# 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
• abs(ZZ), see abs -- absolute value function
• binomial -- binomial coefficient
• binomial(ZZ,ZZ), see binomial -- binomial coefficient
• binomial(List,List) (missing documentation)
• char -- computes the characteristic of the ring or field
• degreeLength -- the number of degrees
• depth -- depth of a net
• depth(Ideal,Ideal) (missing documentation) -- computes the depth of a ring
• depth(Ideal,Module) (missing documentation) -- computes the depth of a ring
• depth(Ideal,Ring) (missing documentation) -- computes the depth of a ring
• depth(Module) (missing documentation) -- computes the depth of a ring
• depth(Ring) (missing documentation) -- computes the depth of a ring
• dim -- 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
• length(Module) -- length
• 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
• pdim -- calculate the projective dimension
• powermod(ZZ,ZZ,ZZ), see powermod -- powers of integers mod N
• random(ZZ) -- random integer
• random(ZZ,ZZ) -- random integer in a range
• regularity -- compute the Castelnuovo-Mumford regularity
• round(ZZ,ZZ) (missing documentation)
• 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(ZZ), see size2 -- number of binary digits to the left of the point
• 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 !, 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
• Matrix * 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
• 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
• CC + ZZ, see + -- a unary or binary operator, usually used for addition
• ChainComplexMap + 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
• 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 + QQ, see + -- a unary or binary operator, usually used for addition
• ZZ + RR, see + -- a unary or binary operator, usually used for addition
• CC - 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
• ZZ - CC, 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
• 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
• 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
• 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
• QQ == ZZ, see == -- equality
• Ring == ZZ, see == -- equality
• RingElement == ZZ, see == -- equality
• RingMap == ZZ, see == -- equality
• RR == 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 == QQ, see == -- equality
• ZZ == Ring, see == -- equality
• ZZ == RingElement, see == -- equality
• ZZ == RingMap, see == -- equality
• ZZ == RR, see == -- equality
• ZZ == ZZ, see == -- equality
• > ZZ -- greater than
• >= ZZ -- greater than or equal
• CC ^ ZZ, see ^ -- a binary operator, usually used for powers
• InexactFieldFamily ^ ZZ, see ^ -- a binary operator, usually used for powers
• help(ZZ), see about -- search the documentation
• acos(ZZ), see acos -- arccosine
• agm(CC,ZZ), see agm -- arithmetic-geometric mean
• agm(QQ,ZZ), see agm -- arithmetic-geometric mean
• agm(RR,ZZ), see agm -- arithmetic-geometric mean
• agm(ZZ,CC), see agm -- arithmetic-geometric mean
• agm(ZZ,QQ), see agm -- arithmetic-geometric mean
• agm(ZZ,RR), see agm -- arithmetic-geometric mean
• agm(ZZ,ZZ), see agm -- arithmetic-geometric mean
• 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}
• asin(ZZ), see asin -- arcsine
• atan(ZZ), see atan(RR) -- compute the arctangent of a number
• atan2(QQ,ZZ), see atan2(RR,RR) -- compute an angle of a certain triangle
• atan2(RR,ZZ), see atan2(RR,RR) -- compute an angle of a certain triangle
• atan2(ZZ,QQ), see atan2(RR,RR) -- compute an angle of a certain triangle
• atan2(ZZ,RR), see atan2(RR,RR) -- compute an angle of a certain triangle
• atan2(ZZ,ZZ), see atan2(RR,RR) -- compute an angle of a certain triangle
• 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,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,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,QQ), see BesselJ -- Bessel function of the first kind
• BesselJ(ZZ,RR), see BesselJ -- Bessel function of the first kind
• BesselJ(ZZ,ZZ), see BesselJ -- Bessel function of the first kind
• BesselY(ZZ,QQ), see BesselY -- Bessel function of the second kind
• BesselY(ZZ,RR), see BesselY -- Bessel function of the second kind
• BesselY(ZZ,ZZ), 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) (missing documentation)
• 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
• check(ZZ,Package), see check -- perform tests of a package
• check(ZZ,String), see check -- perform tests of a package
• code(ZZ) (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
• 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
• concatenate(ZZ), see concatenate -- join strings
• conwayPolynomial(ZZ), see conwayPolynomial -- provide a Conway polynomial
• conwayPolynomial(ZZ,ZZ), see conwayPolynomial -- provide a Conway polynomial
• cos(ZZ), see cos -- compute the cosine
• cosh(ZZ), see cosh -- compute the hyperbolic cosine
• cot(ZZ), see cot -- cotangent
• cotangentSheaf(ZZ,ProjectiveVariety) -- exterior powers of the cotangent sheaf of a projective variety
• coth(ZZ), see coth -- hyperbolic cotangent
• csc(ZZ), see csc -- cosecant
• csch(ZZ), see csch -- hyperbolic cosecant
• debug(ZZ) (missing documentation)
• degreesMonoid(ZZ), see degreesMonoid -- get the monoid of degrees
• degreesRing(ZZ), see degreesRing(List) -- the ring of degrees
• denominator(ZZ) (missing documentation)
• 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)
• 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.
• eint(ZZ), see eint -- exponential integral
• erf(ZZ), see erf -- error function
• erfc(ZZ), see erfc -- complementary error function
• exp(ZZ), see exp -- exponential function
• expm1(ZZ), see expm1 -- exponential minus 1
• 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
• Gamma(ZZ), see Gamma -- Gamma function
• 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 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(Sequence) (missing documentation)
• 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
• hilbertPolynomial(ZZ) (missing documentation)
• 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,ZZ) -- integral closure of an ideal in an affine domain
• inverseSystem(ZZ,Ideal), see inverseSystem -- Inverse systems with equivariance
• inverseSystem(ZZ,Matrix), see inverseSystem -- Inverse systems with equivariance
• isPrime(ZZ), see isPrime -- whether a integer, polynomial, or ideal is prime
• isPseudoprime(ZZ) -- whether an integer is a pseudoprime
• isReal(ZZ), see isReal -- whether a number is real
• isUnit(ZZ), see isUnit -- whether a ring element is a unit
• kernel(RingMap,ZZ) (missing documentation) -- homogeneous components of the kernel of a homogeneous ring map
• kernel(ZZ,RingMap) (missing documentation)
• 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,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
• 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 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(ZZ,type of ZZ), see lift -- lift to another ring
• lngamma(ZZ), see lngamma -- logarithm of the Gamma function
• log(QQ,ZZ), see log -- logarithm function
• log(RR,ZZ), see log -- logarithm function
• log(ZZ), see log -- logarithm function
• log(ZZ,CC), see log -- logarithm function
• log(ZZ,QQ), see log -- logarithm function
• log(ZZ,RR), see log -- logarithm function
• log(ZZ,ZZ), see log -- logarithm function
• log1p(ZZ), see log1p -- logarithm of 1+x
• 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
• 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
• matrix(BettiTally,ZZ) (missing documentation) -- Betti diagram to matrix
• matrix(BettiTally,ZZ,ZZ) (missing documentation) -- Betti diagram to matrix
• matrix(Matrix,ZZ,ZZ) (missing documentation)
• minors(ZZ,Matrix) -- ideal generated by minors
• Module * ZZ (missing documentation)
• Module ^ ZZ -- direct sum
• Module ^** ZZ -- tensor power
• 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
• numerator(ZZ) (missing documentation)
• 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,VisibleList), see numeric -- convert to floating point
• numeric(ZZ,Vector) (missing documentation)
• pack(BasicList,ZZ), see pack -- pack elements of a list into several shorter lists
• pack(ZZ,BasicList), see pack -- pack elements of a list into several shorter lists
• 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
• partitions(ZZ,List) (missing documentation)
• 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
• 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 (missing documentation)
• ProjectiveHilbertPolynomial - ZZ (missing documentation)
• ProjectiveHilbertPolynomial == ZZ (missing documentation)
• ProjectiveHilbertPolynomial ZZ -- value of polynomial
• random(ZZ,Ring), see random(Type) -- random element of a type
• randomMutableMatrix(ZZ,ZZ,RR,ZZ) -- a random mutable matrix of integers
• regex(String,ZZ,String), see regex -- define and evaluate a regular expression
• regex(String,ZZ,ZZ,String), see regex -- define and evaluate a regular expression
• replace(ZZ,Thing,VisibleList) -- copy a list, replacing an element
• CC >> ZZ, see right shift
• RR >> ZZ, see right shift
• ZZ >> ZZ, see right shift
• 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
• round(ZZ), see round -- round a number
• round(ZZ,RR), see round -- round a number
• round(ZZ,BasicList) (missing documentation)
• round(ZZ,CC) (missing documentation)
• round(ZZ,Matrix) (missing documentation)
• round(ZZ,RingElement) (missing documentation)
• 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 Pluecker ideal of a Schubert variety
• sec(ZZ), see sec -- secant
• sech(ZZ), see sech -- hyperbolic secant
• 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
• separateRegexp(String,ZZ,String), see separateRegexp -- split a string into substrings, using a regular expression to define the separators
• 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
• simpleDocFrob(ZZ,Matrix) -- a sample documentation node
• 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 element from string
• submatrixByDegrees(Matrix,ZZ,ZZ) (missing documentation)
• 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.
• tan(ZZ), see tan -- compute the tangent
• tanh(ZZ), see tanh -- compute the hyperbolic tangent
• 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
• 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
• truncate(ZZ,Ideal), see truncate -- truncation of the graded ring, ideal or module at a specified degree or set of degrees
• truncate(ZZ,Module), see truncate -- truncation of the graded ring, ideal or module at a specified degree or set of degrees
• truncate(ZZ,Ring), see truncate -- truncation of the graded ring, ideal or module at a specified degree or set of degrees
• truncate(ChainComplex,ZZ) (missing documentation) -- compute the hard truncation of a chain complex
• truncate(ZZ,Matrix), see truncate(List,Matrix) -- truncation of a matrix
• truncateOutput(ZZ), see truncateOutput
• 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
• zeta(ZZ), see zeta -- Riemann zeta function
• ZZ * MutableMatrix (missing documentation)
• ZZ * Symbol (missing documentation)
• ZZ * Tally (missing documentation)
• ZZ * VirtualTally (missing documentation)
• ZZ + ProjectiveHilbertPolynomial (missing documentation)
• ZZ - ProjectiveHilbertPolynomial (missing documentation)
• ZZ .. ZZ -- sequences of consecutive integers
• ZZ ..< ZZ -- sequences of consecutive integers
• ZZ : Thing -- repeat an item
• ZZ == VirtualTally (missing documentation)
• ZZ ? VirtualTally (missing documentation)
• ZZ _ Module -- integers or zero element
• ZZ | ZZ -- logical or

## For the programmer

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