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::
ZZ
ZZ -- the class of all integers
Functions and methods returning an integer :
# BasicList
-- length or cardinality
"# HashTable"
-- see
# BasicList
-- length or cardinality
"# Set"
-- see
# BasicList
-- length or cardinality
"# String"
-- see
# BasicList
-- length or cardinality
"ZZ * ZZ"
-- see
*
-- a binary operator, usually used for multiplication
"+ ZZ"
-- see
+
-- a unary or binary operator, usually used for addition
"ZZ + ZZ"
-- see
+
-- a unary or binary operator, usually used for addition
"- ZZ"
-- see
-
-- a unary or binary operator, usually used for negation or subtraction
"ZZ - ZZ"
-- see
-
-- a unary or binary operator, usually used for negation or subtraction
"ZZ / ZZ"
-- see
/
-- a binary operator, usually used for division
"ZZ // ZZ"
-- see
//
-- a binary operator, usually used for quotient
"analyticSpread(Ideal)"
-- see
analyticSpread
-- Compute the analytic spread of a module or ideal
"analyticSpread(Ideal,RingElement)"
-- see
analyticSpread
-- Compute the analytic spread of a module or ideal
"analyticSpread(Module)"
-- see
analyticSpread
-- Compute the analytic spread of a module or ideal
"analyticSpread(Module,RingElement)"
-- see
analyticSpread
-- Compute the analytic spread of a module or ideal
binomial
-- binomial coefficient
"binomial(Constant,Constant)"
-- see
binomial
-- binomial coefficient
"binomial(Constant,Number)"
-- see
binomial
-- binomial coefficient
"binomial(Number,Constant)"
-- see
binomial
-- binomial coefficient
"binomial(Number,Number)"
-- see
binomial
-- binomial coefficient
"changeBase(String,ZZ)"
-- see
changeBase
-- change integer base
char
-- computes the characteristic of the ring or field
degreeLength
-- the length of the degree vector
depth
-- depth of a net
dim
-- compute the Krull dimension
"dim(GaloisField)"
-- see
dim(Ring)
-- compute the Krull dimension
euler
-- Euler characteristic
fileMode(String)
-- get file mode
"gcd(ZZ,ZZ)"
-- see
gcd
-- greatest common divisor
genus
-- arithmetic genus
height
-- height of a net
length
-- length of an object
length(Module)
-- Computes the length of a module
"maxPosition(BasicList)"
-- see
maxPosition
-- position of the largest element
"minPosition(BasicList)"
-- see
minPosition
-- position of the smallest element
"multiplicity(Ideal)"
-- see
multiplicity
-- Compute the Hilbert-Samuel multiplicity of an ideal
"multiplicity(Ideal,RingElement)"
-- see
multiplicity
-- Compute the Hilbert-Samuel multiplicity of an ideal
nextPrime
-- compute the smallest prime greater than or equal to a given number
"numColumns"
-- see
numColumns(Matrix)
-- number of columns in a matrix or mutable matrix
numgens
-- the number of generators
"numRows"
-- see
numRows(Matrix)
-- number of rows in a matrix or mutable matrix
pdim
-- calculate the projective dimension
"powermod(ZZ,ZZ,ZZ)"
-- see
powermod
-- powers of integers mod N
"random(ZZ)"
-- see
random(ZZ,ZZ)
-- get a random integer or real number
random(ZZ,ZZ)
-- get a random integer or real number
rank
-- compute the rank
regularity
-- compute the Castelnuovo-Mumford regularity
"size2(CC)"
-- see
size2
-- number of binary digits to the left of the point
"size2(RR)"
-- see
size2
-- number of binary digits to the left of the point
"size2(RRi)"
-- see
size2
-- number of binary digits to the left of the point
"size2(ZZ)"
-- see
size2
-- number of binary digits to the left of the point
width
-- width of a file or net
ZZ & ZZ
-- logical and
Methods that use an integer :
"ZZ !"
-- see
!
-- factorial
"BasicList #? ZZ"
-- see
#?
-- check existence of value in a list, hash table, database, or string
"String #? ZZ"
-- see
#?
-- check existence of value in a list, hash table, database, or string
"CC % ZZ"
-- see
%
-- a binary operator, usually used for remainder and reduction
"QQ % ZZ"
-- see
%
-- a binary operator, usually used for remainder and reduction
"RR % ZZ"
-- see
%
-- a binary operator, usually used for remainder and reduction
"ZZ % MonomialIdeal"
-- see
%
-- a binary operator, usually used for remainder and reduction
"ZZ % ZZ"
-- see
%
-- a binary operator, usually used for remainder and reduction
"CC * ZZ"
-- see
*
-- a binary operator, usually used for multiplication
"ProjectiveHilbertPolynomial * ZZ"
-- see
*
-- a binary operator, usually used for multiplication
"QQ * ZZ"
-- see
*
-- a binary operator, usually used for multiplication
"RR * ZZ"
-- see
*
-- a binary operator, usually used for multiplication
"RRi * ZZ"
-- see
*
-- a binary operator, usually used for multiplication
"ZZ * CC"
-- see
*
-- a binary operator, usually used for multiplication
"ZZ * ProjectiveHilbertPolynomial"
-- see
*
-- a binary operator, usually used for multiplication
"ZZ * QQ"
-- see
*
-- a binary operator, usually used for multiplication
"ZZ * RR"
-- see
*
-- a binary operator, usually used for multiplication
"ZZ * RRi"
-- see
*
-- a binary operator, usually used for multiplication
"CC + ZZ"
-- see
+
-- a unary or binary operator, usually used for addition
"ChainComplexMap + ZZ"
-- see
+
-- a unary or binary operator, usually used for addition
"ProjectiveHilbertPolynomial + ZZ"
-- see
+
-- a unary or binary operator, usually used for addition
"QQ + ZZ"
-- see
+
-- a unary or binary operator, usually used for addition
"RR + ZZ"
-- see
+
-- a unary or binary operator, usually used for addition
"RRi + ZZ"
-- see
+
-- a unary or binary operator, usually used for addition
"ZZ + CC"
-- see
+
-- a unary or binary operator, usually used for addition
"ZZ + ChainComplexMap"
-- see
+
-- a unary or binary operator, usually used for addition
"ZZ + ProjectiveHilbertPolynomial"
-- see
+
-- a unary or binary operator, usually used for addition
"ZZ + QQ"
-- see
+
-- a unary or binary operator, usually used for addition
"ZZ + RR"
-- see
+
-- a unary or binary operator, usually used for addition
"ZZ + RRi"
-- see
+
-- a unary or binary operator, usually used for addition
"CC - ZZ"
-- see
-
-- a unary or binary operator, usually used for negation or subtraction
"ProjectiveHilbertPolynomial - ZZ"
-- see
-
-- a unary or binary operator, usually used for negation or subtraction
"QQ - ZZ"
-- see
-
-- a unary or binary operator, usually used for negation or subtraction
"RR - ZZ"
-- see
-
-- a unary or binary operator, usually used for negation or subtraction
"RRi - ZZ"
-- see
-
-- a unary or binary operator, usually used for negation or subtraction
"ZZ - CC"
-- see
-
-- a unary or binary operator, usually used for negation or subtraction
"ZZ - ProjectiveHilbertPolynomial"
-- see
-
-- a unary or binary operator, usually used for negation or subtraction
"ZZ - QQ"
-- see
-
-- a unary or binary operator, usually used for negation or subtraction
"ZZ - RR"
-- see
-
-- a unary or binary operator, usually used for negation or subtraction
"ZZ - RRi"
-- see
-
-- a unary or binary operator, usually used for negation or subtraction
"CC / ZZ"
-- see
/
-- a binary operator, usually used for division
"QQ / ZZ"
-- see
/
-- a binary operator, usually used for division
"RR / ZZ"
-- see
/
-- a binary operator, usually used for division
"RRi / ZZ"
-- see
/
-- a binary operator, usually used for division
"ZZ / CC"
-- see
/
-- a binary operator, usually used for division
"ZZ / QQ"
-- see
/
-- a binary operator, usually used for division
"ZZ / RR"
-- see
/
-- a binary operator, usually used for division
"ZZ / RRi"
-- see
/
-- a binary operator, usually used for division
"CC // ZZ"
-- see
//
-- a binary operator, usually used for quotient
"InfiniteNumber // ZZ"
-- see
//
-- a binary operator, usually used for quotient
"QQ // ZZ"
-- see
//
-- a binary operator, usually used for quotient
"RR // ZZ"
-- see
//
-- a binary operator, usually used for quotient
"ZZ // MonomialIdeal"
-- see
//
-- a binary operator, usually used for quotient
"CC == ZZ"
-- see
==
-- equality
"ChainComplex == ZZ"
-- see
==
-- equality
"ChainComplexMap == ZZ"
-- see
==
-- equality
"GradedModuleMap == ZZ"
-- see
==
-- equality
"Ideal == ZZ"
-- see
==
-- equality
"Matrix == ZZ"
-- see
==
-- equality
"Module == ZZ"
-- see
==
-- equality
"MonomialIdeal == ZZ"
-- see
==
-- equality
"MutableMatrix == ZZ"
-- see
==
-- equality
"ProjectiveHilbertPolynomial == ZZ"
-- see
==
-- equality
"QQ == ZZ"
-- see
==
-- equality
"Ring == ZZ"
-- see
==
-- equality
"RingElement == ZZ"
-- see
==
-- equality
"RingMap == ZZ"
-- see
==
-- equality
"RR == ZZ"
-- see
==
-- equality
"RRi == ZZ"
-- see
==
-- equality
"ZZ == CC"
-- see
==
-- equality
"ZZ == ChainComplex"
-- see
==
-- equality
"ZZ == ChainComplexMap"
-- see
==
-- equality
"ZZ == GradedModuleMap"
-- see
==
-- equality
"ZZ == Ideal"
-- see
==
-- equality
"ZZ == Module"
-- see
==
-- equality
"ZZ == MonomialIdeal"
-- see
==
-- equality
"ZZ == MutableMatrix"
-- see
==
-- equality
"ZZ == ProjectiveHilbertPolynomial"
-- see
==
-- equality
"ZZ == QQ"
-- see
==
-- equality
"ZZ == Ring"
-- see
==
-- equality
"ZZ == RingElement"
-- see
==
-- equality
"ZZ == RingMap"
-- see
==
-- equality
"ZZ == RR"
-- see
==
-- equality
"ZZ == RRi"
-- see
==
-- equality
"ZZ == ZZ"
-- see
==
-- equality
> ZZ
-- greater than
>= ZZ
-- greater than or equal
"ZZ ^^ ZZ"
-- see
^^
-- logical exclusive-or
"help(ZZ)"
-- see
about
-- search the documentation
"infoHelp(ZZ)"
-- see
about
-- search the documentation
"viewHelp(ZZ)"
-- see
about
-- search the documentation
"abs(ZZ)"
-- see
abs
-- absolute value function
"all(ZZ,Function)"
-- see
all(BasicList,Function)
-- whether all elements of a list satisfy a specified condition
"any(ZZ,Function)"
-- see
any(BasicList,Function)
-- whether any elements of a list satisfy a specified condition
apply(ZZ,Function)
-- apply a function to {0,..., n-1}
BasicList # ZZ
-- get value from list, hash table, database, or string
"String # ZZ"
-- see
BasicList # ZZ
-- get value from list, hash table, database, or string
"basis(InfiniteNumber,ZZ,Ideal)"
-- see
basis
-- basis or generating set of all or part of a ring, ideal or module
"basis(InfiniteNumber,ZZ,Matrix)"
-- see
basis
-- basis or generating set of all or part of a ring, ideal or module
"basis(InfiniteNumber,ZZ,Module)"
-- see
basis
-- basis or generating set of all or part of a ring, ideal or module
"basis(InfiniteNumber,ZZ,Ring)"
-- see
basis
-- basis or generating set of all or part of a ring, ideal or module
"basis(List,ZZ,Ideal)"
-- see
basis
-- basis or generating set of all or part of a ring, ideal or module
"basis(List,ZZ,Matrix)"
-- see
basis
-- basis or generating set of all or part of a ring, ideal or module
"basis(List,ZZ,Module)"
-- see
basis
-- basis or generating set of all or part of a ring, ideal or module
"basis(List,ZZ,Ring)"
-- see
basis
-- basis or generating set of all or part of a ring, ideal or module
"basis(ZZ,Ideal)"
-- see
basis
-- basis or generating set of all or part of a ring, ideal or module
"basis(ZZ,InfiniteNumber,Ideal)"
-- see
basis
-- basis or generating set of all or part of a ring, ideal or module
"basis(ZZ,InfiniteNumber,Matrix)"
-- see
basis
-- basis or generating set of all or part of a ring, ideal or module
"basis(ZZ,InfiniteNumber,Module)"
-- see
basis
-- basis or generating set of all or part of a ring, ideal or module
"basis(ZZ,InfiniteNumber,Ring)"
-- see
basis
-- basis or generating set of all or part of a ring, ideal or module
"basis(ZZ,List,Ideal)"
-- see
basis
-- basis or generating set of all or part of a ring, ideal or module
"basis(ZZ,List,Matrix)"
-- see
basis
-- basis or generating set of all or part of a ring, ideal or module
"basis(ZZ,List,Module)"
-- see
basis
-- basis or generating set of all or part of a ring, ideal or module
"basis(ZZ,List,Ring)"
-- see
basis
-- basis or generating set of all or part of a ring, ideal or module
"basis(ZZ,Matrix)"
-- see
basis
-- basis or generating set of all or part of a ring, ideal or module
"basis(ZZ,Module)"
-- see
basis
-- basis or generating set of all or part of a ring, ideal or module
"basis(ZZ,Ring)"
-- see
basis
-- basis or generating set of all or part of a ring, ideal or module
"basis(ZZ,ZZ,Ideal)"
-- see
basis
-- basis or generating set of all or part of a ring, ideal or module
"basis(ZZ,ZZ,Matrix)"
-- see
basis
-- basis or generating set of all or part of a ring, ideal or module
"basis(ZZ,ZZ,Module)"
-- see
basis
-- basis or generating set of all or part of a ring, ideal or module
"basis(ZZ,ZZ,Ring)"
-- see
basis
-- basis or generating set of all or part of a ring, ideal or module
"BesselJ(ZZ,Constant)"
-- see
BesselJ
-- Bessel function of the first kind
"BesselJ(ZZ,Number)"
-- see
BesselJ
-- Bessel function of the first kind
"BesselY(ZZ,Constant)"
-- see
BesselY
-- Bessel function of the second kind
"BesselY(ZZ,Number)"
-- see
BesselY
-- Bessel function of the second kind
"BettiTally ZZ"
-- see
BettiTally
-- the class of all Betti tallies
"hilbertPolynomial(ZZ,BettiTally)"
-- see
BettiTally
-- the class of all Betti tallies
"hilbertSeries(ZZ,BettiTally)"
-- see
BettiTally
-- the class of all Betti tallies
"lift(BettiTally,type of ZZ)"
-- see
BettiTally
-- the class of all Betti tallies
"ZZ * BettiTally"
-- see
BettiTally
-- the class of all Betti tallies
"binomial(RingElement,ZZ)"
-- see
binomial
-- binomial coefficient
ChainComplex ^ ZZ
-- access member, cohomological degree
ChainComplex _ ZZ
-- component
"GradedModule _ ZZ"
-- see
ChainComplex _ ZZ
-- component
ChainComplex _ ZZ = Thing
-- install component of chain complex
ChainComplexMap ^ ZZ
-- iterated composition
"GradedModuleMap ^ ZZ"
-- see
ChainComplexMap ^ ZZ
-- iterated composition
ChainComplexMap _ ZZ
-- component map
"GradedModuleMap _ ZZ"
-- see
ChainComplexMap _ ZZ
-- component map
ChainComplexMap _ ZZ = Thing
-- install component of chain complex map
"changeBase(String,ZZ,ZZ)"
-- see
changeBase
-- change integer base
"changeBase(ZZ,ZZ)"
-- see
changeBase
-- change integer base
"check(ZZ,Package)"
-- see
check
-- perform tests of a package
"check(ZZ,String)"
-- see
check
-- perform tests of a package
"code(ZZ)"
-- see
code
-- display source code
CoherentSheaf ^ ZZ
-- direct sum
"SheafOfRings ^ ZZ"
-- see
CoherentSheaf ^ ZZ
-- direct sum
CoherentSheaf ^** ZZ
-- tensor power
CoherentSheaf ZZ
-- canonical twist of a coherent sheaf
"SheafOfRings ZZ"
-- see
CoherentSheaf ZZ
-- canonical twist of a coherent sheaf
"columnAdd(MutableMatrix,ZZ,Number,ZZ)"
-- see
columnAdd
-- add a multiple of one column to another
"columnAdd(MutableMatrix,ZZ,RingElement,ZZ)"
-- see
columnAdd
-- add a multiple of one column to another
"columnMult(MutableMatrix,ZZ,Number)"
-- see
columnMult
-- multiply a column by a ring element
"columnMult(MutableMatrix,ZZ,RingElement)"
-- see
columnMult
-- multiply a column by a ring element
"columnPermute(MutableMatrix,ZZ,List)"
-- see
columnPermute
-- permute some columns
"columnSwap(MutableMatrix,ZZ,ZZ)"
-- see
columnSwap
-- interchange columns
"compositions(ZZ,ZZ)"
-- see
compositions
-- list the compositions of an integer
compositions(ZZ)
(missing documentation)
"concatenate(ZZ)"
-- see
concatenate
-- join strings
"conwayPolynomial(ZZ)"
-- see
conwayPolynomial
-- provide a Conway polynomial
"conwayPolynomial(ZZ,ZZ)"
-- see
conwayPolynomial
-- provide a Conway polynomial
cotangentSheaf(ZZ,ProjectiveVariety)
-- exterior powers of the cotangent sheaf of a projective variety
debug(ZZ)
(missing documentation)
"degreesMonoid(ZZ)"
-- see
degreesRing(List)
-- the ring or monoid of degrees
"degreesRing(ZZ)"
-- see
degreesRing(List)
-- the ring or monoid of degrees
diagonalMatrix(Ring,ZZ,ZZ,List)
-- make a diagonal matrix from a list
"diagonalMatrix(RingFamily,ZZ,ZZ,List)"
-- see
diagonalMatrix(Ring,ZZ,ZZ,List)
-- make a diagonal matrix from a list
"diagonalMatrix(ZZ,ZZ,List)"
-- see
diagonalMatrix(Ring,ZZ,ZZ,List)
-- make a diagonal matrix from a list
diff(ProjectiveHilbertPolynomial,ZZ)
-- differentiate or take difference
"divideByVariable(Matrix,RingElement,ZZ)"
-- see
divideByVariable
-- divide all columns by a (power of a) variable
"drop(BasicList,ZZ)"
-- see
drop
-- drop some elements from a list or sequence.
Ext^ZZ(CoherentSheaf,CoherentSheaf)
-- global Ext
"Ext^ZZ(CoherentSheaf,SheafOfRings)"
-- see
Ext^ZZ(CoherentSheaf,CoherentSheaf)
-- global Ext
"Ext^ZZ(SheafOfRings,CoherentSheaf)"
-- see
Ext^ZZ(CoherentSheaf,CoherentSheaf)
-- global Ext
"Ext^ZZ(SheafOfRings,SheafOfRings)"
-- see
Ext^ZZ(CoherentSheaf,CoherentSheaf)
-- global Ext
Ext^ZZ(CoherentSheaf,SumOfTwists)
-- global Ext
"Ext^ZZ(SheafOfRings,SumOfTwists)"
-- see
Ext^ZZ(CoherentSheaf,SumOfTwists)
-- global Ext
"Ext^ZZ(Matrix,Ideal)"
-- see
Ext^ZZ(Matrix,Module)
-- map between Ext modules
Ext^ZZ(Matrix,Module)
-- map between Ext modules
"Ext^ZZ(Matrix,Ring)"
-- see
Ext^ZZ(Matrix,Module)
-- map between Ext modules
"Ext^ZZ(Ideal,Matrix)"
-- see
Ext^ZZ(Module,Matrix)
-- map between Ext modules
Ext^ZZ(Module,Matrix)
-- map between Ext modules
"Ext^ZZ(Ideal,Ideal)"
-- see
Ext^ZZ(Module,Module)
-- Ext module
"Ext^ZZ(Ideal,Module)"
-- see
Ext^ZZ(Module,Module)
-- Ext module
"Ext^ZZ(Ideal,Ring)"
-- see
Ext^ZZ(Module,Module)
-- Ext module
"Ext^ZZ(Module,Ideal)"
-- see
Ext^ZZ(Module,Module)
-- Ext module
Ext^ZZ(Module,Module)
-- Ext module
"Ext^ZZ(Module,Ring)"
-- see
Ext^ZZ(Module,Module)
-- Ext module
exteriorPower(ZZ,CoherentSheaf)
-- exterior power
exteriorPower(ZZ,Matrix)
-- exterior power of a matrix
exteriorPower(ZZ,Module)
-- exterior power of a module
"factor(ZZ)"
-- see
factor(RingElement)
-- factor a ring element
Fano(ZZ,Ideal)
-- Fano scheme
Fano(ZZ,Ideal,Ring)
-- Fano scheme
fileMode(ZZ,File)
-- set file mode
fileMode(ZZ,String)
-- set file mode
"fillMatrix(MutableMatrix,ZZ)"
-- see
fillMatrix
-- fill a mutable matrix with random numbers
"fittingIdeal(ZZ,Module)"
-- see
fittingIdeal
-- Fitting ideal of a module
fittingIdeal(ZZ,Matrix)
(missing documentation)
"gcd(QQ,ZZ)"
-- see
gcd
-- greatest common divisor
"gcd(RingElement,ZZ)"
-- see
gcd
-- greatest common divisor
"gcd(ZZ,QQ)"
-- see
gcd
-- greatest common divisor
"gcd(ZZ,RingElement)"
-- see
gcd
-- greatest common divisor
"gcdCoefficients(ZZ,ZZ)"
-- see
gcdCoefficients
-- gcd with coefficients
"Ideal _ ZZ"
-- see
generators of ideals and modules
"Matrix _ ZZ"
-- see
generators of ideals and modules
"Module _ ZZ"
-- see
generators of ideals and modules
"MonomialIdeal _ ZZ"
-- see
generators of ideals and modules
"genericMatrix(Ring,RingElement,ZZ,ZZ)"
-- see
genericMatrix
-- make a generic matrix of variables
"genericMatrix(Ring,ZZ,ZZ)"
-- see
genericMatrix
-- make a generic matrix of variables
"genericSkewMatrix(Ring,RingElement,ZZ)"
-- see
genericSkewMatrix
-- make a generic skew symmetric matrix of variables
"genericSkewMatrix(Ring,ZZ)"
-- see
genericSkewMatrix
-- make a generic skew symmetric matrix of variables
"genericSymmetricMatrix(Ring,RingElement,ZZ)"
-- see
genericSymmetricMatrix
-- make a generic symmetric matrix
"genericSymmetricMatrix(Ring,ZZ)"
-- see
genericSymmetricMatrix
-- make a generic symmetric matrix
"getPrimeWithRootOfUnity(ZZ,ZZ)"
-- see
getPrimeWithRootOfUnity
-- find a prime p with a primitive n-th root of unity r in ZZ/p
"GF(ZZ)"
-- see
GF
-- make a finite field
"GF(ZZ,ZZ)"
-- see
GF
-- make a finite field
Grassmannian(ZZ,ZZ)
-- the Grassmannian of linear subspaces of a vector space
"Grassmannian(ZZ,ZZ,PolynomialRing)"
-- see
Grassmannian(ZZ,ZZ)
-- the Grassmannian of linear subspaces of a vector space
HH^ZZ ChainComplex
-- cohomology of a chain complex
HH^ZZ ChainComplexMap
-- cohomology of a chain complex map
HH^ZZ Module
-- local cohomology of a module
HH^ZZ SheafOfRings
-- cohomology of a sheaf of rings on a projective variety
HH^ZZ SumOfTwists
-- coherent sheaf cohomology module
"HH^ZZ CoherentSheaf"
-- see
HH^ZZ(ProjectiveVariety,CoherentSheaf)
-- cohomology of a coherent sheaf on a projective variety
HH^ZZ(ProjectiveVariety,CoherentSheaf)
-- cohomology of a coherent sheaf on a projective variety
HH_ZZ ChainComplex
-- homology of a chain complex
HH_ZZ ChainComplexMap
-- homology of a chain complex map
HH_ZZ Sequence
(missing documentation)
"hilbertFunction(ZZ,CoherentSheaf)"
-- see
hilbertFunction
-- the Hilbert function
"hilbertFunction(ZZ,Ideal)"
-- see
hilbertFunction
-- the Hilbert function
"hilbertFunction(ZZ,Module)"
-- see
hilbertFunction
-- the Hilbert function
"hilbertFunction(ZZ,ProjectiveVariety)"
-- see
hilbertFunction
-- the Hilbert function
"hilbertFunction(ZZ,Ring)"
-- see
hilbertFunction
-- the Hilbert function
"hooks(ZZ)"
-- see
hooks
-- list hooks attached to a key
"icPIdeal(RingElement,RingElement,ZZ)"
-- see
icPIdeal
-- compute the integral closure in prime characteristic of a principal ideal
Ideal * ZZ
(missing documentation)
Ideal ^ ZZ
-- power of an ideal
"MonomialIdeal ^ ZZ"
-- see
Ideal ^ ZZ
-- power of an ideal
"insert(ZZ,Thing,VisibleList)"
-- see
insert
-- copy a list, inserting an element
integralClosure(Ideal,RingElement,ZZ)
-- integral closure of an ideal in an affine domain
"integralClosure(Ideal,ZZ)"
-- see
integralClosure(Ideal,RingElement,ZZ)
-- integral closure of an ideal in an affine domain
"interval(QQ,ZZ)"
-- see
interval
-- construct an interval
"interval(RR,ZZ)"
-- see
interval
-- construct an interval
"interval(ZZ)"
-- see
interval
-- construct an interval
"interval(ZZ,QQ)"
-- see
interval
-- construct an interval
"interval(ZZ,RR)"
-- see
interval
-- construct an interval
"interval(ZZ,ZZ)"
-- see
interval
-- construct an interval
"inverseSystem(ZZ,Ideal)"
-- see
inverseSystem
-- Inverse systems with equivariance
"inverseSystem(ZZ,Matrix)"
-- see
inverseSystem
-- Inverse systems with equivariance
"isMember(ZZ,RRi)"
-- see
isMember(QQ,RRi)
-- membership test in an interval
"isPrime(ZZ)"
-- see
isPrime
-- whether a integer or polynomial is prime
isPseudoprime(ZZ)
-- whether an integer is probably prime
"isReal(ZZ)"
-- see
isReal
-- whether a number is real
"isUnit(ZZ)"
-- see
isUnit
-- whether a ring element is a unit
kill(ZZ)
-- kill a process
koszul(ZZ,Matrix)
-- a differential in a Koszul complex
"lcm(QQ,ZZ)"
-- see
lcm
-- least common multiple
"lcm(RingElement,ZZ)"
-- see
lcm
-- least common multiple
"lcm(ZZ,QQ)"
-- see
lcm
-- least common multiple
"lcm(ZZ,RingElement)"
-- see
lcm
-- least common multiple
"lcm(ZZ,ZZ)"
-- see
lcm
-- least common multiple
leadTerm(ZZ,Ideal)
-- get the ideal of lead polynomials
leadTerm(ZZ,Matrix)
-- get the matrix of lead polynomials of each column
leadTerm(ZZ,RingElement)
-- get the lead polynomials using part of the monomial order
"CC << ZZ"
-- see
left shift
"RR << ZZ"
-- see
left shift
"RRi << ZZ"
-- see
left shift
"ZZ << ZZ"
-- see
left shift
"lift(CC,type of ZZ)"
-- see
lift
-- lift to another ring
"lift(Ideal,type of ZZ)"
-- see
lift
-- lift to another ring
"lift(Matrix,type of CC_*,type of ZZ)"
-- see
lift
-- lift to another ring
"lift(Matrix,type of QQ,type of ZZ)"
-- see
lift
-- lift to another ring
"lift(Matrix,type of RR_*,type of ZZ)"
-- see
lift
-- lift to another ring
"lift(Matrix,type of RRi_*,type of ZZ)"
-- see
lift
-- lift to another ring
"lift(Matrix,type of ZZ,type of ZZ)"
-- see
lift
-- lift to another ring
"lift(QQ,type of ZZ)"
-- see
lift
-- lift to another ring
"lift(RR,type of ZZ)"
-- see
lift
-- lift to another ring
"lift(RRi,type of ZZ)"
-- see
lift
-- lift to another ring
"lift(ZZ,type of ZZ)"
-- see
lift
-- lift to another ring
"lngamma(ZZ)"
-- see
lngamma
-- logarithm of the Gamma function
"locate(ZZ)"
-- see
locate
-- locate source code
"map(Module,Module,ZZ)"
-- see
map(Module,Module,RingElement)
-- construct the map induced by multiplication by a ring element on the generators
"map(Module,ZZ,ZZ)"
-- see
map(Module,Module,RingElement)
-- construct the map induced by multiplication by a ring element on the generators
map(Module,ZZ,Function)
-- create a matrix from a free module by specifying a function that gives each entry
map(Module,ZZ,List)
-- create a matrix by giving a sparse or dense list of entries
Matrix ^ ZZ
-- power
minors(ZZ,Matrix)
-- ideal generated by minors
Module * ZZ
(missing documentation)
Module ^ ZZ
-- direct sum
Module ^** ZZ
-- tensor power
MonoidElement * ZZ
(missing documentation)
MonoidElement == ZZ
(missing documentation)
MonoidElement ^ ZZ
(missing documentation)
mutableIdentity(Ring,ZZ)
-- make a mutable identity matrix
"mutableIdentity(RingFamily,ZZ)"
-- see
mutableIdentity(Ring,ZZ)
-- make a mutable identity matrix
MutableMatrix * ZZ
(missing documentation)
MutableMatrix ^ ZZ
(missing documentation)
mutableMatrix(Ring,ZZ,ZZ)
-- make a mutable matrix filled with zeroes
"mutableMatrix(RingFamily,ZZ,ZZ)"
-- see
mutableMatrix(Ring,ZZ,ZZ)
-- make a mutable matrix filled with zeroes
Net ^ ZZ
-- raise a net or string
"String ^ ZZ"
-- see
Net ^ ZZ
-- raise a net or string
"String | ZZ"
-- see
Net | Net
-- join strings or nets
"ZZ | String"
-- see
Net | Net
-- join strings or nets
"numeric(ZZ,CC)"
-- see
numeric
-- convert to floating point
"numeric(ZZ,Constant)"
-- see
numeric
-- convert to floating point
"numeric(ZZ,InfiniteNumber)"
-- see
numeric
-- convert to floating point
"numeric(ZZ,Matrix)"
-- see
numeric
-- convert to floating point
"numeric(ZZ,Number)"
-- see
numeric
-- convert to floating point
"numeric(ZZ,RRi)"
-- see
numeric
-- convert to floating point
"numeric(ZZ,Vector)"
-- see
numeric
-- convert to floating point
"numeric(ZZ,VisibleList)"
-- see
numeric
-- convert to floating point
"numericInterval(ZZ,Constant)"
-- see
numericInterval
-- converts to an interval
"options(ZZ)"
-- see
options(Command)
-- get optional arguments and default values of functions and methods
"pack(BasicList,ZZ)"
-- see
pack
-- pack elements of a list or string into several shorter lists or strings
"pack(String,ZZ)"
-- see
pack
-- pack elements of a list or string into several shorter lists or strings
"pack(ZZ,BasicList)"
-- see
pack
-- pack elements of a list or string into several shorter lists or strings
"pack(ZZ,String)"
-- see
pack
-- pack elements of a list or string into several shorter lists or strings
"pad(Net,ZZ)"
-- see
pad
-- pad a string or net with spaces
"pad(String,ZZ)"
-- see
pad
-- pad a string or net with spaces
"pad(ZZ,Net)"
-- see
pad
-- pad a string or net with spaces
"pad(ZZ,String)"
-- see
pad
-- pad a string or net with spaces
"part(InfiniteNumber,ZZ,RingElement)"
-- see
part
-- select terms of a polynomial by degree(s) or weight(s)
"part(InfiniteNumber,ZZ,VisibleList,RingElement)"
-- see
part
-- select terms of a polynomial by degree(s) or weight(s)
"part(Nothing,ZZ,RingElement)"
-- see
part
-- select terms of a polynomial by degree(s) or weight(s)
"part(Nothing,ZZ,VisibleList,RingElement)"
-- see
part
-- select terms of a polynomial by degree(s) or weight(s)
"part(ZZ,InfiniteNumber,RingElement)"
-- see
part
-- select terms of a polynomial by degree(s) or weight(s)
"part(ZZ,InfiniteNumber,VisibleList,RingElement)"
-- see
part
-- select terms of a polynomial by degree(s) or weight(s)
"part(ZZ,Nothing,RingElement)"
-- see
part
-- select terms of a polynomial by degree(s) or weight(s)
"part(ZZ,Nothing,VisibleList,RingElement)"
-- see
part
-- select terms of a polynomial by degree(s) or weight(s)
"part(ZZ,RingElement)"
-- see
part
-- select terms of a polynomial by degree(s) or weight(s)
"part(ZZ,VisibleList,RingElement)"
-- see
part
-- select terms of a polynomial by degree(s) or weight(s)
"part(ZZ,ZZ,RingElement)"
-- see
part
-- select terms of a polynomial by degree(s) or weight(s)
"part(ZZ,ZZ,VisibleList,RingElement)"
-- see
part
-- select terms of a polynomial by degree(s) or weight(s)
part(InfiniteNumber,ZZ,Number)
(missing documentation)
part(ZZ,InfiniteNumber,Number)
(missing documentation)
part(ZZ,Number)
(missing documentation)
part(ZZ,ZZ,Number)
(missing documentation)
Partition _ ZZ
-- get element
"partitions(ZZ)"
-- see
partitions
-- list the partitions of an integer
"partitions(ZZ,ZZ)"
-- see
partitions
-- list the partitions of an integer
"peek'(ZZ,Thing)"
-- see
peek'
-- examine contents of an object
"permanents(ZZ,Matrix)"
-- see
permanents
-- ideal generated by square permanents of a matrix
"permutations(ZZ)"
-- see
permutations
-- produce all permutations of a list
"pfaffians(ZZ,Matrix)"
-- see
pfaffians
-- ideal generated by Pfaffians
"RingMap ^ ZZ"
-- see
powers
product(ZZ,Function)
-- product of consecutive values of a function
"projectiveHilbertPolynomial(ZZ)"
-- see
projectiveHilbertPolynomial
-- Hilbert polynomial of projective space
"projectiveHilbertPolynomial(ZZ,ZZ)"
-- see
projectiveHilbertPolynomial
-- Hilbert polynomial of projective space
ProjectiveHilbertPolynomial ZZ
-- value of polynomial
"quotientRemainder(ZZ,ZZ)"
-- see
quotientRemainder(RingElement,RingElement)
-- quotient and remainder
random(ZZ,Ideal)
-- get a random homogeneous element from a graded ideal
random(ZZ,Ring)
-- get a random homogeneous element from a graded ring
randomMutableMatrix(ZZ,ZZ,RR,ZZ)
-- a random mutable matrix of integers
read(File,ZZ)
-- read from a file
"regex(String,ZZ,String)"
-- see
regex
-- evaluate a regular expression search
"regex(String,ZZ,ZZ,String)"
-- see
regex
-- evaluate a regular expression search
"regSeqInIdeal(Ideal,ZZ)"
-- see
regSeqInIdeal
-- a regular sequence contained in an ideal
"regSeqInIdeal(Ideal,ZZ,ZZ,ZZ)"
-- see
regSeqInIdeal
-- a regular sequence contained in an ideal
"remove(MutableList,ZZ)"
-- see
remove
-- remove an entry from a mutable hash table, list, or database
replace(ZZ,Thing,VisibleList)
-- copy a list, replacing an element
"CC >> ZZ"
-- see
right shift
"RR >> ZZ"
-- see
right shift
"RRi >> ZZ"
-- see
right shift
"ZZ >> ZZ"
-- see
right shift
"Ring / ZZ"
-- see
Ring / Ideal
-- make a quotient ring
Ring ^ ZZ
-- make a free module
"RingFamily ^ ZZ"
-- see
Ring ^ ZZ
-- make a free module
Ring _ ZZ
-- get a ring variable by index
RingElement ^ ZZ
-- power
rotate(ZZ,VisibleList)
-- rotate a list
"round(ZZ)"
-- see
round
-- round a number
"round(ZZ,RR)"
-- see
round
-- round a number
"rowAdd(MutableMatrix,ZZ,Number,ZZ)"
-- see
rowAdd
-- add a multiple of one row to another
"rowAdd(MutableMatrix,ZZ,RingElement,ZZ)"
-- see
rowAdd
-- add a multiple of one row to another
"rowMult(MutableMatrix,ZZ,Number)"
-- see
rowMult
-- multiply a row by a ring element
"rowMult(MutableMatrix,ZZ,RingElement)"
-- see
rowMult
-- multiply a row by a ring element
"rowPermute(MutableMatrix,ZZ,List)"
-- see
rowPermute
-- permute some rows
"rowSwap(MutableMatrix,ZZ,ZZ)"
-- see
rowSwap
-- interchange rows
"scan(ZZ,Function)"
-- see
scan
-- apply a function to each element in a list or sequence
Schubert(ZZ,ZZ,VisibleList)
-- find the Plücker ideal of a Schubert variety
select(ZZ,BasicList,Function)
-- select a limited number of elements from a list
select(ZZ,Function)
-- select integers
select(ZZ,HashTable,Function)
-- select a limited number of pairs from a hash table
"selectInSubring(ZZ,Matrix)"
-- see
selectInSubring
-- select columns in a subring
"separate(String,ZZ,String)"
-- see
separate
-- split a string into substrings using a regular expression
"setRandomSeed(ZZ)"
-- see
setRandomSeed
-- set starting point for random number generator
sheafExt^ZZ(CoherentSheaf,CoherentSheaf)
-- sheaf Ext of coherent sheaves
"sheafExt^ZZ(CoherentSheaf,SheafOfRings)"
-- see
sheafExt^ZZ(CoherentSheaf,CoherentSheaf)
-- sheaf Ext of coherent sheaves
"sheafExt^ZZ(SheafOfRings,CoherentSheaf)"
-- see
sheafExt^ZZ(CoherentSheaf,CoherentSheaf)
-- sheaf Ext of coherent sheaves
"sheafExt^ZZ(SheafOfRings,SheafOfRings)"
-- see
sheafExt^ZZ(CoherentSheaf,CoherentSheaf)
-- sheaf Ext of coherent sheaves
"simpleDocFrob(ZZ,Matrix)"
-- see
simpleDocFrob
-- a sample documentation node
"someTerms(RingElement,ZZ,ZZ)"
-- see
someTerms
-- select some terms of a polynomial
String _ ZZ
-- get element from string
"submatrixByDegrees(Matrix,ZZ,ZZ)"
-- see
submatrixByDegrees
-- submatrix consisting of rows and columns in an interval or box of degrees
"subsets(List,ZZ)"
-- see
subsets
-- produce the subsets of a set or list
"subsets(Sequence,ZZ)"
-- see
subsets
-- produce the subsets of a set or list
"subsets(Set,ZZ)"
-- see
subsets
-- produce the subsets of a set or list
"subsets(ZZ)"
-- see
subsets
-- produce the subsets of a set or list
"subsets(ZZ,ZZ)"
-- see
subsets
-- produce the subsets of a set or list
"substitute(Matrix,ZZ)"
-- see
substitute
-- substituting values for variables
"substring(String,ZZ)"
-- see
substring
-- extract part of a string
"substring(String,ZZ,ZZ)"
-- see
substring
-- extract part of a string
"substring(ZZ,String)"
-- see
substring
-- extract part of a string
"substring(ZZ,ZZ,String)"
-- see
substring
-- extract part of a string
sum(ZZ,Function)
-- sum consecutive values of a function
"switch(ZZ,ZZ,VisibleList)"
-- see
switch
-- copy a list, switching two elements
"symmetricPower(ZZ,Matrix)"
-- see
symmetricPower
-- symmetric power
"symmetricPower(ZZ,Module)"
-- see
symmetricPower
-- symmetric power
"take(BasicList,ZZ)"
-- see
take
-- Take some elements from a list or sequence.
"take(Thing,ZZ)"
-- see
take
-- Take some elements from a list or sequence.
Tally == ZZ
(missing documentation)
Tally ? ZZ
(missing documentation)
"Monoid ^** ZZ"
-- see
tensor(Monoid,Monoid)
-- tensor product of monoids
"Ring ^** ZZ"
-- see
tensor(Monoid,Monoid)
-- tensor product of monoids
"tests(ZZ,Package)"
-- see
tests
-- locate a package's tests
"tests(ZZ,String)"
-- see
tests
-- locate a package's tests
"toCC(ZZ)"
-- see
toCC
-- convert to high-precision complex number
"toCC(ZZ,CC)"
-- see
toCC
-- convert to high-precision complex number
"toCC(ZZ,QQ)"
-- see
toCC
-- convert to high-precision complex number
"toCC(ZZ,QQ,QQ)"
-- see
toCC
-- convert to high-precision complex number
"toCC(ZZ,QQ,RR)"
-- see
toCC
-- convert to high-precision complex number
"toCC(ZZ,QQ,ZZ)"
-- see
toCC
-- convert to high-precision complex number
"toCC(ZZ,RR)"
-- see
toCC
-- convert to high-precision complex number
"toCC(ZZ,RR,QQ)"
-- see
toCC
-- convert to high-precision complex number
"toCC(ZZ,RR,RR)"
-- see
toCC
-- convert to high-precision complex number
"toCC(ZZ,RR,ZZ)"
-- see
toCC
-- convert to high-precision complex number
"toCC(ZZ,ZZ)"
-- see
toCC
-- convert to high-precision complex number
"toCC(ZZ,ZZ,QQ)"
-- see
toCC
-- convert to high-precision complex number
"toCC(ZZ,ZZ,RR)"
-- see
toCC
-- convert to high-precision complex number
"toCC(ZZ,ZZ,ZZ)"
-- see
toCC
-- convert to high-precision complex number
"toDual(ZZ,Ideal)"
-- see
toDual
-- finds the inverse system to an ideal up to a given degree
"toDual(ZZ,Matrix)"
-- see
toDual
-- finds the inverse system to an ideal up to a given degree
topComponents(Module,ZZ)
(missing documentation)
Tor_ZZ(Ideal,Matrix)
(missing documentation)
Tor_ZZ(Matrix,Ideal)
(missing documentation)
Tor_ZZ(Matrix,Ring)
(missing documentation)
"Tor_ZZ(Ideal,Ideal)"
-- see
Tor_ZZ(Module,Module)
-- compute a Tor module
"Tor_ZZ(Ideal,Module)"
-- see
Tor_ZZ(Module,Module)
-- compute a Tor module
"Tor_ZZ(Ideal,Ring)"
-- see
Tor_ZZ(Module,Module)
-- compute a Tor module
"Tor_ZZ(Module,Ideal)"
-- see
Tor_ZZ(Module,Module)
-- compute a Tor module
Tor_ZZ(Module,Module)
-- compute a Tor module
"Tor_ZZ(Module,Ring)"
-- see
Tor_ZZ(Module,Module)
-- compute a Tor module
"truncateOutput(ZZ)"
-- see
truncateOutput
"uniquePermutations(ZZ)"
-- see
uniquePermutations
-- produce all unique permutations of a list
"vars(ZZ)"
-- see
vars(List)
-- a sequence of variables
VirtualTally == ZZ
(missing documentation)
VirtualTally ? ZZ
(missing documentation)
VirtualTally ^** ZZ
-- Cartesian power of sets and tallies
VisibleList _ ZZ
-- get element from list
wedgeProduct(ZZ,ZZ,Module)
-- the exterior multiplication map
ZZ * MonoidElement
(missing documentation)
ZZ * MutableMatrix
(missing documentation)
ZZ .. ZZ
-- sequences of consecutive integers
ZZ ..< ZZ
-- sequences of consecutive integers
ZZ : Thing
-- repeat an item
ZZ == MonoidElement
(missing documentation)
ZZ == Tally
(missing documentation)
ZZ == VirtualTally
(missing documentation)
ZZ ? Tally
(missing documentation)
ZZ ? VirtualTally
(missing documentation)
ZZ _ Module
-- integers or zero element
ZZ | ZZ
-- logical or
ZZ ~
-- logical not
Fixed objects of class
ZZ
:
maxAllowableThreads
-- the maximum possible number of simultaneously running tasks
maxExponent
minExponent
printingTimeLimit
For the programmer
The object
ZZ
is
a
ring
, with ancestor classes
Number
<
Thing
.