ZZ -- the class of all integers

Functions and methods returning an integer :

• ZZ !, see ! -- factorial
• # BasicList -- length
• # HashTable -- length
• # Set -- cardinality
• # String -- length
• binomial -- binomial coefficient
• binomial(ZZ,ZZ), see binomial -- binomial coefficient
• char -- computes the characteristic of a field or ring
• codim -- compute the codimension
• degreeLength -- the number of degrees
• depth -- depth of a net
• dim -- compute the Krull dimension
• euler -- Euler characteristic
• fileMode(String) -- get file mode
• floor(RR), see floor -- floor function
• gcd(ZZ,ZZ), see gcd -- greatest common divisor
• genus -- arithmetic genus
• height -- height of a net
• length -- length
• maxPosition(BasicList), see maxPosition -- position of largest element
• minPosition(BasicList), see minPosition -- position of smallest element
• pdim -- calculate the projective dimension
• random(ZZ) -- random integer
• random(ZZ,ZZ) -- random integer in a range
• regularity -- compute the Castelnuovo-Mumford regularity
• width -- width of a file or net
• xor(ZZ,ZZ), see xor -- logical exclusive-or
• ZZ & ZZ -- logical and

Methods that use an integer :

• ZZ % ZZ, see % -- a binary operator, usually used for remainder and reduction
• ZZ * ProjectiveHilbertPolynomial, see * -- a binary operator, usually used for multiplication
• RingElement ** ZZ, see ** -- a binary operator, usually used for tensor product or Cartesian product
• ChainComplexMap + ZZ, see + -- a unary or binary operator, usually used for addition
• InfiniteNumber + ZZ, see + -- a unary or binary operator, usually used for addition
• Matrix + ZZ, see + -- a unary or binary operator, usually used for addition
• ZZ + ChainComplexMap, see + -- a unary or binary operator, usually used for addition
• ZZ + InfiniteNumber, see + -- a unary or binary operator, usually used for addition
• ZZ + Matrix, see + -- a unary or binary operator, usually used for addition
• InfiniteNumber - ZZ, see - -- a unary or binary operator, usually used for negation or subtraction
• Matrix - ZZ, see - -- a unary or binary operator, usually used for negation or subtraction
• ZZ - InfiniteNumber, see - -- a unary or binary operator, usually used for negation or subtraction
• ZZ - Matrix, see - -- a unary or binary operator, usually used for negation or subtraction
• ZZ .. ZZ, see .. -- a binary operator, uses include ranges; sequence of consecutive items
• > ZZ -- greater than
• >= ZZ -- greater than or equal
• acos(ZZ), see acos -- compute the arccosine
• apply(ZZ,Function) -- apply a function to {0,..., n-1}
• asin(ZZ), see asin -- compute the arcsine
• atan(ZZ), see atan(RR) -- compute the arctangent of a number
• BasicList # ZZ -- get element from list
• BasicList #? ZZ -- check for element in list
• basis(InfiniteNumber,ZZ,Ideal), see basis -- basis of all or part of a module or ring
• basis(InfiniteNumber,ZZ,Module), see basis -- basis of all or part of a module or ring
• basis(InfiniteNumber,ZZ,Ring), see basis -- basis of all or part of a module or ring
• basis(List,ZZ,Ideal), see basis -- basis of all or part of a module or ring
• basis(List,ZZ,Ring), see basis -- basis of all or part of a module or ring
• basis(ZZ,Ideal), see basis -- basis of all or part of a module or ring
• basis(ZZ,InfiniteNumber,Ideal), see basis -- basis of all or part of a module or ring
• basis(ZZ,InfiniteNumber,Module), see basis -- basis of all or part of a module or ring
• basis(ZZ,InfiniteNumber,Ring), see basis -- basis of all or part of a module or ring
• basis(ZZ,List,Ideal), see basis -- basis of all or part of a module or ring
• basis(ZZ,List,Ring), see basis -- basis of all or part of a module or ring
• basis(ZZ,Module), see basis -- basis of all or part of a module or ring
• basis(ZZ,Ring), see basis -- basis of all or part of a module or ring
• basis(ZZ,ZZ,Ideal), see basis -- basis of all or part of a module or ring
• basis(ZZ,ZZ,Module), see basis -- basis of all or part of a module or ring
• basis(ZZ,ZZ,Ring), see basis -- basis of all or part of a module or ring
• ChainComplex ^ ZZ -- access member, cohomological degree
• ChainComplex _ ZZ -- component
• GradedModule _ ZZ, see ChainComplex _ ZZ -- component
• ChainComplex _ ZZ = Thing -- missing documentation
• 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 -- missing documentation
• 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,RingElement,ZZ), see columnAdd -- add a multiple of one column to another
• 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 -- compositions of an integer
• concatenate(ZZ), see concatenate -- join strings
• cos(ZZ), see cos -- compute the cosine
• cosh(ZZ), see cosh -- compute the hyperbolic cosine
• cotangentSheaf(ZZ,ProjectiveVariety) -- exterior powers of the cotangent sheaf of a projective variety
• degreesMonoid(ZZ), see degreesMonoid -- get the monoid of degrees
• degreesRing(ZZ) -- the ring of degrees
• diff(ProjectiveHilbertPolynomial,ZZ)
• divideByVariable(Matrix,RingElement,ZZ), see divideByVariable -- divide all columns by a (power of a) variable
• drop(BasicList,ZZ)
• exp(ZZ), see exp -- exponential function
• 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
• fittingIdeal(ZZ,Module), see fittingIdeal -- Fitting ideal of a module
• gcd(ZZ,QQ), see gcd -- greatest common divisor
• gcdCoefficients(ZZ,ZZ), see gcdCoefficients -- gcd with coefficients
• Ideal _ 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
• GF(ZZ) -- make a finite field of a given order
• GF(ZZ,ZZ) -- make a finite field of a given prime power order
• 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 CoherentSheaf -- cohomology of a coherent sheaf on a projective variety
• 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 ChainComplex -- homology of a chain complex
• HH_ZZ ChainComplexMap -- homology of a chain complex map
• HH_ZZ Sequence
• hilbertFunction(ZZ,Ideal), see hilbertFunction(List,Ideal) -- compute the Hilbert function of the quotient of the ambient ring by an ideal
• hilbertFunction(ZZ,Module), see hilbertFunction(List,Module) -- compute the Hilbert function of a module
• hilbertFunction(ZZ,ProjectiveVariety), see hilbertFunction(List,ProjectiveVariety) -- compute the Hilbert function of a projective variety
• hilbertFunction(ZZ,Ring), see hilbertFunction(List,Ring) -- compute the Hilbert function of a ring
• Ideal ^ ZZ -- power
• MonomialIdeal ^ ZZ, see Ideal ^ ZZ -- power
• ideal(ZZ), see ideal(RingElement) -- make an ideal
• isPrime(ZZ), see isPrime -- whether a integer, polynomial, or ideal is prime
• isUnit(ZZ), see isUnit -- whether a ring element is a unit
• koszul(ZZ,Matrix) -- a differential in a Koszul complex
• 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
• log(ZZ), see log -- logarithm function
• map(Module,ZZ,Function) -- create a matrix from a free module by specifying a function which gives each entry
• map(Module,ZZ,List) -- create a matrix by giving a sparse or dense list of entries
• Matrix ^ ZZ -- power
• Matrix _ ZZ -- missing documentation
• Matrix | ZZ, see Matrix | Matrix -- join matrices horizontally
• ZZ | Matrix, see Matrix | Matrix -- join matrices horizontally
• Matrix || ZZ, see Matrix || Matrix -- join matrices vertically
• ZZ || Matrix, see Matrix || Matrix -- join matrices vertically
• minors(ZZ,Matrix)
• Module ^ ZZ -- direct sum
• Module ^** ZZ -- tensor power
• mutableIdentity(Ring,ZZ) -- make a mutable identity matrix
• mutableZero(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
• new Eliminate from ZZ -- missing documentation
• pack(BasicList,ZZ), see pack -- pack elements of a list into shorter ones
• pack(ZZ,BasicList), see pack -- pack elements of a list into shorter ones
• pad(String,ZZ), see pad -- pad a string with spaces
• pad(ZZ,String), see pad -- pad a string with spaces
• part(ZZ,RingElement), see part -- sum of monomials of a polynomial of a given degree
• Partition _ ZZ -- get element
• partitions(ZZ) -- list the partitions of an integer
• partitions(ZZ,ZZ) -- list the partitions of an integer
• peek'(ZZ,Thing) -- 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
• 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
• random(ZZ,Ring) -- a random ring element of a given degree
• randomMutableMatrix(ZZ,ZZ,RR,ZZ) -- a random mutable matrix of integers
• read(File,ZZ) -- read from a file
• regex(String,ZZ,String) -- regular expression matching
• Ring / ZZ, see Ring / Ideal -- make a quotient ring
• Ring ^ ZZ -- make a free module
• Ring _ ZZ -- get a ring variable by index
• RingElement ^ ZZ -- power
• rotate(ZZ,VisibleList) -- rotate a list
• rowAdd(MutableMatrix,ZZ,RingElement,ZZ), see rowAdd -- add a multiple of one row to another
• 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) -- apply a function to 0 .. n-1
• Schubert(ZZ,ZZ,VisibleList) -- find the Pluecker ideal of a Schubert variety
• schurRing(Symbol,ZZ), see schurRing -- make a Schur ring
• select(ZZ,BasicList,Function) -- select a limited number of elements from a list
• select(ZZ,HashTable,Function) -- select a limited number of pairs from a hash table
• selectInSubring(ZZ,Matrix), see selectInSubring -- select columns in a subring
• separateRegexp(String,ZZ,String) -- separate a string into pieces, with separators determined by a regular expression
• setRandomSeed(ZZ) -- 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
• sin(ZZ), see sin -- compute the sine
• sinh(ZZ), see sinh -- compute the hyperbolic sine
• someTerms(RingElement,ZZ,ZZ), see someTerms -- select some terms of a polynomial
• sqrt(ZZ), see sqrt -- square root function
• String # ZZ -- get character from string
• String #? ZZ -- check for character in string
• String _ ZZ -- get element from string
• subsets(List,ZZ) -- produce all the subsets
• subsets(Sequence,ZZ), see subsets(List,ZZ) -- produce all the subsets
• subsets(Set,ZZ) -- produce all the subsets
• subsets(ZZ) -- produce all the subsets
• subsets(ZZ,ZZ) -- produce all the subsets
• 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
• symmetricPower(ZZ,Matrix) -- symmetric power
• take(BasicList,ZZ)
• Tally ^** ZZ -- Cartesian power of sets and tallies
• tan(ZZ), see tan -- compute the tangent
• tanh(ZZ), see tanh -- compute the hyperbolic tangent
• Tor_ZZ(Module,Module) -- compute a Tor module
• truncate(ZZ,Ideal), see truncate -- truncate the module at a specified degree
• truncate(ZZ,Module), see truncate -- truncate the module at a specified degree
• truncateOutput(ZZ), see truncateOutput
• vars(ZZ), see vars(List) -- a sequence of variables
• VisibleList _ ZZ -- get element from list
• wedgeProduct(ZZ,ZZ,Module) -- the exterior multiplication map
• ZZ : Thing -- repeat an item
• ZZ << ZZ -- shift bits leftward
• ZZ >> ZZ -- shift bits rightward
• ZZ _ Module -- missing documentation
• ZZ _ Ring -- missing documentation
• ZZ | ZZ -- logical or

Fixed objects of class ZZ :

• printingTimeLimit

For the programmer

The object ZZ is a ring, with ancestor classes Number < Thing.