id stringlengths 16 152 | package stringclasses 563
values | symbol stringlengths 1 125 | headline stringlengths 0 249 | usage stringlengths 0 722 | description stringclasses 11
values | example_code stringlengths 0 2.87k | example_output stringclasses 42
values | see_also stringlengths 0 1.59k | source_file stringclasses 602
values | branch stringclasses 1
value | text_for_embedding stringlengths 0 4.44k | raw_text stringlengths 14 77.1k |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
doc_48282a9c_[rightKernel,Strategy] | doc | [rightKernel,Strategy] | Right kernel of a matrix | K = rightKernel M | d1*d2
ncMatrixMult(d1,d2) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Right kernel of a matrix
USAGE: K = rightKernel M
INPUTS: M : Matrix
OUTPUTS: N : Matrix
EXAMPLE CODE:
```macaulay2
d1*d2
ncMatrixMult(d1,d2)
``` | Key
rightKernel
(rightKernel,Matrix)
[rightKernel,DegreeLimit]
[rightKernel,Strategy]
Headline
Right kernel of a matrix
Usage
K = rightKernel M
Inputs
M : Matrix
Outputs
N : Matrix
Description
Text
This function computes a minimal generating set of th... | |||
doc_64fb6584_Derivation | doc | Derivation | Derivation defined on a noncommutative algebra | delta = derivation(A,output,sigma) | A = QQ<|x,y|>
sigma = map(A,A,{y,x})
delta = derivation(A,{-x*y,y*x},sigma)
delta y^2 | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Derivation defined on a noncommutative algebra
USAGE: delta = derivation(A,output,sigma)
INPUTS: A : FreeAlgebra
or a @ TO FreeAlgebraQuotient @.
output : List
sigma : RingMap
OUTPUTS: delta : Derivation
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y|>
sigma = map(A,A,{y,x})
delta = derivation(A,{-x... | Key
Derivation
derivation
(derivation,FreeAlgebra,List)
(derivation,FreeAlgebra,List,RingMap)
(derivation,FreeAlgebraQuotient,List)
(derivation,FreeAlgebraQuotient,List,RingMap)
(symbol SPACE, Derivation, RingElement)
(symbol SPACE, Derivation, ZZ)
Headline
Derivation def... | |||
doc_64fb6584_derivation | doc | derivation | Derivation defined on a noncommutative algebra | delta = derivation(A,output,sigma) | A = QQ<|x,y|>
sigma = map(A,A,{y,x})
delta = derivation(A,{-x*y,y*x},sigma)
delta y^2 | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Derivation defined on a noncommutative algebra
USAGE: delta = derivation(A,output,sigma)
INPUTS: A : FreeAlgebra
or a @ TO FreeAlgebraQuotient @.
output : List
sigma : RingMap
OUTPUTS: delta : Derivation
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y|>
sigma = map(A,A,{y,x})
delta = derivation(A,{-x... | Key
Derivation
derivation
(derivation,FreeAlgebra,List)
(derivation,FreeAlgebra,List,RingMap)
(derivation,FreeAlgebraQuotient,List)
(derivation,FreeAlgebraQuotient,List,RingMap)
(symbol SPACE, Derivation, RingElement)
(symbol SPACE, Derivation, ZZ)
Headline
Derivation def... | |||
doc_64fb6584_(derivation,FreeAlgebra,List) | doc | (derivation,FreeAlgebra,List) | Derivation defined on a noncommutative algebra | delta = derivation(A,output,sigma) | A = QQ<|x,y|>
sigma = map(A,A,{y,x})
delta = derivation(A,{-x*y,y*x},sigma)
delta y^2 | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Derivation defined on a noncommutative algebra
USAGE: delta = derivation(A,output,sigma)
INPUTS: A : FreeAlgebra
or a @ TO FreeAlgebraQuotient @.
output : List
sigma : RingMap
OUTPUTS: delta : Derivation
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y|>
sigma = map(A,A,{y,x})
delta = derivation(A,{-x... | Key
Derivation
derivation
(derivation,FreeAlgebra,List)
(derivation,FreeAlgebra,List,RingMap)
(derivation,FreeAlgebraQuotient,List)
(derivation,FreeAlgebraQuotient,List,RingMap)
(symbol SPACE, Derivation, RingElement)
(symbol SPACE, Derivation, ZZ)
Headline
Derivation def... | |||
doc_64fb6584_(derivation,FreeAlgebra,List,RingMap) | doc | (derivation,FreeAlgebra,List,RingMap) | Derivation defined on a noncommutative algebra | delta = derivation(A,output,sigma) | A = QQ<|x,y|>
sigma = map(A,A,{y,x})
delta = derivation(A,{-x*y,y*x},sigma)
delta y^2 | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Derivation defined on a noncommutative algebra
USAGE: delta = derivation(A,output,sigma)
INPUTS: A : FreeAlgebra
or a @ TO FreeAlgebraQuotient @.
output : List
sigma : RingMap
OUTPUTS: delta : Derivation
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y|>
sigma = map(A,A,{y,x})
delta = derivation(A,{-x... | Key
Derivation
derivation
(derivation,FreeAlgebra,List)
(derivation,FreeAlgebra,List,RingMap)
(derivation,FreeAlgebraQuotient,List)
(derivation,FreeAlgebraQuotient,List,RingMap)
(symbol SPACE, Derivation, RingElement)
(symbol SPACE, Derivation, ZZ)
Headline
Derivation def... | |||
doc_64fb6584_(derivation,FreeAlgebraQuotient,List) | doc | (derivation,FreeAlgebraQuotient,List) | Derivation defined on a noncommutative algebra | delta = derivation(A,output,sigma) | A = QQ<|x,y|>
sigma = map(A,A,{y,x})
delta = derivation(A,{-x*y,y*x},sigma)
delta y^2 | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Derivation defined on a noncommutative algebra
USAGE: delta = derivation(A,output,sigma)
INPUTS: A : FreeAlgebra
or a @ TO FreeAlgebraQuotient @.
output : List
sigma : RingMap
OUTPUTS: delta : Derivation
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y|>
sigma = map(A,A,{y,x})
delta = derivation(A,{-x... | Key
Derivation
derivation
(derivation,FreeAlgebra,List)
(derivation,FreeAlgebra,List,RingMap)
(derivation,FreeAlgebraQuotient,List)
(derivation,FreeAlgebraQuotient,List,RingMap)
(symbol SPACE, Derivation, RingElement)
(symbol SPACE, Derivation, ZZ)
Headline
Derivation def... | |||
doc_64fb6584_(derivation,FreeAlgebraQuotient,List,RingMap) | doc | (derivation,FreeAlgebraQuotient,List,RingMap) | Derivation defined on a noncommutative algebra | delta = derivation(A,output,sigma) | A = QQ<|x,y|>
sigma = map(A,A,{y,x})
delta = derivation(A,{-x*y,y*x},sigma)
delta y^2 | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Derivation defined on a noncommutative algebra
USAGE: delta = derivation(A,output,sigma)
INPUTS: A : FreeAlgebra
or a @ TO FreeAlgebraQuotient @.
output : List
sigma : RingMap
OUTPUTS: delta : Derivation
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y|>
sigma = map(A,A,{y,x})
delta = derivation(A,{-x... | Key
Derivation
derivation
(derivation,FreeAlgebra,List)
(derivation,FreeAlgebra,List,RingMap)
(derivation,FreeAlgebraQuotient,List)
(derivation,FreeAlgebraQuotient,List,RingMap)
(symbol SPACE, Derivation, RingElement)
(symbol SPACE, Derivation, ZZ)
Headline
Derivation def... | |||
doc_64fb6584_(symbol_SPACE,_Derivation,_RingElement) | doc | (symbol SPACE, Derivation, RingElement) | Derivation defined on a noncommutative algebra | delta = derivation(A,output,sigma) | A = QQ<|x,y|>
sigma = map(A,A,{y,x})
delta = derivation(A,{-x*y,y*x},sigma)
delta y^2 | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Derivation defined on a noncommutative algebra
USAGE: delta = derivation(A,output,sigma)
INPUTS: A : FreeAlgebra
or a @ TO FreeAlgebraQuotient @.
output : List
sigma : RingMap
OUTPUTS: delta : Derivation
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y|>
sigma = map(A,A,{y,x})
delta = derivation(A,{-x... | Key
Derivation
derivation
(derivation,FreeAlgebra,List)
(derivation,FreeAlgebra,List,RingMap)
(derivation,FreeAlgebraQuotient,List)
(derivation,FreeAlgebraQuotient,List,RingMap)
(symbol SPACE, Derivation, RingElement)
(symbol SPACE, Derivation, ZZ)
Headline
Derivation def... | |||
doc_64fb6584_(symbol_SPACE,_Derivation,_ZZ) | doc | (symbol SPACE, Derivation, ZZ) | Derivation defined on a noncommutative algebra | delta = derivation(A,output,sigma) | A = QQ<|x,y|>
sigma = map(A,A,{y,x})
delta = derivation(A,{-x*y,y*x},sigma)
delta y^2 | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Derivation defined on a noncommutative algebra
USAGE: delta = derivation(A,output,sigma)
INPUTS: A : FreeAlgebra
or a @ TO FreeAlgebraQuotient @.
output : List
sigma : RingMap
OUTPUTS: delta : Derivation
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y|>
sigma = map(A,A,{y,x})
delta = derivation(A,{-x... | Key
Derivation
derivation
(derivation,FreeAlgebra,List)
(derivation,FreeAlgebra,List,RingMap)
(derivation,FreeAlgebraQuotient,List)
(derivation,FreeAlgebraQuotient,List,RingMap)
(symbol SPACE, Derivation, RingElement)
(symbol SPACE, Derivation, ZZ)
Headline
Derivation def... | |||
doc_a9c85626_nakayamaAut | doc | nakayamaAut | Computes the Nakayama automorphism using the superpotential | nu = nakayamaAut B | wR = superpotential(R, Strategy => "Naive")
A = ambient R
nak3 = nakayamaAut wR | superpotential | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Computes the Nakayama automorphism using the superpotential
USAGE: nu = nakayamaAut B
INPUTS: B : FreeAlgebraQuotient
w : RingElement
OUTPUTS: nu : RingMap
EXAMPLE CODE:
```macaulay2
wR = superpotential(R, Strategy => "Naive")
A = ambient R
nak3 = nakayamaAut wR
```
SEEALSO: superp... | Key
nakayamaAut
(nakayamaAut, FreeAlgebraQuotient)
(nakayamaAut, RingElement)
[nakayamaAut, Strategy]
Headline
Computes the Nakayama automorphism using the superpotential
Usage
nu = nakayamaAut B
Inputs
B : FreeAlgebraQuotient
w : RingElement
Outputs
nu : RingMap... | ||
doc_a9c85626_(nakayamaAut,_FreeAlgebraQuotient) | doc | (nakayamaAut, FreeAlgebraQuotient) | Computes the Nakayama automorphism using the superpotential | nu = nakayamaAut B | wR = superpotential(R, Strategy => "Naive")
A = ambient R
nak3 = nakayamaAut wR | superpotential | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Computes the Nakayama automorphism using the superpotential
USAGE: nu = nakayamaAut B
INPUTS: B : FreeAlgebraQuotient
w : RingElement
OUTPUTS: nu : RingMap
EXAMPLE CODE:
```macaulay2
wR = superpotential(R, Strategy => "Naive")
A = ambient R
nak3 = nakayamaAut wR
```
SEEALSO: superp... | Key
nakayamaAut
(nakayamaAut, FreeAlgebraQuotient)
(nakayamaAut, RingElement)
[nakayamaAut, Strategy]
Headline
Computes the Nakayama automorphism using the superpotential
Usage
nu = nakayamaAut B
Inputs
B : FreeAlgebraQuotient
w : RingElement
Outputs
nu : RingMap... | ||
doc_a9c85626_(nakayamaAut,_RingElement) | doc | (nakayamaAut, RingElement) | Computes the Nakayama automorphism using the superpotential | nu = nakayamaAut B | wR = superpotential(R, Strategy => "Naive")
A = ambient R
nak3 = nakayamaAut wR | superpotential | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Computes the Nakayama automorphism using the superpotential
USAGE: nu = nakayamaAut B
INPUTS: B : FreeAlgebraQuotient
w : RingElement
OUTPUTS: nu : RingMap
EXAMPLE CODE:
```macaulay2
wR = superpotential(R, Strategy => "Naive")
A = ambient R
nak3 = nakayamaAut wR
```
SEEALSO: superp... | Key
nakayamaAut
(nakayamaAut, FreeAlgebraQuotient)
(nakayamaAut, RingElement)
[nakayamaAut, Strategy]
Headline
Computes the Nakayama automorphism using the superpotential
Usage
nu = nakayamaAut B
Inputs
B : FreeAlgebraQuotient
w : RingElement
Outputs
nu : RingMap... | ||
doc_a9c85626_[nakayamaAut,_Strategy] | doc | [nakayamaAut, Strategy] | Computes the Nakayama automorphism using the superpotential | nu = nakayamaAut B | wR = superpotential(R, Strategy => "Naive")
A = ambient R
nak3 = nakayamaAut wR | superpotential | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Computes the Nakayama automorphism using the superpotential
USAGE: nu = nakayamaAut B
INPUTS: B : FreeAlgebraQuotient
w : RingElement
OUTPUTS: nu : RingMap
EXAMPLE CODE:
```macaulay2
wR = superpotential(R, Strategy => "Naive")
A = ambient R
nak3 = nakayamaAut wR
```
SEEALSO: superp... | Key
nakayamaAut
(nakayamaAut, FreeAlgebraQuotient)
(nakayamaAut, RingElement)
[nakayamaAut, Strategy]
Headline
Computes the Nakayama automorphism using the superpotential
Usage
nu = nakayamaAut B
Inputs
B : FreeAlgebraQuotient
w : RingElement
Outputs
nu : RingMap... | ||
doc_1b8918cc_superpotential | doc | superpotential | Computes the (twisted) superpotential of an m-Koszul AS regular algebra | w = superpotential B | A = kk <| u,v |>
I = ideal (u^2*v + v*u^2, v^2*u + u*v^2 )
Igb = NCGB(I, 10)
U = A/I
wU = superpotential U | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Computes the (twisted) superpotential of an m-Koszul AS regular algebra
USAGE: w = superpotential B
INPUTS: B : FreeAlgebraQuotient
OUTPUTS: w : RingElement
EXAMPLE CODE:
```macaulay2
A = kk <| u,v |>
I = ideal (u^2*v + v*u^2, v^2*u + u*v^2 )
Igb = NCGB(I, 10)
U = A/I
w... | Key
superpotential
(superpotential, FreeAlgebraQuotient)
[superpotential, Strategy]
Headline
Computes the (twisted) superpotential of an m-Koszul AS regular algebra
Usage
w = superpotential B
Inputs
B : FreeAlgebraQuotient
Outputs
w : RingElement
Description
Text
... | |||
doc_1b8918cc_(superpotential,_FreeAlgebraQuotient) | doc | (superpotential, FreeAlgebraQuotient) | Computes the (twisted) superpotential of an m-Koszul AS regular algebra | w = superpotential B | A = kk <| u,v |>
I = ideal (u^2*v + v*u^2, v^2*u + u*v^2 )
Igb = NCGB(I, 10)
U = A/I
wU = superpotential U | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Computes the (twisted) superpotential of an m-Koszul AS regular algebra
USAGE: w = superpotential B
INPUTS: B : FreeAlgebraQuotient
OUTPUTS: w : RingElement
EXAMPLE CODE:
```macaulay2
A = kk <| u,v |>
I = ideal (u^2*v + v*u^2, v^2*u + u*v^2 )
Igb = NCGB(I, 10)
U = A/I
w... | Key
superpotential
(superpotential, FreeAlgebraQuotient)
[superpotential, Strategy]
Headline
Computes the (twisted) superpotential of an m-Koszul AS regular algebra
Usage
w = superpotential B
Inputs
B : FreeAlgebraQuotient
Outputs
w : RingElement
Description
Text
... | |||
doc_1b8918cc_[superpotential,_Strategy] | doc | [superpotential, Strategy] | Computes the (twisted) superpotential of an m-Koszul AS regular algebra | w = superpotential B | A = kk <| u,v |>
I = ideal (u^2*v + v*u^2, v^2*u + u*v^2 )
Igb = NCGB(I, 10)
U = A/I
wU = superpotential U | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Computes the (twisted) superpotential of an m-Koszul AS regular algebra
USAGE: w = superpotential B
INPUTS: B : FreeAlgebraQuotient
OUTPUTS: w : RingElement
EXAMPLE CODE:
```macaulay2
A = kk <| u,v |>
I = ideal (u^2*v + v*u^2, v^2*u + u*v^2 )
Igb = NCGB(I, 10)
U = A/I
w... | Key
superpotential
(superpotential, FreeAlgebraQuotient)
[superpotential, Strategy]
Headline
Computes the (twisted) superpotential of an m-Koszul AS regular algebra
Usage
w = superpotential B
Inputs
B : FreeAlgebraQuotient
Outputs
w : RingElement
Description
Text
... | |||
doc_177a9017_derivationQuotientIdeal | doc | derivationQuotientIdeal | Computes the derivation-quotient algebra of a superpotential | w = superpotential B | kk = ZZ/32009
R = skewPolynomialRing(kk,(-1)_kk,{a,b,c,d,e})
A = ambient R
wR = superpotential R
I = derivationQuotientIdeal(wR,3)
Igb = NCGB(I,10)
R' = A/I
phi = map(R,R',gens R)
ncKernel phi | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Computes the derivation-quotient algebra of a superpotential
USAGE: w = superpotential B
INPUTS: w : RingElement
l : ZZ
OUTPUTS: B : FreeAlgebraQuotient
EXAMPLE CODE:
```macaulay2
kk = ZZ/32009
R = skewPolynomialRing(kk,(-1)_kk,{a,b,c,d,e})
A = ambient R
wR = superpotential R
... | Key
derivationQuotientIdeal
(derivationQuotientIdeal, RingElement, ZZ)
derivationQuotient
(derivationQuotient, RingElement, ZZ)
[derivationQuotient, Strategy]
Headline
Computes the derivation-quotient algebra of a superpotential
Usage
w = superpotential B
Inputs
w : Ring... | |||
doc_177a9017_(derivationQuotientIdeal,_RingElement,_ZZ) | doc | (derivationQuotientIdeal, RingElement, ZZ) | Computes the derivation-quotient algebra of a superpotential | w = superpotential B | kk = ZZ/32009
R = skewPolynomialRing(kk,(-1)_kk,{a,b,c,d,e})
A = ambient R
wR = superpotential R
I = derivationQuotientIdeal(wR,3)
Igb = NCGB(I,10)
R' = A/I
phi = map(R,R',gens R)
ncKernel phi | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Computes the derivation-quotient algebra of a superpotential
USAGE: w = superpotential B
INPUTS: w : RingElement
l : ZZ
OUTPUTS: B : FreeAlgebraQuotient
EXAMPLE CODE:
```macaulay2
kk = ZZ/32009
R = skewPolynomialRing(kk,(-1)_kk,{a,b,c,d,e})
A = ambient R
wR = superpotential R
... | Key
derivationQuotientIdeal
(derivationQuotientIdeal, RingElement, ZZ)
derivationQuotient
(derivationQuotient, RingElement, ZZ)
[derivationQuotient, Strategy]
Headline
Computes the derivation-quotient algebra of a superpotential
Usage
w = superpotential B
Inputs
w : Ring... | |||
doc_177a9017_derivationQuotient | doc | derivationQuotient | Computes the derivation-quotient algebra of a superpotential | w = superpotential B | kk = ZZ/32009
R = skewPolynomialRing(kk,(-1)_kk,{a,b,c,d,e})
A = ambient R
wR = superpotential R
I = derivationQuotientIdeal(wR,3)
Igb = NCGB(I,10)
R' = A/I
phi = map(R,R',gens R)
ncKernel phi | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Computes the derivation-quotient algebra of a superpotential
USAGE: w = superpotential B
INPUTS: w : RingElement
l : ZZ
OUTPUTS: B : FreeAlgebraQuotient
EXAMPLE CODE:
```macaulay2
kk = ZZ/32009
R = skewPolynomialRing(kk,(-1)_kk,{a,b,c,d,e})
A = ambient R
wR = superpotential R
... | Key
derivationQuotientIdeal
(derivationQuotientIdeal, RingElement, ZZ)
derivationQuotient
(derivationQuotient, RingElement, ZZ)
[derivationQuotient, Strategy]
Headline
Computes the derivation-quotient algebra of a superpotential
Usage
w = superpotential B
Inputs
w : Ring... | |||
doc_177a9017_(derivationQuotient,_RingElement,_ZZ) | doc | (derivationQuotient, RingElement, ZZ) | Computes the derivation-quotient algebra of a superpotential | w = superpotential B | kk = ZZ/32009
R = skewPolynomialRing(kk,(-1)_kk,{a,b,c,d,e})
A = ambient R
wR = superpotential R
I = derivationQuotientIdeal(wR,3)
Igb = NCGB(I,10)
R' = A/I
phi = map(R,R',gens R)
ncKernel phi | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Computes the derivation-quotient algebra of a superpotential
USAGE: w = superpotential B
INPUTS: w : RingElement
l : ZZ
OUTPUTS: B : FreeAlgebraQuotient
EXAMPLE CODE:
```macaulay2
kk = ZZ/32009
R = skewPolynomialRing(kk,(-1)_kk,{a,b,c,d,e})
A = ambient R
wR = superpotential R
... | Key
derivationQuotientIdeal
(derivationQuotientIdeal, RingElement, ZZ)
derivationQuotient
(derivationQuotient, RingElement, ZZ)
[derivationQuotient, Strategy]
Headline
Computes the derivation-quotient algebra of a superpotential
Usage
w = superpotential B
Inputs
w : Ring... | |||
doc_177a9017_[derivationQuotient,_Strategy] | doc | [derivationQuotient, Strategy] | Computes the derivation-quotient algebra of a superpotential | w = superpotential B | kk = ZZ/32009
R = skewPolynomialRing(kk,(-1)_kk,{a,b,c,d,e})
A = ambient R
wR = superpotential R
I = derivationQuotientIdeal(wR,3)
Igb = NCGB(I,10)
R' = A/I
phi = map(R,R',gens R)
ncKernel phi | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Computes the derivation-quotient algebra of a superpotential
USAGE: w = superpotential B
INPUTS: w : RingElement
l : ZZ
OUTPUTS: B : FreeAlgebraQuotient
EXAMPLE CODE:
```macaulay2
kk = ZZ/32009
R = skewPolynomialRing(kk,(-1)_kk,{a,b,c,d,e})
A = ambient R
wR = superpotential R
... | Key
derivationQuotientIdeal
(derivationQuotientIdeal, RingElement, ZZ)
derivationQuotient
(derivationQuotient, RingElement, ZZ)
[derivationQuotient, Strategy]
Headline
Computes the derivation-quotient algebra of a superpotential
Usage
w = superpotential B
Inputs
w : Ring... | |||
BGG_4db34818_directImageComplex | BGG | directImageComplex | direct image complex | directImageComplex F | M2_git/M2/M2/Macaulay2/packages/BGG.m2 | stable | HEADLINE: direct image complex
USAGE: directImageComplex F | Key
directImageComplex
[directImageComplex, Regularity]
Headline
direct image complex
Usage
directImageComplex F
Description
Text
Forms a minimal free complex representing the direct image complex of $F$ in the
derived category, where $F$ is a module, chain complex or map of... | ||||
BGG_4db34818_[directImageComplex,_Regularity] | BGG | [directImageComplex, Regularity] | direct image complex | directImageComplex F | M2_git/M2/M2/Macaulay2/packages/BGG.m2 | stable | HEADLINE: direct image complex
USAGE: directImageComplex F | Key
directImageComplex
[directImageComplex, Regularity]
Headline
direct image complex
Usage
directImageComplex F
Description
Text
Forms a minimal free complex representing the direct image complex of $F$ in the
derived category, where $F$ is a module, chain complex or map of... | ||||
BGG_3d1a0bd8_Regularity | BGG | Regularity | Option for directImageComplex
Caveat
Currently not supported | M2_git/M2/M2/Macaulay2/packages/BGG.m2 | stable | HEADLINE: Option for directImageComplex
Caveat
Currently not supported | Key
Regularity
Headline
Option for directImageComplex
Caveat
Currently not supported | |||||
BGG_9bf32b4b_Exterior | BGG | Exterior | dual exterior algebra cached in a polynomial ring | M2_git/M2/M2/Macaulay2/packages/BGG.m2 | stable | HEADLINE: dual exterior algebra cached in a polynomial ring | Key
Exterior
Headline
dual exterior algebra cached in a polynomial ring
Description
Text
checked for, and possibly installed, by "symmetricToExterior" | |||||
BGG_b071de35_(directImageComplex,_Module) | BGG | (directImageComplex, Module) | Complex representing the direct image | F = directImageComplex M | r=3;
d=6;
M=universalExtension(splice {(r-1):0}, {d})
S = ring M;
A = coefficientRing S;
L = for i from -d to 0 list directImageComplex(S^{{i,0}}**M);
netList L
maps = apply(L, F-> F.dd_0) | universalExtension
(directImageComplex, Matrix) | M2_git/M2/M2/Macaulay2/packages/BGG.m2 | stable | HEADLINE: Complex representing the direct image
USAGE: F = directImageComplex M
INPUTS: M: Module
graded over a ring of the form S = A[y_0..y_n], representing a sheaf on $\PP^n_A$,
where A is a polynomial ring.
Regularity=>ZZ
the Castelnuovo-Mumford regularity of {\tt M}. If not provided, t... | Key
(directImageComplex, Module)
Headline
Complex representing the direct image
Usage
F = directImageComplex M
Inputs
M: Module
graded over a ring of the form S = A[y_0..y_n], representing a sheaf on $\PP^n_A$,
where A is a polynomial ring.
Regularity=>ZZ
the Ca... | ||
BGG_648c3ced_(directImageComplex,_Matrix) | BGG | (directImageComplex, Matrix) | map of direct image complexes | piF = directImageComplex F | m = transpose D_(-1)
betti freeResolution coker m
(dual oo)[-3] | directImageComplex
pureResolution
universalExtension | M2_git/M2/M2/Macaulay2/packages/BGG.m2 | stable | HEADLINE: map of direct image complexes
USAGE: piF = directImageComplex F
INPUTS: F:Matrix
a homomorphism $F : M \rightarrow N$ of graded $S = A[y_0..y_n]$ modules, graded of degree 0,
where we think of $M$ and $N$ as representing sheaves on $\PP^n_A$
OUTPUTS: piF:ComplexMap
the induced map on c... | Key
(directImageComplex, Matrix)
Headline
map of direct image complexes
Usage
piF = directImageComplex F
Inputs
F:Matrix
a homomorphism $F : M \rightarrow N$ of graded $S = A[y_0..y_n]$ modules, graded of degree 0,
where we think of $M$ and $N$ as representing sheaves on $\PP^... | ||
BGG_2ded08a8_(directImageComplex,_Complex) | BGG | (directImageComplex, Complex) | direct image of a chain complex | piC = directImageComplex C | (p,q) = (2,5) -- number of rows and columns
A=ZZ/101[a_(0,0)..a_(p-1,q-1)];
S = A [x_0..x_(p-1)];
M = sub(map(A^p, A^{q:-1},transpose genericMatrix(A,a_(0,0),q,p)), S)
Y = map(S^1, S^{q:{-1,-1}}, (vars S)*M)
F = koszulComplex Y
L = for i from -1 to q-p+1 list directImageComplex(F**S^... | directImageComplex | M2_git/M2/M2/Macaulay2/packages/BGG.m2 | stable | HEADLINE: direct image of a chain complex
USAGE: piC = directImageComplex C
INPUTS: C:Complex
over a bigraded ring $S = A[y_0..y_n]$, where $A$ is singly graded
OUTPUTS: piC:Complex
a minimal free complex representing the direct image complex in the derived category
EXAMPLE CODE:
```macaulay2
(p,q) = ... | Key
(directImageComplex, Complex)
Headline
direct image of a chain complex
Usage
piC = directImageComplex C
Inputs
C:Complex
over a bigraded ring $S = A[y_0..y_n]$, where $A$ is singly graded
Outputs
piC:Complex
a minimal free complex representing the direct image comp... | ||
BGG_13663a15_universalExtension | BGG | universalExtension | Universal extension of vector bundles on P^1 | E = universalExtension(La, Lb) | M = universalExtension({-2}, {2})
M = universalExtension({-2,-3}, {2,3}) | directImageComplex | M2_git/M2/M2/Macaulay2/packages/BGG.m2 | stable | HEADLINE: Universal extension of vector bundles on P^1
USAGE: E = universalExtension(La, Lb)
INPUTS: La: List
of integers
Lb: List
of integers
OUTPUTS: E: Module
representing the extension
EXAMPLE CODE:
```macaulay2
M = universalExtension({-2}, {2})
M = universalExtension({-2,-3}, ... | Key
universalExtension
(universalExtension, List, List)
Headline
Universal extension of vector bundles on P^1
Usage
E = universalExtension(La, Lb)
Inputs
La: List
of integers
Lb: List
of integers
Outputs
E: Module
representing the extension
Des... | ||
BGG_13663a15_(universalExtension,_List,_List) | BGG | (universalExtension, List, List) | Universal extension of vector bundles on P^1 | E = universalExtension(La, Lb) | M = universalExtension({-2}, {2})
M = universalExtension({-2,-3}, {2,3}) | directImageComplex | M2_git/M2/M2/Macaulay2/packages/BGG.m2 | stable | HEADLINE: Universal extension of vector bundles on P^1
USAGE: E = universalExtension(La, Lb)
INPUTS: La: List
of integers
Lb: List
of integers
OUTPUTS: E: Module
representing the extension
EXAMPLE CODE:
```macaulay2
M = universalExtension({-2}, {2})
M = universalExtension({-2,-3}, ... | Key
universalExtension
(universalExtension, List, List)
Headline
Universal extension of vector bundles on P^1
Usage
E = universalExtension(La, Lb)
Inputs
La: List
of integers
Lb: List
of integers
Outputs
E: Module
representing the extension
Des... | ||
BGG_bc31ef5a_pureResolution | BGG | pureResolution | creates a pure resolution as an iterated direct image | F = pureResolution(A, D)
F = pureResolution(M, D)
F = pureResolution(p, D)
F = pureResolution(p,q,D) | time betti (F = pureResolution(11,4,{0,2,4}))
ring F | directImageComplex
universalExtension | M2_git/M2/M2/Macaulay2/packages/BGG.m2 | stable | HEADLINE: creates a pure resolution as an iterated direct image
USAGE: F = pureResolution(A, D)
F = pureResolution(M, D)
F = pureResolution(p, D)
F = pureResolution(p,q,D)
INPUTS: A:Ring
D:List
M:Matrix
D is a strictly increasing list of integers, the degree sequence
A is a rin... | Key
pureResolution
(pureResolution, Ring, List)
(pureResolution, Matrix, List)
(pureResolution, ZZ, List)
(pureResolution, ZZ, ZZ, List)
Headline
creates a pure resolution as an iterated direct image
Usage
F = pureResolution(A, D)
F = pureResolution(M, D)
F =... | ||
BGG_bc31ef5a_(pureResolution,_Ring,_List) | BGG | (pureResolution, Ring, List) | creates a pure resolution as an iterated direct image | F = pureResolution(A, D)
F = pureResolution(M, D)
F = pureResolution(p, D)
F = pureResolution(p,q,D) | time betti (F = pureResolution(11,4,{0,2,4}))
ring F | directImageComplex
universalExtension | M2_git/M2/M2/Macaulay2/packages/BGG.m2 | stable | HEADLINE: creates a pure resolution as an iterated direct image
USAGE: F = pureResolution(A, D)
F = pureResolution(M, D)
F = pureResolution(p, D)
F = pureResolution(p,q,D)
INPUTS: A:Ring
D:List
M:Matrix
D is a strictly increasing list of integers, the degree sequence
A is a rin... | Key
pureResolution
(pureResolution, Ring, List)
(pureResolution, Matrix, List)
(pureResolution, ZZ, List)
(pureResolution, ZZ, ZZ, List)
Headline
creates a pure resolution as an iterated direct image
Usage
F = pureResolution(A, D)
F = pureResolution(M, D)
F =... | ||
BGG_bc31ef5a_(pureResolution,_Matrix,_List) | BGG | (pureResolution, Matrix, List) | creates a pure resolution as an iterated direct image | F = pureResolution(A, D)
F = pureResolution(M, D)
F = pureResolution(p, D)
F = pureResolution(p,q,D) | time betti (F = pureResolution(11,4,{0,2,4}))
ring F | directImageComplex
universalExtension | M2_git/M2/M2/Macaulay2/packages/BGG.m2 | stable | HEADLINE: creates a pure resolution as an iterated direct image
USAGE: F = pureResolution(A, D)
F = pureResolution(M, D)
F = pureResolution(p, D)
F = pureResolution(p,q,D)
INPUTS: A:Ring
D:List
M:Matrix
D is a strictly increasing list of integers, the degree sequence
A is a rin... | Key
pureResolution
(pureResolution, Ring, List)
(pureResolution, Matrix, List)
(pureResolution, ZZ, List)
(pureResolution, ZZ, ZZ, List)
Headline
creates a pure resolution as an iterated direct image
Usage
F = pureResolution(A, D)
F = pureResolution(M, D)
F =... | ||
BGG_bc31ef5a_(pureResolution,_ZZ,_List) | BGG | (pureResolution, ZZ, List) | creates a pure resolution as an iterated direct image | F = pureResolution(A, D)
F = pureResolution(M, D)
F = pureResolution(p, D)
F = pureResolution(p,q,D) | time betti (F = pureResolution(11,4,{0,2,4}))
ring F | directImageComplex
universalExtension | M2_git/M2/M2/Macaulay2/packages/BGG.m2 | stable | HEADLINE: creates a pure resolution as an iterated direct image
USAGE: F = pureResolution(A, D)
F = pureResolution(M, D)
F = pureResolution(p, D)
F = pureResolution(p,q,D)
INPUTS: A:Ring
D:List
M:Matrix
D is a strictly increasing list of integers, the degree sequence
A is a rin... | Key
pureResolution
(pureResolution, Ring, List)
(pureResolution, Matrix, List)
(pureResolution, ZZ, List)
(pureResolution, ZZ, ZZ, List)
Headline
creates a pure resolution as an iterated direct image
Usage
F = pureResolution(A, D)
F = pureResolution(M, D)
F =... | ||
BGG_bc31ef5a_(pureResolution,_ZZ,_ZZ,_List) | BGG | (pureResolution, ZZ, ZZ, List) | creates a pure resolution as an iterated direct image | F = pureResolution(A, D)
F = pureResolution(M, D)
F = pureResolution(p, D)
F = pureResolution(p,q,D) | time betti (F = pureResolution(11,4,{0,2,4}))
ring F | directImageComplex
universalExtension | M2_git/M2/M2/Macaulay2/packages/BGG.m2 | stable | HEADLINE: creates a pure resolution as an iterated direct image
USAGE: F = pureResolution(A, D)
F = pureResolution(M, D)
F = pureResolution(p, D)
F = pureResolution(p,q,D)
INPUTS: A:Ring
D:List
M:Matrix
D is a strictly increasing list of integers, the degree sequence
A is a rin... | Key
pureResolution
(pureResolution, Ring, List)
(pureResolution, Matrix, List)
(pureResolution, ZZ, List)
(pureResolution, ZZ, ZZ, List)
Headline
creates a pure resolution as an iterated direct image
Usage
F = pureResolution(A, D)
F = pureResolution(M, D)
F =... | ||
BGG_0e565d18_projectiveProduct | BGG | projectiveProduct | Makes a product of projective spaces and a system of parameters | (S, params) = projectiveProduct(A,dimList) | pureResolution | M2_git/M2/M2/Macaulay2/packages/BGG.m2 | stable | HEADLINE: Makes a product of projective spaces and a system of parameters
USAGE: (S, params) = projectiveProduct(A,dimList)
INPUTS: A: Ring
dimList: List
A the desired base ring. dimList a list of dimensions {d1..dn}
OUTPUTS: S: Ring
params: Matrix
S is the iterated tower ring A[x_(0,0)..x_(0... | Key
projectiveProduct
(projectiveProduct, Ring, List)
(projectiveProduct, Matrix, List)
Headline
Makes a product of projective spaces and a system of parameters
Usage
(S, params) = projectiveProduct(A,dimList)
Inputs
A: Ring
dimList: List
A the desired base ring. dim... | |||
BGG_0e565d18_(projectiveProduct,_Ring,_List) | BGG | (projectiveProduct, Ring, List) | Makes a product of projective spaces and a system of parameters | (S, params) = projectiveProduct(A,dimList) | pureResolution | M2_git/M2/M2/Macaulay2/packages/BGG.m2 | stable | HEADLINE: Makes a product of projective spaces and a system of parameters
USAGE: (S, params) = projectiveProduct(A,dimList)
INPUTS: A: Ring
dimList: List
A the desired base ring. dimList a list of dimensions {d1..dn}
OUTPUTS: S: Ring
params: Matrix
S is the iterated tower ring A[x_(0,0)..x_(0... | Key
projectiveProduct
(projectiveProduct, Ring, List)
(projectiveProduct, Matrix, List)
Headline
Makes a product of projective spaces and a system of parameters
Usage
(S, params) = projectiveProduct(A,dimList)
Inputs
A: Ring
dimList: List
A the desired base ring. dim... | |||
BGG_0e565d18_(projectiveProduct,_Matrix,_List) | BGG | (projectiveProduct, Matrix, List) | Makes a product of projective spaces and a system of parameters | (S, params) = projectiveProduct(A,dimList) | pureResolution | M2_git/M2/M2/Macaulay2/packages/BGG.m2 | stable | HEADLINE: Makes a product of projective spaces and a system of parameters
USAGE: (S, params) = projectiveProduct(A,dimList)
INPUTS: A: Ring
dimList: List
A the desired base ring. dimList a list of dimensions {d1..dn}
OUTPUTS: S: Ring
params: Matrix
S is the iterated tower ring A[x_(0,0)..x_(0... | Key
projectiveProduct
(projectiveProduct, Ring, List)
(projectiveProduct, Matrix, List)
Headline
Makes a product of projective spaces and a system of parameters
Usage
(S, params) = projectiveProduct(A,dimList)
Inputs
A: Ring
dimList: List
A the desired base ring. dim... | |||
Benchmark_96d80862_runBenchmarks | Benchmark | runBenchmarks | standard Macaulay2 benchmarks | runBenchmarks
runBenchmarks x
runBenchmarks all | runBenchmarks "res39" | M2_git/M2/M2/Macaulay2/packages/Benchmark.m2 | stable | HEADLINE: standard Macaulay2 benchmarks
USAGE: runBenchmarks
runBenchmarks x
runBenchmarks all
INPUTS: x:
a string or list of strings
Consequences
Item
the benchmarks whose names occur in @ TT "x" @ are run. The output is in a standard
format that can be inserted into a Macaulay2 source file as a com... | Node
Key
Benchmark
Headline
standard Macaulay2 benchmarks
Description
Text
This package provides some standard benchmarks for gauging the speed of
algorithms and machines.
Node
Key
runBenchmarks
Usage
runBenchmarks
runBenchmarks x
runBenchmarks all
Inputs
x:
a string or list of strings
C... | |||
DHom_51c85456_polynomialSolutions | DHom | polynomialSolutions | polynomial solutions of a holonomic system | polynomialSolutions I
polynomialSolutions M
polynomialSolutions(I,w)
polynomialSolutions(M,w) | makeWA(QQ[x])
I = ideal(dx^2, (x-1)*dx-1)
polynomialSolutions I | rationalFunctionSolutions
Dintegration | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/DHom.m2 | stable | HEADLINE: polynomial solutions of a holonomic system
USAGE: polynomialSolutions I
polynomialSolutions M
polynomialSolutions(I,w)
polynomialSolutions(M,w)
INPUTS: M:Module
over the Weyl algebra $D$
I:Ideal
holonomic ideal in the Weyl algebra $D$
w:List
a weight vector
OUTPUTS: :L... | Key
polynomialSolutions
(polynomialSolutions,Module)
(polynomialSolutions,Ideal,List)
(polynomialSolutions,Module,List)
(polynomialSolutions,Ideal)
Headline
polynomial solutions of a holonomic system
Usage
polynomialSolutions I
polynomialSolutions M
polynomialSolutions(I,w)
p... | ||
DHom_51c85456_(polynomialSolutions,Module) | DHom | (polynomialSolutions,Module) | polynomial solutions of a holonomic system | polynomialSolutions I
polynomialSolutions M
polynomialSolutions(I,w)
polynomialSolutions(M,w) | makeWA(QQ[x])
I = ideal(dx^2, (x-1)*dx-1)
polynomialSolutions I | rationalFunctionSolutions
Dintegration | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/DHom.m2 | stable | HEADLINE: polynomial solutions of a holonomic system
USAGE: polynomialSolutions I
polynomialSolutions M
polynomialSolutions(I,w)
polynomialSolutions(M,w)
INPUTS: M:Module
over the Weyl algebra $D$
I:Ideal
holonomic ideal in the Weyl algebra $D$
w:List
a weight vector
OUTPUTS: :L... | Key
polynomialSolutions
(polynomialSolutions,Module)
(polynomialSolutions,Ideal,List)
(polynomialSolutions,Module,List)
(polynomialSolutions,Ideal)
Headline
polynomial solutions of a holonomic system
Usage
polynomialSolutions I
polynomialSolutions M
polynomialSolutions(I,w)
p... | ||
DHom_51c85456_(polynomialSolutions,Ideal,List) | DHom | (polynomialSolutions,Ideal,List) | polynomial solutions of a holonomic system | polynomialSolutions I
polynomialSolutions M
polynomialSolutions(I,w)
polynomialSolutions(M,w) | makeWA(QQ[x])
I = ideal(dx^2, (x-1)*dx-1)
polynomialSolutions I | rationalFunctionSolutions
Dintegration | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/DHom.m2 | stable | HEADLINE: polynomial solutions of a holonomic system
USAGE: polynomialSolutions I
polynomialSolutions M
polynomialSolutions(I,w)
polynomialSolutions(M,w)
INPUTS: M:Module
over the Weyl algebra $D$
I:Ideal
holonomic ideal in the Weyl algebra $D$
w:List
a weight vector
OUTPUTS: :L... | Key
polynomialSolutions
(polynomialSolutions,Module)
(polynomialSolutions,Ideal,List)
(polynomialSolutions,Module,List)
(polynomialSolutions,Ideal)
Headline
polynomial solutions of a holonomic system
Usage
polynomialSolutions I
polynomialSolutions M
polynomialSolutions(I,w)
p... | ||
DHom_51c85456_(polynomialSolutions,Module,List) | DHom | (polynomialSolutions,Module,List) | polynomial solutions of a holonomic system | polynomialSolutions I
polynomialSolutions M
polynomialSolutions(I,w)
polynomialSolutions(M,w) | makeWA(QQ[x])
I = ideal(dx^2, (x-1)*dx-1)
polynomialSolutions I | rationalFunctionSolutions
Dintegration | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/DHom.m2 | stable | HEADLINE: polynomial solutions of a holonomic system
USAGE: polynomialSolutions I
polynomialSolutions M
polynomialSolutions(I,w)
polynomialSolutions(M,w)
INPUTS: M:Module
over the Weyl algebra $D$
I:Ideal
holonomic ideal in the Weyl algebra $D$
w:List
a weight vector
OUTPUTS: :L... | Key
polynomialSolutions
(polynomialSolutions,Module)
(polynomialSolutions,Ideal,List)
(polynomialSolutions,Module,List)
(polynomialSolutions,Ideal)
Headline
polynomial solutions of a holonomic system
Usage
polynomialSolutions I
polynomialSolutions M
polynomialSolutions(I,w)
p... | ||
DHom_51c85456_(polynomialSolutions,Ideal) | DHom | (polynomialSolutions,Ideal) | polynomial solutions of a holonomic system | polynomialSolutions I
polynomialSolutions M
polynomialSolutions(I,w)
polynomialSolutions(M,w) | makeWA(QQ[x])
I = ideal(dx^2, (x-1)*dx-1)
polynomialSolutions I | rationalFunctionSolutions
Dintegration | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/DHom.m2 | stable | HEADLINE: polynomial solutions of a holonomic system
USAGE: polynomialSolutions I
polynomialSolutions M
polynomialSolutions(I,w)
polynomialSolutions(M,w)
INPUTS: M:Module
over the Weyl algebra $D$
I:Ideal
holonomic ideal in the Weyl algebra $D$
w:List
a weight vector
OUTPUTS: :L... | Key
polynomialSolutions
(polynomialSolutions,Module)
(polynomialSolutions,Ideal,List)
(polynomialSolutions,Module,List)
(polynomialSolutions,Ideal)
Headline
polynomial solutions of a holonomic system
Usage
polynomialSolutions I
polynomialSolutions M
polynomialSolutions(I,w)
p... | ||
DHom_b2813d12_rationalFunctionSolutions | DHom | rationalFunctionSolutions | rational solutions of a holonomic system | rationalFunctionSolutions I
rationalFunctionSolutions(I,f)
rationalFunctionSolutions(I,f,w)
rationalFunctionSolutions(I,ff)
rationalFunctionSolutions(I,ff,w) | makeWA(QQ[x])
I = ideal((x+1)*dx+5)
rationalFunctionSolutions I
Caveat
The most efficient method to find rational solutions of a system of differential
equations is
to find the singular locus, then try to find its irreducible
factors. With these, call rationalFunctionSolutions(I, ff, w), ... | polynomialSolutions
rationalFunctionExt
DHom | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/DHom.m2 | stable | HEADLINE: rational solutions of a holonomic system
USAGE: rationalFunctionSolutions I
rationalFunctionSolutions(I,f)
rationalFunctionSolutions(I,f,w)
rationalFunctionSolutions(I,ff)
rationalFunctionSolutions(I,ff,w)
INPUTS: I:Ideal
holonomic ideal in the Weyl algebra @EM "D"@
f:RingElement
... | Key
rationalFunctionSolutions
(rationalFunctionSolutions,Ideal,List,List)
(rationalFunctionSolutions,Ideal,RingElement,List)
(rationalFunctionSolutions,Ideal,List)
(rationalFunctionSolutions,Ideal,RingElement)
(rationalFunctionSolutions,Ideal)
Headline
rational solutions of a holonomic sys... | ||
DHom_b2813d12_(rationalFunctionSolutions,Ideal,List,List) | DHom | (rationalFunctionSolutions,Ideal,List,List) | rational solutions of a holonomic system | rationalFunctionSolutions I
rationalFunctionSolutions(I,f)
rationalFunctionSolutions(I,f,w)
rationalFunctionSolutions(I,ff)
rationalFunctionSolutions(I,ff,w) | makeWA(QQ[x])
I = ideal((x+1)*dx+5)
rationalFunctionSolutions I
Caveat
The most efficient method to find rational solutions of a system of differential
equations is
to find the singular locus, then try to find its irreducible
factors. With these, call rationalFunctionSolutions(I, ff, w), ... | polynomialSolutions
rationalFunctionExt
DHom | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/DHom.m2 | stable | HEADLINE: rational solutions of a holonomic system
USAGE: rationalFunctionSolutions I
rationalFunctionSolutions(I,f)
rationalFunctionSolutions(I,f,w)
rationalFunctionSolutions(I,ff)
rationalFunctionSolutions(I,ff,w)
INPUTS: I:Ideal
holonomic ideal in the Weyl algebra @EM "D"@
f:RingElement
... | Key
rationalFunctionSolutions
(rationalFunctionSolutions,Ideal,List,List)
(rationalFunctionSolutions,Ideal,RingElement,List)
(rationalFunctionSolutions,Ideal,List)
(rationalFunctionSolutions,Ideal,RingElement)
(rationalFunctionSolutions,Ideal)
Headline
rational solutions of a holonomic sys... | ||
DHom_b2813d12_(rationalFunctionSolutions,Ideal,RingElement,List) | DHom | (rationalFunctionSolutions,Ideal,RingElement,List) | rational solutions of a holonomic system | rationalFunctionSolutions I
rationalFunctionSolutions(I,f)
rationalFunctionSolutions(I,f,w)
rationalFunctionSolutions(I,ff)
rationalFunctionSolutions(I,ff,w) | makeWA(QQ[x])
I = ideal((x+1)*dx+5)
rationalFunctionSolutions I
Caveat
The most efficient method to find rational solutions of a system of differential
equations is
to find the singular locus, then try to find its irreducible
factors. With these, call rationalFunctionSolutions(I, ff, w), ... | polynomialSolutions
rationalFunctionExt
DHom | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/DHom.m2 | stable | HEADLINE: rational solutions of a holonomic system
USAGE: rationalFunctionSolutions I
rationalFunctionSolutions(I,f)
rationalFunctionSolutions(I,f,w)
rationalFunctionSolutions(I,ff)
rationalFunctionSolutions(I,ff,w)
INPUTS: I:Ideal
holonomic ideal in the Weyl algebra @EM "D"@
f:RingElement
... | Key
rationalFunctionSolutions
(rationalFunctionSolutions,Ideal,List,List)
(rationalFunctionSolutions,Ideal,RingElement,List)
(rationalFunctionSolutions,Ideal,List)
(rationalFunctionSolutions,Ideal,RingElement)
(rationalFunctionSolutions,Ideal)
Headline
rational solutions of a holonomic sys... | ||
DHom_b2813d12_(rationalFunctionSolutions,Ideal,List) | DHom | (rationalFunctionSolutions,Ideal,List) | rational solutions of a holonomic system | rationalFunctionSolutions I
rationalFunctionSolutions(I,f)
rationalFunctionSolutions(I,f,w)
rationalFunctionSolutions(I,ff)
rationalFunctionSolutions(I,ff,w) | makeWA(QQ[x])
I = ideal((x+1)*dx+5)
rationalFunctionSolutions I
Caveat
The most efficient method to find rational solutions of a system of differential
equations is
to find the singular locus, then try to find its irreducible
factors. With these, call rationalFunctionSolutions(I, ff, w), ... | polynomialSolutions
rationalFunctionExt
DHom | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/DHom.m2 | stable | HEADLINE: rational solutions of a holonomic system
USAGE: rationalFunctionSolutions I
rationalFunctionSolutions(I,f)
rationalFunctionSolutions(I,f,w)
rationalFunctionSolutions(I,ff)
rationalFunctionSolutions(I,ff,w)
INPUTS: I:Ideal
holonomic ideal in the Weyl algebra @EM "D"@
f:RingElement
... | Key
rationalFunctionSolutions
(rationalFunctionSolutions,Ideal,List,List)
(rationalFunctionSolutions,Ideal,RingElement,List)
(rationalFunctionSolutions,Ideal,List)
(rationalFunctionSolutions,Ideal,RingElement)
(rationalFunctionSolutions,Ideal)
Headline
rational solutions of a holonomic sys... | ||
DHom_b2813d12_(rationalFunctionSolutions,Ideal,RingElement) | DHom | (rationalFunctionSolutions,Ideal,RingElement) | rational solutions of a holonomic system | rationalFunctionSolutions I
rationalFunctionSolutions(I,f)
rationalFunctionSolutions(I,f,w)
rationalFunctionSolutions(I,ff)
rationalFunctionSolutions(I,ff,w) | makeWA(QQ[x])
I = ideal((x+1)*dx+5)
rationalFunctionSolutions I
Caveat
The most efficient method to find rational solutions of a system of differential
equations is
to find the singular locus, then try to find its irreducible
factors. With these, call rationalFunctionSolutions(I, ff, w), ... | polynomialSolutions
rationalFunctionExt
DHom | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/DHom.m2 | stable | HEADLINE: rational solutions of a holonomic system
USAGE: rationalFunctionSolutions I
rationalFunctionSolutions(I,f)
rationalFunctionSolutions(I,f,w)
rationalFunctionSolutions(I,ff)
rationalFunctionSolutions(I,ff,w)
INPUTS: I:Ideal
holonomic ideal in the Weyl algebra @EM "D"@
f:RingElement
... | Key
rationalFunctionSolutions
(rationalFunctionSolutions,Ideal,List,List)
(rationalFunctionSolutions,Ideal,RingElement,List)
(rationalFunctionSolutions,Ideal,List)
(rationalFunctionSolutions,Ideal,RingElement)
(rationalFunctionSolutions,Ideal)
Headline
rational solutions of a holonomic sys... | ||
DHom_b2813d12_(rationalFunctionSolutions,Ideal) | DHom | (rationalFunctionSolutions,Ideal) | rational solutions of a holonomic system | rationalFunctionSolutions I
rationalFunctionSolutions(I,f)
rationalFunctionSolutions(I,f,w)
rationalFunctionSolutions(I,ff)
rationalFunctionSolutions(I,ff,w) | makeWA(QQ[x])
I = ideal((x+1)*dx+5)
rationalFunctionSolutions I
Caveat
The most efficient method to find rational solutions of a system of differential
equations is
to find the singular locus, then try to find its irreducible
factors. With these, call rationalFunctionSolutions(I, ff, w), ... | polynomialSolutions
rationalFunctionExt
DHom | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/DHom.m2 | stable | HEADLINE: rational solutions of a holonomic system
USAGE: rationalFunctionSolutions I
rationalFunctionSolutions(I,f)
rationalFunctionSolutions(I,f,w)
rationalFunctionSolutions(I,ff)
rationalFunctionSolutions(I,ff,w)
INPUTS: I:Ideal
holonomic ideal in the Weyl algebra @EM "D"@
f:RingElement
... | Key
rationalFunctionSolutions
(rationalFunctionSolutions,Ideal,List,List)
(rationalFunctionSolutions,Ideal,RingElement,List)
(rationalFunctionSolutions,Ideal,List)
(rationalFunctionSolutions,Ideal,RingElement)
(rationalFunctionSolutions,Ideal)
Headline
rational solutions of a holonomic sys... | ||
Dlocalize_f936f1c9_DlocalizeAll | Dlocalize | DlocalizeAll | localization of a D-module (extended version) | DlocalizeAll(M,f)
DlocalizeAll(I,f) | W = makeWeylAlgebra(QQ[x,y])
M = W^1/ideal(x*dx + 1, dy)
f = x^2 - y^3
Mfall = DlocalizeAll(M, f)
gens image Mfall.LocMap == f^(-Mfall.GeneratorPower) * gens Mfall.LocModule | Dlocalize
AnnFs
Dintegration | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/Dlocalize.m2 | stable | HEADLINE: localization of a D-module (extended version)
USAGE: DlocalizeAll(M,f)
DlocalizeAll(I,f)
INPUTS: M:Module
over the Weyl algebra $D$
I:Ideal
which represents the module $M=D/I$
f:RingElement
a polynomial
OUTPUTS: :HashTable
which contains the localized module $M_f = M[f^{-1}]$ and som... | Key
DlocalizeAll
(DlocalizeAll,Ideal,RingElement)
(DlocalizeAll,Module,RingElement)
LocModule
GeneratorPower
LocMap
annFS
IntegrateBfunction
Bfunction
Headline
localization of a D-module (extended version)
Usage
DlocalizeAll(M,f)
DlocalizeAll(I,f)
Inputs
M:Module
over the W... | ||
Dlocalize_f936f1c9_(DlocalizeAll,Ideal,RingElement) | Dlocalize | (DlocalizeAll,Ideal,RingElement) | localization of a D-module (extended version) | DlocalizeAll(M,f)
DlocalizeAll(I,f) | W = makeWeylAlgebra(QQ[x,y])
M = W^1/ideal(x*dx + 1, dy)
f = x^2 - y^3
Mfall = DlocalizeAll(M, f)
gens image Mfall.LocMap == f^(-Mfall.GeneratorPower) * gens Mfall.LocModule | Dlocalize
AnnFs
Dintegration | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/Dlocalize.m2 | stable | HEADLINE: localization of a D-module (extended version)
USAGE: DlocalizeAll(M,f)
DlocalizeAll(I,f)
INPUTS: M:Module
over the Weyl algebra $D$
I:Ideal
which represents the module $M=D/I$
f:RingElement
a polynomial
OUTPUTS: :HashTable
which contains the localized module $M_f = M[f^{-1}]$ and som... | Key
DlocalizeAll
(DlocalizeAll,Ideal,RingElement)
(DlocalizeAll,Module,RingElement)
LocModule
GeneratorPower
LocMap
annFS
IntegrateBfunction
Bfunction
Headline
localization of a D-module (extended version)
Usage
DlocalizeAll(M,f)
DlocalizeAll(I,f)
Inputs
M:Module
over the W... | ||
Dlocalize_f936f1c9_(DlocalizeAll,Module,RingElement) | Dlocalize | (DlocalizeAll,Module,RingElement) | localization of a D-module (extended version) | DlocalizeAll(M,f)
DlocalizeAll(I,f) | W = makeWeylAlgebra(QQ[x,y])
M = W^1/ideal(x*dx + 1, dy)
f = x^2 - y^3
Mfall = DlocalizeAll(M, f)
gens image Mfall.LocMap == f^(-Mfall.GeneratorPower) * gens Mfall.LocModule | Dlocalize
AnnFs
Dintegration | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/Dlocalize.m2 | stable | HEADLINE: localization of a D-module (extended version)
USAGE: DlocalizeAll(M,f)
DlocalizeAll(I,f)
INPUTS: M:Module
over the Weyl algebra $D$
I:Ideal
which represents the module $M=D/I$
f:RingElement
a polynomial
OUTPUTS: :HashTable
which contains the localized module $M_f = M[f^{-1}]$ and som... | Key
DlocalizeAll
(DlocalizeAll,Ideal,RingElement)
(DlocalizeAll,Module,RingElement)
LocModule
GeneratorPower
LocMap
annFS
IntegrateBfunction
Bfunction
Headline
localization of a D-module (extended version)
Usage
DlocalizeAll(M,f)
DlocalizeAll(I,f)
Inputs
M:Module
over the W... | ||
Dlocalize_f936f1c9_LocModule | Dlocalize | LocModule | localization of a D-module (extended version) | DlocalizeAll(M,f)
DlocalizeAll(I,f) | W = makeWeylAlgebra(QQ[x,y])
M = W^1/ideal(x*dx + 1, dy)
f = x^2 - y^3
Mfall = DlocalizeAll(M, f)
gens image Mfall.LocMap == f^(-Mfall.GeneratorPower) * gens Mfall.LocModule | Dlocalize
AnnFs
Dintegration | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/Dlocalize.m2 | stable | HEADLINE: localization of a D-module (extended version)
USAGE: DlocalizeAll(M,f)
DlocalizeAll(I,f)
INPUTS: M:Module
over the Weyl algebra $D$
I:Ideal
which represents the module $M=D/I$
f:RingElement
a polynomial
OUTPUTS: :HashTable
which contains the localized module $M_f = M[f^{-1}]$ and som... | Key
DlocalizeAll
(DlocalizeAll,Ideal,RingElement)
(DlocalizeAll,Module,RingElement)
LocModule
GeneratorPower
LocMap
annFS
IntegrateBfunction
Bfunction
Headline
localization of a D-module (extended version)
Usage
DlocalizeAll(M,f)
DlocalizeAll(I,f)
Inputs
M:Module
over the W... | ||
Dlocalize_f936f1c9_GeneratorPower | Dlocalize | GeneratorPower | localization of a D-module (extended version) | DlocalizeAll(M,f)
DlocalizeAll(I,f) | W = makeWeylAlgebra(QQ[x,y])
M = W^1/ideal(x*dx + 1, dy)
f = x^2 - y^3
Mfall = DlocalizeAll(M, f)
gens image Mfall.LocMap == f^(-Mfall.GeneratorPower) * gens Mfall.LocModule | Dlocalize
AnnFs
Dintegration | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/Dlocalize.m2 | stable | HEADLINE: localization of a D-module (extended version)
USAGE: DlocalizeAll(M,f)
DlocalizeAll(I,f)
INPUTS: M:Module
over the Weyl algebra $D$
I:Ideal
which represents the module $M=D/I$
f:RingElement
a polynomial
OUTPUTS: :HashTable
which contains the localized module $M_f = M[f^{-1}]$ and som... | Key
DlocalizeAll
(DlocalizeAll,Ideal,RingElement)
(DlocalizeAll,Module,RingElement)
LocModule
GeneratorPower
LocMap
annFS
IntegrateBfunction
Bfunction
Headline
localization of a D-module (extended version)
Usage
DlocalizeAll(M,f)
DlocalizeAll(I,f)
Inputs
M:Module
over the W... | ||
Dlocalize_f936f1c9_LocMap | Dlocalize | LocMap | localization of a D-module (extended version) | DlocalizeAll(M,f)
DlocalizeAll(I,f) | W = makeWeylAlgebra(QQ[x,y])
M = W^1/ideal(x*dx + 1, dy)
f = x^2 - y^3
Mfall = DlocalizeAll(M, f)
gens image Mfall.LocMap == f^(-Mfall.GeneratorPower) * gens Mfall.LocModule | Dlocalize
AnnFs
Dintegration | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/Dlocalize.m2 | stable | HEADLINE: localization of a D-module (extended version)
USAGE: DlocalizeAll(M,f)
DlocalizeAll(I,f)
INPUTS: M:Module
over the Weyl algebra $D$
I:Ideal
which represents the module $M=D/I$
f:RingElement
a polynomial
OUTPUTS: :HashTable
which contains the localized module $M_f = M[f^{-1}]$ and som... | Key
DlocalizeAll
(DlocalizeAll,Ideal,RingElement)
(DlocalizeAll,Module,RingElement)
LocModule
GeneratorPower
LocMap
annFS
IntegrateBfunction
Bfunction
Headline
localization of a D-module (extended version)
Usage
DlocalizeAll(M,f)
DlocalizeAll(I,f)
Inputs
M:Module
over the W... | ||
Dlocalize_f936f1c9_annFS | Dlocalize | annFS | localization of a D-module (extended version) | DlocalizeAll(M,f)
DlocalizeAll(I,f) | W = makeWeylAlgebra(QQ[x,y])
M = W^1/ideal(x*dx + 1, dy)
f = x^2 - y^3
Mfall = DlocalizeAll(M, f)
gens image Mfall.LocMap == f^(-Mfall.GeneratorPower) * gens Mfall.LocModule | Dlocalize
AnnFs
Dintegration | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/Dlocalize.m2 | stable | HEADLINE: localization of a D-module (extended version)
USAGE: DlocalizeAll(M,f)
DlocalizeAll(I,f)
INPUTS: M:Module
over the Weyl algebra $D$
I:Ideal
which represents the module $M=D/I$
f:RingElement
a polynomial
OUTPUTS: :HashTable
which contains the localized module $M_f = M[f^{-1}]$ and som... | Key
DlocalizeAll
(DlocalizeAll,Ideal,RingElement)
(DlocalizeAll,Module,RingElement)
LocModule
GeneratorPower
LocMap
annFS
IntegrateBfunction
Bfunction
Headline
localization of a D-module (extended version)
Usage
DlocalizeAll(M,f)
DlocalizeAll(I,f)
Inputs
M:Module
over the W... | ||
Dlocalize_f936f1c9_IntegrateBfunction | Dlocalize | IntegrateBfunction | localization of a D-module (extended version) | DlocalizeAll(M,f)
DlocalizeAll(I,f) | W = makeWeylAlgebra(QQ[x,y])
M = W^1/ideal(x*dx + 1, dy)
f = x^2 - y^3
Mfall = DlocalizeAll(M, f)
gens image Mfall.LocMap == f^(-Mfall.GeneratorPower) * gens Mfall.LocModule | Dlocalize
AnnFs
Dintegration | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/Dlocalize.m2 | stable | HEADLINE: localization of a D-module (extended version)
USAGE: DlocalizeAll(M,f)
DlocalizeAll(I,f)
INPUTS: M:Module
over the Weyl algebra $D$
I:Ideal
which represents the module $M=D/I$
f:RingElement
a polynomial
OUTPUTS: :HashTable
which contains the localized module $M_f = M[f^{-1}]$ and som... | Key
DlocalizeAll
(DlocalizeAll,Ideal,RingElement)
(DlocalizeAll,Module,RingElement)
LocModule
GeneratorPower
LocMap
annFS
IntegrateBfunction
Bfunction
Headline
localization of a D-module (extended version)
Usage
DlocalizeAll(M,f)
DlocalizeAll(I,f)
Inputs
M:Module
over the W... | ||
Dlocalize_f936f1c9_Bfunction | Dlocalize | Bfunction | localization of a D-module (extended version) | DlocalizeAll(M,f)
DlocalizeAll(I,f) | W = makeWeylAlgebra(QQ[x,y])
M = W^1/ideal(x*dx + 1, dy)
f = x^2 - y^3
Mfall = DlocalizeAll(M, f)
gens image Mfall.LocMap == f^(-Mfall.GeneratorPower) * gens Mfall.LocModule | Dlocalize
AnnFs
Dintegration | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/Dlocalize.m2 | stable | HEADLINE: localization of a D-module (extended version)
USAGE: DlocalizeAll(M,f)
DlocalizeAll(I,f)
INPUTS: M:Module
over the Weyl algebra $D$
I:Ideal
which represents the module $M=D/I$
f:RingElement
a polynomial
OUTPUTS: :HashTable
which contains the localized module $M_f = M[f^{-1}]$ and som... | Key
DlocalizeAll
(DlocalizeAll,Ideal,RingElement)
(DlocalizeAll,Module,RingElement)
LocModule
GeneratorPower
LocMap
annFS
IntegrateBfunction
Bfunction
Headline
localization of a D-module (extended version)
Usage
DlocalizeAll(M,f)
DlocalizeAll(I,f)
Inputs
M:Module
over the W... | ||
WeylClosure_a03d04cf_WeylClosure | WeylClosure | WeylClosure | Weyl closure of an ideal | WeylClosure I
WeylClosure(I,f) | makeWA(QQ[x])
I = ideal(x*dx-2)
holonomicRank I
WeylClosure I
Caveat
The ideal I should be of finite holonomic rank, which can be tested
manually by using the function holonomicRank. The Weyl closure of non-finite rank
ideals or arbitrary submodules has not been implemented. | Dlocalize
singLocus
holonomicRank | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/WeylClosure.m2 | stable | HEADLINE: Weyl closure of an ideal
USAGE: WeylClosure I
WeylClosure(I,f)
INPUTS: I:Ideal
a left ideal of the Weyl Algebra
f:RingElement
a polynomial
OUTPUTS: :Ideal
the Weyl closure (w.r.t. $f$) of $I$
EXAMPLE CODE:
```macaulay2
makeWA(QQ[x])
I = ideal(x*dx-2)
holonomicRank I
... | Key
WeylClosure
(WeylClosure, Ideal)
(WeylClosure, Ideal, RingElement)
Headline
Weyl closure of an ideal
Usage
WeylClosure I
WeylClosure(I,f)
Inputs
I:Ideal
a left ideal of the Weyl Algebra
f:RingElement
a polynomial
Outputs
:Ideal
the Weyl closure (w.r.t. $... | ||
WeylClosure_a03d04cf_(WeylClosure,_Ideal) | WeylClosure | (WeylClosure, Ideal) | Weyl closure of an ideal | WeylClosure I
WeylClosure(I,f) | makeWA(QQ[x])
I = ideal(x*dx-2)
holonomicRank I
WeylClosure I
Caveat
The ideal I should be of finite holonomic rank, which can be tested
manually by using the function holonomicRank. The Weyl closure of non-finite rank
ideals or arbitrary submodules has not been implemented. | Dlocalize
singLocus
holonomicRank | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/WeylClosure.m2 | stable | HEADLINE: Weyl closure of an ideal
USAGE: WeylClosure I
WeylClosure(I,f)
INPUTS: I:Ideal
a left ideal of the Weyl Algebra
f:RingElement
a polynomial
OUTPUTS: :Ideal
the Weyl closure (w.r.t. $f$) of $I$
EXAMPLE CODE:
```macaulay2
makeWA(QQ[x])
I = ideal(x*dx-2)
holonomicRank I
... | Key
WeylClosure
(WeylClosure, Ideal)
(WeylClosure, Ideal, RingElement)
Headline
Weyl closure of an ideal
Usage
WeylClosure I
WeylClosure(I,f)
Inputs
I:Ideal
a left ideal of the Weyl Algebra
f:RingElement
a polynomial
Outputs
:Ideal
the Weyl closure (w.r.t. $... | ||
WeylClosure_a03d04cf_(WeylClosure,_Ideal,_RingElement) | WeylClosure | (WeylClosure, Ideal, RingElement) | Weyl closure of an ideal | WeylClosure I
WeylClosure(I,f) | makeWA(QQ[x])
I = ideal(x*dx-2)
holonomicRank I
WeylClosure I
Caveat
The ideal I should be of finite holonomic rank, which can be tested
manually by using the function holonomicRank. The Weyl closure of non-finite rank
ideals or arbitrary submodules has not been implemented. | Dlocalize
singLocus
holonomicRank | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/WeylClosure.m2 | stable | HEADLINE: Weyl closure of an ideal
USAGE: WeylClosure I
WeylClosure(I,f)
INPUTS: I:Ideal
a left ideal of the Weyl Algebra
f:RingElement
a polynomial
OUTPUTS: :Ideal
the Weyl closure (w.r.t. $f$) of $I$
EXAMPLE CODE:
```macaulay2
makeWA(QQ[x])
I = ideal(x*dx-2)
holonomicRank I
... | Key
WeylClosure
(WeylClosure, Ideal)
(WeylClosure, Ideal, RingElement)
Headline
Weyl closure of an ideal
Usage
WeylClosure I
WeylClosure(I,f)
Inputs
I:Ideal
a left ideal of the Weyl Algebra
f:RingElement
a polynomial
Outputs
:Ideal
the Weyl closure (w.r.t. $... | ||
annFs_f2e15d57_AnnFs | annFs | AnnFs | differential annihilator of a polynomial in a Weyl algebra | AnnFs(f) | makeWA(QQ[x,y])
f = x^2+y
AnnFs f
Caveat
Must be over a ring of characteristic $0$. | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/annFs.m2 | stable | HEADLINE: differential annihilator of a polynomial in a Weyl algebra
USAGE: AnnFs(f)
INPUTS: f:RingElement
polynomial in a Weyl algebra $D$
OUTPUTS: :Ideal
the differential annihilator of $f$ in $D[s]$, for a new variable $s$.
EXAMPLE CODE:
```macaulay2
makeWA(QQ[x,y])
f = x^2+y
AnnFs f
Ca... | Key
AnnFs
(AnnFs, RingElement)
Headline
differential annihilator of a polynomial in a Weyl algebra
Usage
AnnFs(f)
Inputs
f:RingElement
polynomial in a Weyl algebra $D$
Outputs
:Ideal
the differential annihilator of $f$ in $D[s]$, for a new variable $s$.
Description
Te... | |||
annFs_f2e15d57_(AnnFs,_RingElement) | annFs | (AnnFs, RingElement) | differential annihilator of a polynomial in a Weyl algebra | AnnFs(f) | makeWA(QQ[x,y])
f = x^2+y
AnnFs f
Caveat
Must be over a ring of characteristic $0$. | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/annFs.m2 | stable | HEADLINE: differential annihilator of a polynomial in a Weyl algebra
USAGE: AnnFs(f)
INPUTS: f:RingElement
polynomial in a Weyl algebra $D$
OUTPUTS: :Ideal
the differential annihilator of $f$ in $D[s]$, for a new variable $s$.
EXAMPLE CODE:
```macaulay2
makeWA(QQ[x,y])
f = x^2+y
AnnFs f
Ca... | Key
AnnFs
(AnnFs, RingElement)
Headline
differential annihilator of a polynomial in a Weyl algebra
Usage
AnnFs(f)
Inputs
f:RingElement
polynomial in a Weyl algebra $D$
Outputs
:Ideal
the differential annihilator of $f$ in $D[s]$, for a new variable $s$.
Description
Te... | |||
annFs_33d26f00_diffRatFun | annFs | diffRatFun | derivative of a rational function in a Weyl algebra | diffRatFun(m,f)
diffRatFun(m,g,f,a) | makeWA(QQ[x,y,z])
m = {1,2,1}
g = z
f = x*y
a = 3
diffRatFun(m,g,f,a)
Caveat
Must be over a ring of characteristic $0$. | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/annFs.m2 | stable | HEADLINE: derivative of a rational function in a Weyl algebra
USAGE: diffRatFun(m,f)
diffRatFun(m,g,f,a)
INPUTS: f:RingElement
polynomial in a Weyl algebra $D$ in $n$ variables or rational function in the fraction
field of a polynomial ring in $n$ variables
g:RingElement
polynomial in a Weyl... | Key
diffRatFun
(diffRatFun, List, RingElement)
(diffRatFun, List, RingElement, RingElement, ZZ)
Headline
derivative of a rational function in a Weyl algebra
Usage
diffRatFun(m,f)
diffRatFun(m,g,f,a)
Inputs
f:RingElement
polynomial in a Weyl algebra $D$ in $n$ variables or rational ... | |||
annFs_33d26f00_(diffRatFun,_List,_RingElement) | annFs | (diffRatFun, List, RingElement) | derivative of a rational function in a Weyl algebra | diffRatFun(m,f)
diffRatFun(m,g,f,a) | makeWA(QQ[x,y,z])
m = {1,2,1}
g = z
f = x*y
a = 3
diffRatFun(m,g,f,a)
Caveat
Must be over a ring of characteristic $0$. | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/annFs.m2 | stable | HEADLINE: derivative of a rational function in a Weyl algebra
USAGE: diffRatFun(m,f)
diffRatFun(m,g,f,a)
INPUTS: f:RingElement
polynomial in a Weyl algebra $D$ in $n$ variables or rational function in the fraction
field of a polynomial ring in $n$ variables
g:RingElement
polynomial in a Weyl... | Key
diffRatFun
(diffRatFun, List, RingElement)
(diffRatFun, List, RingElement, RingElement, ZZ)
Headline
derivative of a rational function in a Weyl algebra
Usage
diffRatFun(m,f)
diffRatFun(m,g,f,a)
Inputs
f:RingElement
polynomial in a Weyl algebra $D$ in $n$ variables or rational ... | |||
annFs_33d26f00_(diffRatFun,_List,_RingElement,_RingElement,_ZZ) | annFs | (diffRatFun, List, RingElement, RingElement, ZZ) | derivative of a rational function in a Weyl algebra | diffRatFun(m,f)
diffRatFun(m,g,f,a) | makeWA(QQ[x,y,z])
m = {1,2,1}
g = z
f = x*y
a = 3
diffRatFun(m,g,f,a)
Caveat
Must be over a ring of characteristic $0$. | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/annFs.m2 | stable | HEADLINE: derivative of a rational function in a Weyl algebra
USAGE: diffRatFun(m,f)
diffRatFun(m,g,f,a)
INPUTS: f:RingElement
polynomial in a Weyl algebra $D$ in $n$ variables or rational function in the fraction
field of a polynomial ring in $n$ variables
g:RingElement
polynomial in a Weyl... | Key
diffRatFun
(diffRatFun, List, RingElement)
(diffRatFun, List, RingElement, RingElement, ZZ)
Headline
derivative of a rational function in a Weyl algebra
Usage
diffRatFun(m,f)
diffRatFun(m,g,f,a)
Inputs
f:RingElement
polynomial in a Weyl algebra $D$ in $n$ variables or rational ... | |||
annFs_fa068dac_polynomialAnnihilator | annFs | polynomialAnnihilator | annihilator of a polynomial in the Weyl algebra | polynomialAnnihilator f | makeWA(QQ[x,y])
f = x^2-y^3
I = polynomialAnnihilator f
Caveat
The input f should be an element of a Weyl algebra, and not an element of a commutative polynomial ring.
However, f should only involve commutative variables. | rationalFunctionAnnihilator | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/annFs.m2 | stable | HEADLINE: annihilator of a polynomial in the Weyl algebra
USAGE: polynomialAnnihilator f
INPUTS: f:RingElement
polynomial
OUTPUTS: :Ideal
the annihilating (left) ideal of @{EM "f"}@ in the Weyl algebra
EXAMPLE CODE:
```macaulay2
makeWA(QQ[x,y])
f = x^2-y^3
I = polynomialAnnihilator f
Cavea... | Key
polynomialAnnihilator
(polynomialAnnihilator, RingElement)
Headline
annihilator of a polynomial in the Weyl algebra
Usage
polynomialAnnihilator f
Inputs
f:RingElement
polynomial
Outputs
:Ideal
the annihilating (left) ideal of @{EM "f"}@ in the Weyl algebra
Description
... | ||
annFs_fa068dac_(polynomialAnnihilator,_RingElement) | annFs | (polynomialAnnihilator, RingElement) | annihilator of a polynomial in the Weyl algebra | polynomialAnnihilator f | makeWA(QQ[x,y])
f = x^2-y^3
I = polynomialAnnihilator f
Caveat
The input f should be an element of a Weyl algebra, and not an element of a commutative polynomial ring.
However, f should only involve commutative variables. | rationalFunctionAnnihilator | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/annFs.m2 | stable | HEADLINE: annihilator of a polynomial in the Weyl algebra
USAGE: polynomialAnnihilator f
INPUTS: f:RingElement
polynomial
OUTPUTS: :Ideal
the annihilating (left) ideal of @{EM "f"}@ in the Weyl algebra
EXAMPLE CODE:
```macaulay2
makeWA(QQ[x,y])
f = x^2-y^3
I = polynomialAnnihilator f
Cavea... | Key
polynomialAnnihilator
(polynomialAnnihilator, RingElement)
Headline
annihilator of a polynomial in the Weyl algebra
Usage
polynomialAnnihilator f
Inputs
f:RingElement
polynomial
Outputs
:Ideal
the annihilating (left) ideal of @{EM "f"}@ in the Weyl algebra
Description
... | ||
annFs_f2cc2468_rationalFunctionAnnihilator | annFs | rationalFunctionAnnihilator | annihilator of a rational function in Weyl algebra | rationalFunctionAnnihilator f
rationalFunctionAnnihilator(g,f) | makeWA(QQ[x,y])
f = x^2-y^3
g = 2*x*y
I = rationalFunctionAnnihilator (g,f)
Caveat
The inputs f and g should be elements of a Weyl algebra, and not elements of a commutative polynomial ring.
However, f and g should only use the commutative variables. | polynomialAnnihilator | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/annFs.m2 | stable | HEADLINE: annihilator of a rational function in Weyl algebra
USAGE: rationalFunctionAnnihilator f
rationalFunctionAnnihilator(g,f)
INPUTS: f:RingElement
polynomial
g:RingElement
polynomial
OUTPUTS: :Ideal
left ideal of the Weyl algebra
EXAMPLE CODE:
```macaulay2
makeWA(QQ[x,y])
f = x... | Key
rationalFunctionAnnihilator
(rationalFunctionAnnihilator, RingElement, RingElement)
(rationalFunctionAnnihilator, RingElement)
Headline
annihilator of a rational function in Weyl algebra
Usage
rationalFunctionAnnihilator f
rationalFunctionAnnihilator(g,f)
Inputs
f:RingElement
... | ||
annFs_f2cc2468_(rationalFunctionAnnihilator,_RingElement,_RingElement) | annFs | (rationalFunctionAnnihilator, RingElement, RingElement) | annihilator of a rational function in Weyl algebra | rationalFunctionAnnihilator f
rationalFunctionAnnihilator(g,f) | makeWA(QQ[x,y])
f = x^2-y^3
g = 2*x*y
I = rationalFunctionAnnihilator (g,f)
Caveat
The inputs f and g should be elements of a Weyl algebra, and not elements of a commutative polynomial ring.
However, f and g should only use the commutative variables. | polynomialAnnihilator | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/annFs.m2 | stable | HEADLINE: annihilator of a rational function in Weyl algebra
USAGE: rationalFunctionAnnihilator f
rationalFunctionAnnihilator(g,f)
INPUTS: f:RingElement
polynomial
g:RingElement
polynomial
OUTPUTS: :Ideal
left ideal of the Weyl algebra
EXAMPLE CODE:
```macaulay2
makeWA(QQ[x,y])
f = x... | Key
rationalFunctionAnnihilator
(rationalFunctionAnnihilator, RingElement, RingElement)
(rationalFunctionAnnihilator, RingElement)
Headline
annihilator of a rational function in Weyl algebra
Usage
rationalFunctionAnnihilator f
rationalFunctionAnnihilator(g,f)
Inputs
f:RingElement
... | ||
annFs_f2cc2468_(rationalFunctionAnnihilator,_RingElement) | annFs | (rationalFunctionAnnihilator, RingElement) | annihilator of a rational function in Weyl algebra | rationalFunctionAnnihilator f
rationalFunctionAnnihilator(g,f) | makeWA(QQ[x,y])
f = x^2-y^3
g = 2*x*y
I = rationalFunctionAnnihilator (g,f)
Caveat
The inputs f and g should be elements of a Weyl algebra, and not elements of a commutative polynomial ring.
However, f and g should only use the commutative variables. | polynomialAnnihilator | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/annFs.m2 | stable | HEADLINE: annihilator of a rational function in Weyl algebra
USAGE: rationalFunctionAnnihilator f
rationalFunctionAnnihilator(g,f)
INPUTS: f:RingElement
polynomial
g:RingElement
polynomial
OUTPUTS: :Ideal
left ideal of the Weyl algebra
EXAMPLE CODE:
```macaulay2
makeWA(QQ[x,y])
f = x... | Key
rationalFunctionAnnihilator
(rationalFunctionAnnihilator, RingElement, RingElement)
(rationalFunctionAnnihilator, RingElement)
Headline
annihilator of a rational function in Weyl algebra
Usage
rationalFunctionAnnihilator f
rationalFunctionAnnihilator(g,f)
Inputs
f:RingElement
... | ||
intersectionCohom_d2e555e8_IHmodule | intersectionCohom | IHmodule | intersection (co)homology module of an irreducible closed subvariety | IHmodule(I) | R=QQ[x,y]
I=ideal(x^2+y^3)
IHmodule(I, LocStrategy=>CompleteIntersection)
Caveat
Must be a ring of characteristic 0. The ideal $I$ should have only 1 minimal prime. | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/intersectionCohom.m2 | stable | HEADLINE: intersection (co)homology module of an irreducible closed subvariety
USAGE: IHmodule(I)
INPUTS: I:Ideal
ideal in the polynomial ring $R$
LocCohomStrategy=>Sequence -- (String, String)
see @TO [localCohom, Strategy]@ and @TO[localCohom, LocStrategy]@
LocStrategy=>String
see @TO... | Node
Key
intersectionCohomology
IH
(intersectionCohomology, Ideal)
(intersectionCohomology, ZZ, Ideal)
[intersectionCohomology, Strategy]
[intersectionCohomology, LocStrategy]
[intersectionCohomology, LocCohomStrategy]
Headline
intersection cohomology of an irreducible affine variety... | |||
intersectionCohom_d2e555e8_(IHmodule,_Ideal) | intersectionCohom | (IHmodule, Ideal) | intersection (co)homology module of an irreducible closed subvariety | IHmodule(I) | R=QQ[x,y]
I=ideal(x^2+y^3)
IHmodule(I, LocStrategy=>CompleteIntersection)
Caveat
Must be a ring of characteristic 0. The ideal $I$ should have only 1 minimal prime. | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/intersectionCohom.m2 | stable | HEADLINE: intersection (co)homology module of an irreducible closed subvariety
USAGE: IHmodule(I)
INPUTS: I:Ideal
ideal in the polynomial ring $R$
LocCohomStrategy=>Sequence -- (String, String)
see @TO [localCohom, Strategy]@ and @TO[localCohom, LocStrategy]@
LocStrategy=>String
see @TO... | Node
Key
intersectionCohomology
IH
(intersectionCohomology, Ideal)
(intersectionCohomology, ZZ, Ideal)
[intersectionCohomology, Strategy]
[intersectionCohomology, LocStrategy]
[intersectionCohomology, LocCohomStrategy]
Headline
intersection cohomology of an irreducible affine variety... | |||
intersectionCohom_d2e555e8_[IHmodule,_LocStrategy] | intersectionCohom | [IHmodule, LocStrategy] | intersection (co)homology module of an irreducible closed subvariety | IHmodule(I) | R=QQ[x,y]
I=ideal(x^2+y^3)
IHmodule(I, LocStrategy=>CompleteIntersection)
Caveat
Must be a ring of characteristic 0. The ideal $I$ should have only 1 minimal prime. | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/intersectionCohom.m2 | stable | HEADLINE: intersection (co)homology module of an irreducible closed subvariety
USAGE: IHmodule(I)
INPUTS: I:Ideal
ideal in the polynomial ring $R$
LocCohomStrategy=>Sequence -- (String, String)
see @TO [localCohom, Strategy]@ and @TO[localCohom, LocStrategy]@
LocStrategy=>String
see @TO... | Node
Key
intersectionCohomology
IH
(intersectionCohomology, Ideal)
(intersectionCohomology, ZZ, Ideal)
[intersectionCohomology, Strategy]
[intersectionCohomology, LocStrategy]
[intersectionCohomology, LocCohomStrategy]
Headline
intersection cohomology of an irreducible affine variety... | |||
intersectionCohom_d2e555e8_[IHmodule,_LocCohomStrategy] | intersectionCohom | [IHmodule, LocCohomStrategy] | intersection (co)homology module of an irreducible closed subvariety | IHmodule(I) | R=QQ[x,y]
I=ideal(x^2+y^3)
IHmodule(I, LocStrategy=>CompleteIntersection)
Caveat
Must be a ring of characteristic 0. The ideal $I$ should have only 1 minimal prime. | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/intersectionCohom.m2 | stable | HEADLINE: intersection (co)homology module of an irreducible closed subvariety
USAGE: IHmodule(I)
INPUTS: I:Ideal
ideal in the polynomial ring $R$
LocCohomStrategy=>Sequence -- (String, String)
see @TO [localCohom, Strategy]@ and @TO[localCohom, LocStrategy]@
LocStrategy=>String
see @TO... | Node
Key
intersectionCohomology
IH
(intersectionCohomology, Ideal)
(intersectionCohomology, ZZ, Ideal)
[intersectionCohomology, Strategy]
[intersectionCohomology, LocStrategy]
[intersectionCohomology, LocCohomStrategy]
Headline
intersection cohomology of an irreducible affine variety... | |||
intersectionCohom_d2e555e8_LocCohomStrategy | intersectionCohom | LocCohomStrategy | intersection (co)homology module of an irreducible closed subvariety | IHmodule(I) | R=QQ[x,y]
I=ideal(x^2+y^3)
IHmodule(I, LocStrategy=>CompleteIntersection)
Caveat
Must be a ring of characteristic 0. The ideal $I$ should have only 1 minimal prime. | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/intersectionCohom.m2 | stable | HEADLINE: intersection (co)homology module of an irreducible closed subvariety
USAGE: IHmodule(I)
INPUTS: I:Ideal
ideal in the polynomial ring $R$
LocCohomStrategy=>Sequence -- (String, String)
see @TO [localCohom, Strategy]@ and @TO[localCohom, LocStrategy]@
LocStrategy=>String
see @TO... | Node
Key
intersectionCohomology
IH
(intersectionCohomology, Ideal)
(intersectionCohomology, ZZ, Ideal)
[intersectionCohomology, Strategy]
[intersectionCohomology, LocStrategy]
[intersectionCohomology, LocCohomStrategy]
Headline
intersection cohomology of an irreducible affine variety... | |||
main_8726d546_BernsteinSato | main | BernsteinSato | algorithms for b-functions, local cohomology, and intersection cohomology | Tree
:B-functions
@TOH "bFunction"@
@TOH "generalB"@
@TOH "globalB"@
-- TODO: capital F but lowercase o?
@TOH "globalBFunction"@
@TOH "globalBoperator"@
@TOH "localBFunction"@
@TOH "factorBFunction"@
@TOH "bFunctionRoots"@
@TOH "getIntRoots"@
@TOH "paramBpoly"@
@TOH "AnnFs"@
@TOH "AnnIFs"@
:Re... | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/main.m2 | stable | HEADLINE: algorithms for b-functions, local cohomology, and intersection cohomology
DESCRIPTION: Tree
:B-functions
@TOH "bFunction"@
@TOH "generalB"@
@TOH "globalB"@
-- TODO: capital F but lowercase o?
@TOH "globalBFunction"@
@TOH "globalBoperator"@
@TOH "localBFunction"@
@TOH "factorBFunction"@
@TOH "b... | Node
Key
BernsteinSato
Headline
algorithms for b-functions, local cohomology, and intersection cohomology
Description
Tree
:B-functions
@TOH "bFunction"@
@TOH "generalB"@
@TOH "globalB"@
-- TODO: capital F but lowercase o?
@TOH "globalBFunction"@
@TOH "globalBoperator"@
@TOH "localBFuncti... | ||||
paco-anton-paper_0b403826_reiffen | paco-anton-paper | reiffen | Reiffen's curve | reiffen(p,q) | reiffen(4,5) | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/paco-anton-paper.m2 | stable | HEADLINE: Reiffen's curve
USAGE: reiffen(p,q)
INPUTS: p:ZZ
q:ZZ
OUTPUTS: :RingElement
the defining polynomial of Reiffen's (p,q)-curve
EXAMPLE CODE:
```macaulay2
reiffen(4,5)
``` | Key
reiffen
(reiffen,ZZ,ZZ)
Headline
Reiffen's curve
Usage
reiffen(p,q)
Inputs
p:ZZ
q:ZZ
Outputs
:RingElement
the defining polynomial of Reiffen's (p,q)-curve
Description
Text
Reiffen's (p,q)-curve is a singular curve defined by $f_{p,q} = x_1^p+x_2^q+x_1x_2^{q-1} =... | |||
paco-anton-paper_0b403826_(reiffen,ZZ,ZZ) | paco-anton-paper | (reiffen,ZZ,ZZ) | Reiffen's curve | reiffen(p,q) | reiffen(4,5) | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/paco-anton-paper.m2 | stable | HEADLINE: Reiffen's curve
USAGE: reiffen(p,q)
INPUTS: p:ZZ
q:ZZ
OUTPUTS: :RingElement
the defining polynomial of Reiffen's (p,q)-curve
EXAMPLE CODE:
```macaulay2
reiffen(4,5)
``` | Key
reiffen
(reiffen,ZZ,ZZ)
Headline
Reiffen's curve
Usage
reiffen(p,q)
Inputs
p:ZZ
q:ZZ
Outputs
:RingElement
the defining polynomial of Reiffen's (p,q)-curve
Description
Text
Reiffen's (p,q)-curve is a singular curve defined by $f_{p,q} = x_1^p+x_2^q+x_1x_2^{q-1} =... | |||
paco-anton-paper_826968ed_kappaAnnF1PlanarCurve | paco-anton-paper | kappaAnnF1PlanarCurve | D-annihilator of 1/f for a planar curve | kappaAnnF1PlanarCurve f | f = reiffen(4,5)
As = AnnFs f
A = sub(As, {last gens ring As => -1});
(kappa,A') = kappaAnnF1PlanarCurve f
A == sub(A', ring A) | kOrderAnnFa
kOrderAnnFs
AnnFs | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/paco-anton-paper.m2 | stable | HEADLINE: D-annihilator of 1/f for a planar curve
USAGE: kappaAnnF1PlanarCurve f
INPUTS: k:ZZ
positive
f:RingElement
a polynomial in $R = K[x_1,x_2]$
OUTPUTS: I:Ideal
an ideal in the Weyl algebra $D = K<x_1,x_2,\partial_1,\partial_2>$
EXAMPLE CODE:
```macaulay2
f = reiffen(4,5)
As = AnnF... | Key
kappaAnnF1PlanarCurve
(kappaAnnF1PlanarCurve,RingElement)
Headline
D-annihilator of 1/f for a planar curve
Usage
kappaAnnF1PlanarCurve f
Inputs
k:ZZ
positive
f:RingElement
a polynomial in $R = K[x_1,x_2]$
Outputs
I:Ideal
an ideal in the Weyl algebra $D = K<... | ||
paco-anton-paper_826968ed_(kappaAnnF1PlanarCurve,RingElement) | paco-anton-paper | (kappaAnnF1PlanarCurve,RingElement) | D-annihilator of 1/f for a planar curve | kappaAnnF1PlanarCurve f | f = reiffen(4,5)
As = AnnFs f
A = sub(As, {last gens ring As => -1});
(kappa,A') = kappaAnnF1PlanarCurve f
A == sub(A', ring A) | kOrderAnnFa
kOrderAnnFs
AnnFs | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/paco-anton-paper.m2 | stable | HEADLINE: D-annihilator of 1/f for a planar curve
USAGE: kappaAnnF1PlanarCurve f
INPUTS: k:ZZ
positive
f:RingElement
a polynomial in $R = K[x_1,x_2]$
OUTPUTS: I:Ideal
an ideal in the Weyl algebra $D = K<x_1,x_2,\partial_1,\partial_2>$
EXAMPLE CODE:
```macaulay2
f = reiffen(4,5)
As = AnnF... | Key
kappaAnnF1PlanarCurve
(kappaAnnF1PlanarCurve,RingElement)
Headline
D-annihilator of 1/f for a planar curve
Usage
kappaAnnF1PlanarCurve f
Inputs
k:ZZ
positive
f:RingElement
a polynomial in $R = K[x_1,x_2]$
Outputs
I:Ideal
an ideal in the Weyl algebra $D = K<... | ||
paco-anton-paper_38df5a1e_kOrderAnnFa | paco-anton-paper | kOrderAnnFa | k-th order D-annihilator of a power of a polynomial | kOrderAnnFa(k,f,a)
kOrderAnnFs(k,f) | R = QQ[x_1,x_2]; f = x_1^2-x_2^3;
A1 = kOrderAnnFa(1,f,-1)
As = kOrderAnnFs(1,f) | kappaAnnF1PlanarCurve
AnnFs | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/paco-anton-paper.m2 | stable | HEADLINE: k-th order D-annihilator of a power of a polynomial
USAGE: kOrderAnnFa(k,f,a)
kOrderAnnFs(k,f)
INPUTS: k:ZZ
positive
f:RingElement
a polynomial in $K[x_1,...,x_n]$
a:ZZ
(usually negative) exponent
OUTPUTS: I:Ideal
an ideal in the Weyl algebra $K<x_1,...,x_n,\partial_1,..... | Key
kOrderAnnFa
(kOrderAnnFa,ZZ,RingElement,ZZ)
kOrderAnnFs
(kOrderAnnFs,ZZ,RingElement)
Headline
k-th order D-annihilator of a power of a polynomial
Usage
kOrderAnnFa(k,f,a)
kOrderAnnFs(k,f)
Inputs
k:ZZ
positive
f:RingElement
a polynomial in $K[x_1,...,x_n]$
... | ||
paco-anton-paper_38df5a1e_(kOrderAnnFa,ZZ,RingElement,ZZ) | paco-anton-paper | (kOrderAnnFa,ZZ,RingElement,ZZ) | k-th order D-annihilator of a power of a polynomial | kOrderAnnFa(k,f,a)
kOrderAnnFs(k,f) | R = QQ[x_1,x_2]; f = x_1^2-x_2^3;
A1 = kOrderAnnFa(1,f,-1)
As = kOrderAnnFs(1,f) | kappaAnnF1PlanarCurve
AnnFs | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/paco-anton-paper.m2 | stable | HEADLINE: k-th order D-annihilator of a power of a polynomial
USAGE: kOrderAnnFa(k,f,a)
kOrderAnnFs(k,f)
INPUTS: k:ZZ
positive
f:RingElement
a polynomial in $K[x_1,...,x_n]$
a:ZZ
(usually negative) exponent
OUTPUTS: I:Ideal
an ideal in the Weyl algebra $K<x_1,...,x_n,\partial_1,..... | Key
kOrderAnnFa
(kOrderAnnFa,ZZ,RingElement,ZZ)
kOrderAnnFs
(kOrderAnnFs,ZZ,RingElement)
Headline
k-th order D-annihilator of a power of a polynomial
Usage
kOrderAnnFa(k,f,a)
kOrderAnnFs(k,f)
Inputs
k:ZZ
positive
f:RingElement
a polynomial in $K[x_1,...,x_n]$
... | ||
paco-anton-paper_38df5a1e_kOrderAnnFs | paco-anton-paper | kOrderAnnFs | k-th order D-annihilator of a power of a polynomial | kOrderAnnFa(k,f,a)
kOrderAnnFs(k,f) | R = QQ[x_1,x_2]; f = x_1^2-x_2^3;
A1 = kOrderAnnFa(1,f,-1)
As = kOrderAnnFs(1,f) | kappaAnnF1PlanarCurve
AnnFs | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/paco-anton-paper.m2 | stable | HEADLINE: k-th order D-annihilator of a power of a polynomial
USAGE: kOrderAnnFa(k,f,a)
kOrderAnnFs(k,f)
INPUTS: k:ZZ
positive
f:RingElement
a polynomial in $K[x_1,...,x_n]$
a:ZZ
(usually negative) exponent
OUTPUTS: I:Ideal
an ideal in the Weyl algebra $K<x_1,...,x_n,\partial_1,..... | Key
kOrderAnnFa
(kOrderAnnFa,ZZ,RingElement,ZZ)
kOrderAnnFs
(kOrderAnnFs,ZZ,RingElement)
Headline
k-th order D-annihilator of a power of a polynomial
Usage
kOrderAnnFa(k,f,a)
kOrderAnnFs(k,f)
Inputs
k:ZZ
positive
f:RingElement
a polynomial in $K[x_1,...,x_n]$
... | ||
paco-anton-paper_38df5a1e_(kOrderAnnFs,ZZ,RingElement) | paco-anton-paper | (kOrderAnnFs,ZZ,RingElement) | k-th order D-annihilator of a power of a polynomial | kOrderAnnFa(k,f,a)
kOrderAnnFs(k,f) | R = QQ[x_1,x_2]; f = x_1^2-x_2^3;
A1 = kOrderAnnFa(1,f,-1)
As = kOrderAnnFs(1,f) | kappaAnnF1PlanarCurve
AnnFs | M2_git/M2/M2/Macaulay2/packages/BernsteinSato/DOC/paco-anton-paper.m2 | stable | HEADLINE: k-th order D-annihilator of a power of a polynomial
USAGE: kOrderAnnFa(k,f,a)
kOrderAnnFs(k,f)
INPUTS: k:ZZ
positive
f:RingElement
a polynomial in $K[x_1,...,x_n]$
a:ZZ
(usually negative) exponent
OUTPUTS: I:Ideal
an ideal in the Weyl algebra $K<x_1,...,x_n,\partial_1,..... | Key
kOrderAnnFa
(kOrderAnnFa,ZZ,RingElement,ZZ)
kOrderAnnFs
(kOrderAnnFs,ZZ,RingElement)
Headline
k-th order D-annihilator of a power of a polynomial
Usage
kOrderAnnFa(k,f,a)
kOrderAnnFs(k,f)
Inputs
k:ZZ
positive
f:RingElement
a polynomial in $K[x_1,...,x_n]$
... | ||
doc_be731b05_BertiniInputConfiguration | doc | BertiniInputConfiguration | a key to designate a configuration in a Bertini input file | M2_git/M2/M2/Macaulay2/packages/Bertini/doc.m2 | stable | HEADLINE: a key to designate a configuration in a Bertini input file | Key
BertiniInputConfiguration
SecurityLevel
ScreenOut
OutputLevel
StepsForIncrease
MaxNewtonIts
MaxStepSize
MaxNumberSteps
MaxCycleNum
RegenStartLevel
AccuracyEst
MPType
PRECISION
ODEPredictor
TrackTolBeforeEG
TrackTolDuringEG
FinalTol
MaxNorm
MinStepSizeBeforeEG... | |||||
doc_be731b05_SecurityLevel | doc | SecurityLevel | a key to designate a configuration in a Bertini input file | M2_git/M2/M2/Macaulay2/packages/Bertini/doc.m2 | stable | HEADLINE: a key to designate a configuration in a Bertini input file | Key
BertiniInputConfiguration
SecurityLevel
ScreenOut
OutputLevel
StepsForIncrease
MaxNewtonIts
MaxStepSize
MaxNumberSteps
MaxCycleNum
RegenStartLevel
AccuracyEst
MPType
PRECISION
ODEPredictor
TrackTolBeforeEG
TrackTolDuringEG
FinalTol
MaxNorm
MinStepSizeBeforeEG... | |||||
doc_be731b05_ScreenOut | doc | ScreenOut | a key to designate a configuration in a Bertini input file | M2_git/M2/M2/Macaulay2/packages/Bertini/doc.m2 | stable | HEADLINE: a key to designate a configuration in a Bertini input file | Key
BertiniInputConfiguration
SecurityLevel
ScreenOut
OutputLevel
StepsForIncrease
MaxNewtonIts
MaxStepSize
MaxNumberSteps
MaxCycleNum
RegenStartLevel
AccuracyEst
MPType
PRECISION
ODEPredictor
TrackTolBeforeEG
TrackTolDuringEG
FinalTol
MaxNorm
MinStepSizeBeforeEG... | |||||
doc_be731b05_OutputLevel | doc | OutputLevel | a key to designate a configuration in a Bertini input file | M2_git/M2/M2/Macaulay2/packages/Bertini/doc.m2 | stable | HEADLINE: a key to designate a configuration in a Bertini input file | Key
BertiniInputConfiguration
SecurityLevel
ScreenOut
OutputLevel
StepsForIncrease
MaxNewtonIts
MaxStepSize
MaxNumberSteps
MaxCycleNum
RegenStartLevel
AccuracyEst
MPType
PRECISION
ODEPredictor
TrackTolBeforeEG
TrackTolDuringEG
FinalTol
MaxNorm
MinStepSizeBeforeEG... | |||||
doc_be731b05_StepsForIncrease | doc | StepsForIncrease | a key to designate a configuration in a Bertini input file | M2_git/M2/M2/Macaulay2/packages/Bertini/doc.m2 | stable | HEADLINE: a key to designate a configuration in a Bertini input file | Key
BertiniInputConfiguration
SecurityLevel
ScreenOut
OutputLevel
StepsForIncrease
MaxNewtonIts
MaxStepSize
MaxNumberSteps
MaxCycleNum
RegenStartLevel
AccuracyEst
MPType
PRECISION
ODEPredictor
TrackTolBeforeEG
TrackTolDuringEG
FinalTol
MaxNorm
MinStepSizeBeforeEG... | |||||
doc_be731b05_MaxNewtonIts | doc | MaxNewtonIts | a key to designate a configuration in a Bertini input file | M2_git/M2/M2/Macaulay2/packages/Bertini/doc.m2 | stable | HEADLINE: a key to designate a configuration in a Bertini input file | Key
BertiniInputConfiguration
SecurityLevel
ScreenOut
OutputLevel
StepsForIncrease
MaxNewtonIts
MaxStepSize
MaxNumberSteps
MaxCycleNum
RegenStartLevel
AccuracyEst
MPType
PRECISION
ODEPredictor
TrackTolBeforeEG
TrackTolDuringEG
FinalTol
MaxNorm
MinStepSizeBeforeEG... | |||||
doc_be731b05_MaxStepSize | doc | MaxStepSize | a key to designate a configuration in a Bertini input file | M2_git/M2/M2/Macaulay2/packages/Bertini/doc.m2 | stable | HEADLINE: a key to designate a configuration in a Bertini input file | Key
BertiniInputConfiguration
SecurityLevel
ScreenOut
OutputLevel
StepsForIncrease
MaxNewtonIts
MaxStepSize
MaxNumberSteps
MaxCycleNum
RegenStartLevel
AccuracyEst
MPType
PRECISION
ODEPredictor
TrackTolBeforeEG
TrackTolDuringEG
FinalTol
MaxNorm
MinStepSizeBeforeEG... | |||||
doc_be731b05_MaxNumberSteps | doc | MaxNumberSteps | a key to designate a configuration in a Bertini input file | M2_git/M2/M2/Macaulay2/packages/Bertini/doc.m2 | stable | HEADLINE: a key to designate a configuration in a Bertini input file | Key
BertiniInputConfiguration
SecurityLevel
ScreenOut
OutputLevel
StepsForIncrease
MaxNewtonIts
MaxStepSize
MaxNumberSteps
MaxCycleNum
RegenStartLevel
AccuracyEst
MPType
PRECISION
ODEPredictor
TrackTolBeforeEG
TrackTolDuringEG
FinalTol
MaxNorm
MinStepSizeBeforeEG... | |||||
doc_be731b05_MaxCycleNum | doc | MaxCycleNum | a key to designate a configuration in a Bertini input file | M2_git/M2/M2/Macaulay2/packages/Bertini/doc.m2 | stable | HEADLINE: a key to designate a configuration in a Bertini input file | Key
BertiniInputConfiguration
SecurityLevel
ScreenOut
OutputLevel
StepsForIncrease
MaxNewtonIts
MaxStepSize
MaxNumberSteps
MaxCycleNum
RegenStartLevel
AccuracyEst
MPType
PRECISION
ODEPredictor
TrackTolBeforeEG
TrackTolDuringEG
FinalTol
MaxNorm
MinStepSizeBeforeEG... | |||||
doc_be731b05_RegenStartLevel | doc | RegenStartLevel | a key to designate a configuration in a Bertini input file | M2_git/M2/M2/Macaulay2/packages/Bertini/doc.m2 | stable | HEADLINE: a key to designate a configuration in a Bertini input file | Key
BertiniInputConfiguration
SecurityLevel
ScreenOut
OutputLevel
StepsForIncrease
MaxNewtonIts
MaxStepSize
MaxNumberSteps
MaxCycleNum
RegenStartLevel
AccuracyEst
MPType
PRECISION
ODEPredictor
TrackTolBeforeEG
TrackTolDuringEG
FinalTol
MaxNorm
MinStepSizeBeforeEG... | |||||
doc_be731b05_AccuracyEst | doc | AccuracyEst | a key to designate a configuration in a Bertini input file | M2_git/M2/M2/Macaulay2/packages/Bertini/doc.m2 | stable | HEADLINE: a key to designate a configuration in a Bertini input file | Key
BertiniInputConfiguration
SecurityLevel
ScreenOut
OutputLevel
StepsForIncrease
MaxNewtonIts
MaxStepSize
MaxNumberSteps
MaxCycleNum
RegenStartLevel
AccuracyEst
MPType
PRECISION
ODEPredictor
TrackTolBeforeEG
TrackTolDuringEG
FinalTol
MaxNorm
MinStepSizeBeforeEG... | |||||
doc_be731b05_MPType | doc | MPType | a key to designate a configuration in a Bertini input file | M2_git/M2/M2/Macaulay2/packages/Bertini/doc.m2 | stable | HEADLINE: a key to designate a configuration in a Bertini input file | Key
BertiniInputConfiguration
SecurityLevel
ScreenOut
OutputLevel
StepsForIncrease
MaxNewtonIts
MaxStepSize
MaxNumberSteps
MaxCycleNum
RegenStartLevel
AccuracyEst
MPType
PRECISION
ODEPredictor
TrackTolBeforeEG
TrackTolDuringEG
FinalTol
MaxNorm
MinStepSizeBeforeEG... | |||||
doc_be731b05_PRECISION | doc | PRECISION | a key to designate a configuration in a Bertini input file | M2_git/M2/M2/Macaulay2/packages/Bertini/doc.m2 | stable | HEADLINE: a key to designate a configuration in a Bertini input file | Key
BertiniInputConfiguration
SecurityLevel
ScreenOut
OutputLevel
StepsForIncrease
MaxNewtonIts
MaxStepSize
MaxNumberSteps
MaxCycleNum
RegenStartLevel
AccuracyEst
MPType
PRECISION
ODEPredictor
TrackTolBeforeEG
TrackTolDuringEG
FinalTol
MaxNorm
MinStepSizeBeforeEG... | |||||
doc_be731b05_ODEPredictor | doc | ODEPredictor | a key to designate a configuration in a Bertini input file | M2_git/M2/M2/Macaulay2/packages/Bertini/doc.m2 | stable | HEADLINE: a key to designate a configuration in a Bertini input file | Key
BertiniInputConfiguration
SecurityLevel
ScreenOut
OutputLevel
StepsForIncrease
MaxNewtonIts
MaxStepSize
MaxNumberSteps
MaxCycleNum
RegenStartLevel
AccuracyEst
MPType
PRECISION
ODEPredictor
TrackTolBeforeEG
TrackTolDuringEG
FinalTol
MaxNorm
MinStepSizeBeforeEG... |
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