# 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
• "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(ZZ,ZZ)" -- see binomial -- binomial coefficient
• char -- computes the characteristic of the ring or field
• degreeLength -- the number of degrees
• depth -- depth of a net
• 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 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
• 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
• "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
• "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
• "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
• "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 + 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
• "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 - 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
• "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 == 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
• "CC ^ ZZ" -- see ^ -- a binary operator, usually used for powers
• "InexactFieldFamily ^ ZZ" -- see ^ -- a binary operator, usually used for powers
• "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
• "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 -- compute the arctangent of a number
• "atan2(QQ,ZZ)" -- see atan2 -- compute an angle of a certain triangle
• "atan2(RR,ZZ)" -- see atan2 -- compute an angle of a certain triangle
• "atan2(RRi,ZZ)" -- see atan2 -- compute an angle of a certain triangle
• "atan2(ZZ,QQ)" -- see atan2 -- compute an angle of a certain triangle
• "atan2(ZZ,RR)" -- see atan2 -- compute an angle of a certain triangle
• "atan2(ZZ,RRi)" -- see atan2 -- compute an angle of a certain triangle
• "atan2(ZZ,ZZ)" -- see atan2 -- 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)" -- 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
• "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
• "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
• 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 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 -- 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
• "log(QQ,ZZ)" -- see log -- logarithm function
• "log(RR,ZZ)" -- see log -- logarithm function
• "log(RRi,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,RRi)" -- 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
• 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
• "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 into several shorter lists
• "pack(ZZ,BasicList)" -- see pack -- pack elements of a list into several shorter lists
• "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) -- 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
• "random(ZZ,Ring)" -- see random(Type) -- random element of a type
• 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
• 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
• 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
• "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
• "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
• "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)" -- 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.
• Tally == ZZ (missing documentation)
• Tally ? ZZ (missing documentation)
• "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
• 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
• truncate(ZZ,Ideal)
• truncate(ZZ,Matrix)
• truncate(ZZ,Module)
• truncate(ZZ,Ring)
• "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 .. ZZ -- sequences of consecutive integers
• ZZ ..< ZZ -- sequences of consecutive integers
• ZZ : Thing -- repeat an item
• 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

## For the programmer

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