holonomicRank I -- holonomic rank
singLocus I -- singular locus
charIdeal I -- characteristic ideal
-- Appell F1
I = AppellF1 ({2,4,-1,3/2}, Vars=>Local)
holonomicRank I
singLocus I
charIdeal I
-- Polynomial annihilator
W = QQ[x,y,z,Dx,Dy,Dz, WeylAlgebra => {x=>Dx, y=>Dy, z=>Dz}]
f = x^3-y^2*z^2
I = PolyAnn f
holonomicRank I
singLocus I
charIdeal I
-- Rational annihilator
W = QQ[x,y,z,Dx,Dy,Dz, WeylAlgebra => {x=>Dx, y=>Dy, z=>Dz}]
f = x^3-y^2*z^2
I = RatAnn f
holonomicRank I
singLocus I
charIdeal I
f = x^2-y^3
g = y-2*z
J = RatAnn (g,f)
holonomicRank J
singLocus J
charIdeal J
-- Module routines
A = matrix{{1,1,1},{0,1,3}}
A' = matrix{{1,1,1},{0,1,4}}
b = {1,2}
b' = {2,3}
I = gkz(A,b,Vars=>Local)
I' = substitute(gkz(A',b',Vars=>Local), ring I)
M = directSum(cokernel gens I, cokernel gens I')
Ddim M
holonomicRank M
singLocus M
charIdeal M
-- 2. initial ideals and gb wrt weight vectors
A = matrix{{1,1,1},{0,2,3}}
b = {1,5}
I = gkz(A,b,Vars=>Local)
-- weight vector of the form (-u,u)
w1 = {-1,-2,-3,1,2,3}
inw(I, w1)
gbw(I, w1)
-- weight vector (u,v) with u+v > 0
w2 = {0,1,2,3,4,5}
inw(I, w2)
gbw(I, w2)
-- weight vector (u,v) with some comp's of u+v > 0, others equal to 0.
w3 = {1,-3,7,-1,4,-5}
inw(I, w3)
gbw(I, w3)
-- matrix versions
m = directSum(gens I, gens I)
w1 = {-1,-2,-3,1,2,3}
w2 = {0,1,2,3,4,5}
inw(m, w1)
gbw(m, w1)
inw(m, w2)
gbw(m, w2)
W = QQ[x, y, Dx, Dy, WeylAlgebra => {x=>Dx, y=>Dy}]
m = matrix{{x*Dx+y*Dy+x^3},{3+y^2+y*Dx+x^2*Dy}}
inw(m, {-1,-2,1,2})
gbw(m, {-1,-2,1,2})
inw(m, {0,0,1,1})
gbw(m, {0,0,1,1})
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