CoCoA-5.4.1s Manual


Part 1: Alphabetical list of commands
Part 2: CoCoA Language & Tutorials
Part 3: CoCoA Data Types

Part 1: Alphabetical list of commands

A B C D E F G H I J K L M N O P Q R S T U V W X Z

Special Characters

operators, shortcuts -- Special characters equivalent to commands

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A

abs -- absolute value of a number
adj -- classical adjoint matrix (also known as adjugate)
AdjacentMinors -- list of adjacent minors of a matrix
AffHilbert [OBSOLESCENT] -- [OBSOLESCENT] renamed AffHilbertFn
AffHilbertFn -- the affine Hilbert function
AffHilbertSeries -- the affine Hilbert-Poincare series
AffPoincare [OBSOLESCENT] -- [OBSOLESCENT] Renamed AffHilbertSeries
alias -- define aliases for package names
aliases -- list of global aliases
AllReducedGroebnerBases [OBSOLETE] -- [OBSOLETE] Renamed GroebnerFanIdeals
AlmostQR -- QR decomposition of a matrix
and -- boolean "and" operator
append -- append an object to a list
apply [OBSOLESCENT] -- apply [OBSOLESCENT]
ApproxPointsNBM -- Numerical Border Basis of ideal of points
ApproxSolve -- Approximate real solutions for polynomial system
AreGensMonomial -- checks if given gens are monomial
AreGensSqFreeMonomial -- checks if given gens are squarefree monomial
ArrBettiNumbers -- Betti numbers of an arrangement of hyperplanes
ArrBoolean -- boolean arrangement
ArrBraid -- braid arrangement
ArrCatalanA -- Catalan arrangement of type A
ArrCatalanB -- Catalan arrangement of type B
ArrCatalanD -- Catalan arrangement of type D
ArrCharPoly -- characteristic polynomial of an arrangement of hyperplanes
ArrCone -- cone of an arrangement of hyperplanes
ArrDeletion -- deletes a hyperplane from a list of hyperplanes
ArrDerMod [OBSOLESCENT] -- [OBSOLESCENT] renamed ArrDerModule
ArrDerModule -- set of generators of the module of logarithmic derivations of an arrangement of hyperplanes
ArrExponents -- exponents of a free arrangement of hyperplanes
ArrFlats -- list of flats of an arrangement of hyperplanes
ArrGraphical -- graphical arrangement
ArrLattice -- lattice of an arrangement of hyperplanes
ArrPoincarePoly -- Poincare polynomial of an arrangement of hyperplanes
ArrRestriction -- arrangement of hyperplanes A restricted to a hyperplane
ArrShiA -- Shi arrangement of type A
ArrShiB -- Shi arrangement of type B
ArrShiCatalanA -- Shi-Catalan arrangement of type A with multiplicities
ArrShiCatalanB -- Shi-Catalan arrangement of type B with given multiplicities
ArrShiCatalanD -- Shi-Catalan arrangement of type D with given multiplicities
ArrShiD -- Shi arrangement of type D
ArrSignedGraphical -- signed graphical arrangement
ArrToMultiArr -- multiarrangement from an arrangement and a list of multiplicities
ArrTuttePoly -- Tutte polynomial of an arrangement of hyperplanes
ArrTypeB -- reflection arrangement of type B
ArrTypeD -- reflection arrangement of type D
ArtinianOrlikTeraoIdeal -- Artinian Orlik-Terao ideal of an arrangement of hyperplanes
ascii -- convert between characters and ascii code
AsINT -- convert into an INT
AsRAT -- convert into a RAT
assert -- check an assertion

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B

BaseRing -- the base ring of a ring
BBasis5 -- Border Basis of zero dimensional ideal
BettiDiagram -- the diagram of the graded Betti numbers
BettiMatrix -- the matrix of the graded Betti numbers
BettiNumbers -- (Multi-)graded Betti numbers
binomial -- binomial coefficient
BinomialRepr, BinomialReprShift -- binomial representation of integers
block -- group several commands into a single command
BlockMat -- create a block matrix
BlockMat2x2 -- create a block matrix with 4 matrices
Bool01 -- Convert a boolean to an integer
break -- break out of a loop command
BringIn -- bring in objects from another ring

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C

Call [OBSOLETE] -- [OBSOLETE] apply a function to given arguments
CallOnGroebnerFanIdeals -- apply a function to Groebner fan ideals
CanonicalBasis -- canonical basis of a free module
CanonicalHom -- canonical homomorphism
CanonicalRepr -- representative of a class in a quotient ring
CartesianProduct, CartesianProductList -- Cartesian product of lists
Cast [OBSOLETE] -- [OBSOLETE] type conversion
CatalanNumber -- Catalan number
ceil -- round rational up to integer
CFApprox -- continued fraction approximation
CFApproximants -- continued fraction approximants
ChainCanonicalHom -- canonical homomorphism
characteristic -- the characteristic of a ring
CharPoly -- characteristic polynomial of a matrix
ChebyshevPoly -- Orthogonal Polynomials: Chebyshev, Hermite, Laguerre
CheckArgTypes -- Check types in a list
ciao -- quit CoCoA
ClearDenom -- clear common denominator of a polynomial with rational coeffs
close -- close an output stream
CloseLog -- close a log of a CoCoA session
CoCoA-4 mode -- Backward compatibility mode for CoCoA-4
CocoaLimits -- limits on exponents and ring characteristics
CocoaPackagePath -- returns the path to the CoCoA packages
codomain -- codomain of a homomorphism
CoeffEmbeddingHom -- returns the coefficient embedding homomorphism of a polynomial ring
CoeffHeight -- the maximum of the absolute values of the coefficients of a polynomial
coefficients -- list of coefficients of a polynomial
CoefficientsWRT -- list of coeffs and PPs of a poly wrt indet or list of indets
CoeffListWRT -- list of coefficients of a polynomial wrt an indet
CoeffListWRTSupport -- list of coefficients of a polynomial wrt a power-product basis
CoeffOfTerm -- coefficient of a term of a polynomial
CoeffRing -- the ring of coefficients of a polynomial ring
ColMat -- single column matrix
colon -- ideal or module quotient
ColumnVectors [OBSOLETE] -- [OBSOLETE] list of module elements
CommonDenom -- Common denominator of a polynomial with rational coefficients
Comp [OBSOLETE] -- [OBSOLETE] access a component
CompleteToOrd [OBSOLETE] -- [OBSOLETE] renamed MakeTermOrdMat
compts -- list of components of a ModuleElem
ComputeElimFirst -- ComputeElimFirst
concat -- concatenate lists
ConcatAntiDiag -- create a simple block matrix
ConcatDiag -- create a simple block matrix
ConcatHor -- create a simple block matrix
ConcatHorList -- create a simple block matrix
ConcatLists -- concatenate a list of lists
ConcatStrings -- concatenate strings
ConcatVer -- create a simple block matrix
ConcatVerList -- create a simple block matrix
ConstantCoeff -- constant coefficient of a polynomial
content -- content of a polynomial
ContentFreeFactor -- factorization of multivariate polynomial into content-free factors
ContentWRT -- content of a polynomial wrt and indet or a list of indets
ContFrac -- continued fraction quotients
ContFracToRat -- convert continued fraction to rational
continue -- continue directly with next loop iteration
CoprimeFactor -- determine factor of N coprime to given base
CoprimeFactorBasis -- determine coprime factor base for a set of integers or ring elements
count -- count the objects in a list
covers -- a poset description from the list of the strict relations
CpuTime -- Counts cpu time
CRT -- Chinese Remainder Theorem
CRTPoly -- Chinese Remainder Theorem on polynomial coefficients
CurrentRing -- the current ring
CurrentTypes -- lists all data types
cyclotomic -- n-th cyclotomic polynomial
CyclotomicFactors -- find cyclotomic factors of a polynomial
CyclotomicIndex, CyclotomicTest -- computes the index of a given cyclotomic polynomial

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D

dashes -- returns string of dashes
date -- the date
DecimalStr -- convert rational number to decimal string
define -- define a function
DefiningIdeal -- defining ideal of a quotient ring
deg -- the standard degree of a polynomial or moduleelem
den -- denominator
DensePoly -- the sum of all power-products of a given degree
DenSigma -- den of ideal, wrt to ordering sigma
depth -- Depth of a module
deriv -- the derivative of a polynomial or rational function
DerivationAction -- Action of a derivation
describe -- information about an object
det -- the determinant of a matrix
DF -- the degree form of a polynomial
DiagMat -- matrix with given diagonal
DicksonPoly -- Dickson polynomial
diff -- returns the difference between two lists
dim -- the dimension of a (quotient) ring
discriminant -- the discriminant of a polynomial
distrib -- the distribution of objects in a list
div -- quotient for integers
DivAlg -- division algorithm
domain -- domain of a homomorphism

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E

E_ [OBSOLETE] -- [OBSOLETE] vectors of the canonical basis
eigenfactors -- eigenfactors of a matrix
eigenvectors -- eigenvalues and eigenvectors of a matrix
ElapsedTime -- Counts elapsed time
elim -- eliminate variables
ElimHomogMat -- matrix for elimination ordering
ElimMat -- matrix for elimination ordering
EmbeddingHom -- returns the embedding homomorphism of a fraction field
EqSet -- checks if the set of elements in two lists are equal
Equality Operator -- test whether two values are equal or not
EquiIsoDec -- equidimensional isoradical decomposition
error -- throw an error message
EulerTotient -- Euler Totient function for positive integers
eval -- substitute numbers or polynomials for indeterminates
EvalHilbertFn -- evaluate the Hilbert function
EvalQuasiPoly -- Evaluate a quasi-polynomial at an integer
exit -- quit CoCoA
exponents -- the list of exponents of the leading term of a polynomial
export -- export a function from a package
Ext -- presentation Ext modules as quotients of free modules
ExternalLibs -- Linked external libraries

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F

factor -- factor a polynomial
FactorAlgExt [OBSOLESCENT] -- [OBSOLESCENT] factorization over algebraic extensions
factorial -- factorial function
FactorINT -- find prime factors of an integer
FactorMultiplicity -- multiplicity of a factor of an integer
FGLM5 -- perform a FGLM Groebner Basis conversion
fibonacci -- n-th fibonacci number
fields -- list the fields of a record
first -- the first N elements of a list
FirstNonZero -- the first non-zero entry in a MODULEELEM
FirstNonZeroPosn -- the position of the first non-zero entry in a MODULEELEM
FixedDivisor -- compute (integer) fixed divisor for polynomial
flatten -- flatten a list
FloatApprox -- approx. of rational number of the form M*2^E
FloatStr -- convert rational number to a decimal string
floor -- round rational down to integer
FloorLog2, FloorLog10, FloorLogBase -- integer part of the logarithm
FloorRoot -- integer part of r-th root of an integer
FloorSqrt -- (truncated) square root of an integer
fold -- inset newlines into a long string
FoldToListInput -- convert string so it looks like a list of short strings
for -- loop command
foreach -- loop command
format -- convert object to formatted string
FrbAlexanderDual -- Alexander Dual of monomial ideals
FrbAssociatedPrimes -- Associated primes of monomial ideals
FrbIrreducibleDecomposition -- Irreducible decomposition of monomial ideals
FrbMaximalStandardMonomials -- Maximal standard monomials of monomial ideals
FrbPrimaryDecomposition -- Primary decomposition of monomial ideals
FrobeniusMat -- matrix of the Frobenius Map
FrobeniusNormSq -- Frobenius norm of a matrix
func -- Anonymous function
Function [OBSOLETE] -- [OBSOLETE]
functions [OBSOLETE] -- [OBSOLETE] replaced by describe
FVector -- f-vector of a top simplices list

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G

GBasis -- calculate a Groebner basis
GBasis TimeOut -- compute a Groebner basis with a timeout
GBasisByHomog -- calculate a Groebner basis by homogenization
GBM -- intersection of ideals for zero-dimensional schemes
gcd -- greatest common divisor
GenericPoints -- random projective points
GenRepr -- representation in terms of generators
gens -- list of generators of an ideal
GensAsCols, GensAsRows -- matrix of generators of a module
GensJacobian -- set of generators of the Jacobian ideal of a polynomial
Get [OBSOLETE] -- [OBSOLETE] replaced by GetLine
GetCol -- convert a column of a matrix into a list
GetCols -- convert a matrix into a list of lists
GetEnv -- access shell variables
GetErrMesg -- returns the message associated with an error
GetLine -- read a line of input from an in-stream
GetRow -- convert a row of a matrix into a list
GetRows -- convert a matrix into a list of lists
GFanContainsPositiveVector -- ...
GFanGeneratorsOfLinealitySpace -- ...
GFanGeneratorsOfSpan -- ...
GFanGetAmbientDimension -- ...
GFanGetCodimension -- ...
GFanGetDimension -- ...
GFanGetDimensionOfLinealitySpace -- ...
GFanGetFacets -- ...
GFanGetImpliedEquations -- ...
GFanGetUniquePoint -- ...
GFanRelativeInteriorPoint -- relative interior point of a cone
gin -- generic initial ideal
GinJacobian -- generic initial ideal of the Jacobian ideal of a polynomial
GradingDim -- Number of components in weighted degree
GradingMat -- matrix of generalized weights for indeterminates
graeffe -- graeffe transformation (squares the roots)
GraverBasis -- Graver basis
GroebnerFanIdeals -- all reduced Groebner bases of an ideal
GroebnerFanReducedGBases -- Groebner fan reduced GBases

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H

HadamardBoundSq -- Hadamard bound for determinant
HasGBasis -- checks if the argument has a pre-computed GBasis
HColon -- ideal or module quotient
HermitePoly -- Orthogonal Polynomials: Chebyshev, Hermite, Laguerre
HGBM -- intersection of ideals for zero-dimensional schemes
hilbert [OBSOLESCENT] -- [OBSOLESCENT] renamed HilbertFn
HilbertBasisKer -- Hilbert basis for a monoid
HilbertFn -- the Hilbert function
HilbertMat -- create a Hilbert matrix over QQ
HilbertPoly -- the Hilbert polynomial
HilbertSeries -- the Hilbert-Poincare series
HilbertSeriesMultiDeg -- the Hilbert-Poincare series wrt a multigrading
HilbertSeriesShifts -- the Hilbert-Poincare series
homog -- homogenize wrt an indeterminate
HomogCompt -- homogeneous part of given degree
HomogElimMat [OBSOLESCENT] -- [OBSOLESCENT] renamed ElimHomogMat
HSaturation -- saturation of ideals
HVector -- the h-vector of a module or quotient object

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I

ID [OBSOLETE] -- [OBSOLETE] renamed RingID
ideal -- ideal generated by list
IdealAndSeparatorsOfPoints -- ideal and separators for affine points
IdealAndSeparatorsOfProjectivePoints -- ideal and separators for points
IdealOfGBasis -- ideal generated by GBasis
IdealOfMinGens -- ideal generated by minimal generators
IdealOfPoints -- ideal of a set of affine points
IdealOfProjectivePoints -- ideal of a set of projective points
IdentityMat -- the identity matrix
if -- conditional statement
ILogBase -- [OBSOLESCENT] renamed FloorLogBase
image [OBSOLESCENT] -- [OBSOLESCENT] apply ring homomorphism
implicit -- implicitization
ImplicitHypersurface -- implicitization of hypersurface
ImplicitPlot -- outputs the zero locus of a bivariate polynomial to a file
ImplicitPlotOn -- outputs the zero locus of a bivariate polynomial to a file
ImportByRef, ImportByValue -- import an external local variable by reference or value
in -- list element selector in list constructor
incr, decr -- increment/decrement a counter
indent, IndentStr -- prints in a more readable way
indet -- individual indeterminates
IndetIndex -- index of an indeterminate
IndetName -- the name of an indeterminate
indets -- list of indeterminates in a PolyRing
IndetsProd -- (product of) indeterminates actually in a polynomial
IndetSubscripts -- the indices in the name of an indeterminate
IndetSymbols -- the names of the indeterminates in a PolyRing
InducedHom -- homomorphism induced by a homomorphism
InitialIdeal -- Initial ideal
insert [OBSOLESCENT] -- [OBSOLESCENT] insert an object in a list
Interpolate -- interpolating polynomial
interreduce -- interreduce a list of polynomials
interreduced -- interreduce a list of polynomials
intersection -- intersect lists, ideals, or modules
IntersectionList -- intersect lists, ideals, or modules
inverse -- multiplicative inverse of matrix
InverseSystem -- Inverse system of an ideal of derivations
IsAntiSymmetric -- checks if a matrix is anti-symmetric
IsArrCentral -- checks if the arrangement is central
IsArrFree -- checks if the arrangement is free
IsAtEOF -- checks if input stream has reached end of input
IsCommutative -- test whether a ring is commutative
IsConstant -- checks if a ringelem is in the coefficient ring
IsContained -- checks if A is Contained in B
IsCoprime -- checks if two elements are coprime
IsDefined -- check if an expression has been defined
IsDiagonal -- checks if a matrix is diagonal
IsDivisible -- checks if A is divisible by B
IsElem -- checks if A is an element of B
IsEmpty -- checks if a list is empty
IsEven, IsOdd -- test whether an integer is even or odd
IsEvenPoly, IsOddPoly -- test whether a polynomial is even or odd as a function
IsFactorClosed -- test whether a list of PPs is factor closed
IsField -- test whether a ring is a field
IsFiniteField -- test whether a ring is a finite field
IsFractionField -- test whether a ring is a fraction field
IsHomog -- test whether given polynomials are homogeneous
IsIn -- check if one object is contained in another
IsIndet -- check if argument is an indeterminate
IsIndetPosPower -- check if argument is a power of an indeterminate
IsInImage -- check if a RINGELEM is in image of RINGHOM
IsInjective -- check if a RINGHOM is injective
IsInRadical -- check if a polynomial (or ideal) is in a radical
IsInSubalgebra [OBSOLETE] -- [OBSOLETE] check if one polynomial is in a subalgebra
IsInteger -- check if a RINGELEM is integer
IsIntegralDomain -- test whether a ring is integral
IsInvertible -- check if a RINGELEM is invertible
IsIrred -- check if a RINGELEM is irreducible
IsLattice -- checks if the poset is a lattice
IsLexSegment -- checks if an ideal is lex-segment
IsLRSDegenerate -- checks the given polynomial for LRS-degeneracy
IsLRSDegenerateOrder -- checks the given polynomial for n-LRS-degeneracy
IsMaximal -- maximality test
IsMinusOne -- test whether an object is -1
IsMultiArrFree -- checks if the multiarrangement is free
IsNumber [OBSOLETE] -- [OBSOLETE] checks if the argument is a number
IsOne -- test whether an object is one
IsPalindromic -- test whether a univariate polynomial is palindromic
IsPolyRing -- test whether a ring is a polynomial ring
IsPosetGraded -- checks if the poset is graded
IsPositiveGrading -- check if a matrix defines a positive grading
IsPowerOf2 -- check if an integer is a power of 2
IsPrimary -- primary test
IsPrime -- prime integer test
IsPrimitivePoly -- test if polynomial over finite field is primitive
IsProbPrime -- checks if an integer is a probable prime
IsPthPower -- p-th power test
IsQQ -- test whether a ring is the ring of rationals
isqrt [OBSOLETE] -- [OBSOLETE] renamed FloorSqrt
IsQuotientRing -- test whether a ring is a quotient ring
IsRadical -- check if an IDEAL is radical
IsRational -- check if a RINGELEM is rational
IsSigmaGoodPrime -- check if INT is good prime for IDEAL
IsSqFree -- check if an INT or RINGELEM is square-free
IsSquare -- check if an INT is a square
IsStable -- checks if an ideal is stable
IsStdGraded -- checks if the grading is standard
IsStronglyStable -- checks if an ideal is strongly stable
IsSubset -- checks if the elements of one list are a subset of another
IsSurjective -- check if a RINGHOM is surjective
IsSymmetric -- checks if a matrix is symmetric
IsTerm -- checks if the argument is a term
IsTermOrdering -- check if a matrix defines a term-ordering
IsTree5 -- checks if a facet complex is a tree
IsTrueGCDDomain -- test whether a ring is a true GCD domain
IsZero -- test whether an object is zero
IsZeroCol, IsZeroRow -- test whether a column(row) is zero
IsZeroDet -- test whether determinant is zero
IsZeroDim -- test whether an ideal is zero-dimensional
IsZeroDivisor -- test whether a RINGELEM is a zero-divisor
IsZZ -- test whether a ring is the ring of integers
It -- variable containing the last result

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J

JacobianMat -- the Jacobian matrix of a list of polynomials
JanetBasis -- the Janet basis of an ideal

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K

ker -- Kernel of a homomorphism
KroneckerProd -- returns the Kronecker product of two matrices
KroneckerSymbol -- Kronecker symbol of R mod M

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L

LaguerrePoly -- Orthogonal Polynomials: Chebyshev, Hermite, Laguerre
last -- the last N elements of a list
latex -- LaTeX formatting
LawrenceMat -- Lawrence lifting of a matrix
LC -- the leading coefficient of a polynomial or ModuleElem
lcm -- least common multiple
len -- the length of an object
LexMat -- matrices for std. term-orderings
LexSegmentIdeal -- lex-segment ideal containing L, or with the same HilbertFn as I
LF -- the leading form of a polynomial or an ideal
LinearSimplify -- simplifying linear substitution for a univariate poly over QQ
LinKer -- find the kernel of a matrix
LinKerBasis -- find the kernel of a matrix
LinKerModP [OBSOLETE] -- [OBSOLETE] find the kernel of a matrix mod p
LinKerZZ -- find the kernel of a matrix
LinSol [OBSOLETE] -- [OBSOLETE] renamed LinSolve
LinSolve -- find a solution to a linear system
LM -- the leading monomial of a polynomial or ModuleElem
log [OBSOLESCENT] -- [OBSOLESCENT] renamed exponents
LogCardinality -- extension degree of a finite field
LPosn -- the position of the leading power-product in a ModuleElem
LPP -- the leading power-product of a polynomial or ModuleElem
LRSDegeneracyOrder -- the LRS degeneracy order of a polynomial
LT -- the leading term of an object

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M

MakeCheck -- run a series of tests
MakeMatByRows, MakeMatByCols -- convert a list into a matrix
MakeSet -- remove duplicates from a list
MakeTerm -- returns a monomial (power-product) with given exponents
MakeTermOrd [OBSOLESCENT] -- [OBSOLESCENT] renamed MakeTermOrdMat
MakeTermOrdMat -- Make a term order matrix from a given matrix
MantissaAndExponent10 -- convert rational number to a float
MantissaAndExponent2 -- convert rational number to a binary float
Manual -- search online help system
MapDown [OBSOLETE] -- [OBSOLETE] convert a constant polynomial to a number
matrix -- convert a list into a matrix
max -- a maximum element of a sequence or list
MaxBy -- a maximum element of a list
MaxChains -- maximal chains of the poset (from its relations)
MayerVietorisTreeN1 -- N-1st Betti multidegrees of monomial ideals using Mayer-Vietoris trees
min -- a minimum element of a sequence or list
MinBy -- a minimum element of a list
MinGBoverZZ [PROTOTYPE] -- [PROTOTYPE] minimal Groebner basis of polys over ZZ
MinGens -- list of minimal generators
MinGensGeneral [OBSOLESCENT] -- [OBSOLESCENT] renamed MinSubsetOfGens
minimalize [OBSOLESCENT] -- [OBSOLESCENT]
minimalized [OBSOLESCENT] -- [OBSOLESCENT] Renamed IdealOfMinGens and SubmoduleOfMinGens
MinimalPresentation -- minimal presentation for a quotient
minors -- list of minor determinants of a matrix
MinPoly -- minimal polynomial of a matrix
MinPolyQuot -- minimal polynomial in quotient ring
MinPowerInIdeal -- the mininum power of a polynomial is an ideal
MinSubsetOfGens -- list of minimal generators
mod -- remainder for integers
Mod2Rat [OBSOLETE] -- [OBSOLETE] reconstruct rationals from modular integers
ModuleElem -- create a module element
ModuleOf -- the module environment of the object
moebius -- Moebius function of a poset
MoebiusFn -- Moebius function mu on the whole numbers
monic -- divide polynomials by their leading coefficients
monomials -- the list of monomials of a polynomial
MonsInIdeal -- ideal generated by the monomials in an ideal
MSatLinSolve --
MultiArrDerMod [OBSOLESCENT] -- [OBSOLESCENT] renamed MultiArrDerModule
MultiArrDerModule -- set of generators of the module of logarithmic derivations of a multiarrangement of hyperplanes
MultiArrExponents -- exponents of a free multiarrangement of hyperplanes
MultiArrRestrictionZiegler -- Ziegler multirestriction of the arrangement of hyperplanes wrt a hyperplane
MultiArrToArr -- underling arrangement from a multiarrangement
MultiplicationMat -- multiplication matrix of a ringelem
multiplicity -- the multiplicity (degree) of a ring

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N

NewFractionField -- create a new fraction field
NewFreeModule -- create a new FreeModule
NewId [OBSOLETE] -- [OBSOLETE] create a new identifier
NewLine [OBSOLESCENT] -- [OBSOLESCENT] string containing a newline
NewList -- create a new list
NewMat -- Zero matrix
NewMatFilled -- matrix filled with value
NewPolyRing -- create a new PolyRing
NewQuotientRing -- create a new quotient ring
NewRingFp [OBSOLESCENT] -- create a new finite field
NewRingTwinFloat -- create a new twin-float ring
NewWeylAlgebra -- create a new Weyl Algebra
NewZZmod -- create a new finite ring (integers mod N)
NextPrime, NextProbPrime -- find the next prime number
NF -- normal form
NFsAreZero [OBSOLETE] -- [OBSOLETE]test if normal forms are zero
NmzComputation -- flexible access to Normaliz
NmzDiagInvariants -- ring of invariants of a diagonalizable group action
NmzEhrhartRing -- Ehrhart ring
NmzFiniteDiagInvariants -- ring of invariants of a finite group action
NmzHilbertBasis -- Hilbert Basis of a monoid
NmzHilbertBasisKer -- Hilbert basis for a monoid
NmzIntClosureMonIdeal -- integral closure of a monomial ideal
NmzIntClosureToricRing -- integral closure of a toric ring
NmzIntersectionValRings -- intersection of ring of valuations
NmzNormalToricRing -- normalization of a toric ring
NmzSetVerbosityLevel -- Set the verbosity state for Normaliz
NmzTorusInvariants -- ring of invariants of torus action
NmzVerbosityLevel -- Get the verbosity level for Normaliz
NonZero -- remove zeroes from a list
not -- boolean "not" operator
NR -- normal reduction
num -- numerator
NumBChambers -- number of bounded chambers of an arrangement of hyperplanes
NumChambers -- number of chambers of an arrangement of hyperplanes
NumCols -- number of columns in a matrix
NumCompts -- the number of components
NumGens -- number of generators
NumIndets -- number of indeterminates
NumPartitions -- number of partitions of an integer
NumRealRoots -- number of real roots of a polynomial
NumRows -- number of rows in a matrix
NumTerms -- number of terms in a polynomial

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O

one -- one of a ring
OpenIFile -- open input file
OpenIString -- open input string
OpenLog -- open a log of a CoCoA session
OpenOFile -- open output file
OpenOString -- open output string
OpenSocket -- open a socket connection
Option [OBSOLETE] -- [OBSOLETE] status of a panel option
or -- boolean "or" operator
Order Comparison Operators -- less than, greater than, ...
OrdMat -- matrix defining a term-ordering
OrlikTeraoIdeal -- Orlik-Terao ideal of an arrangement of hyperplanes

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P

package -- keyword marking content of a CoCoA package
PackageOf -- package of an identifier
packages -- list of loaded packages
panel [OBSOLETE] -- [OBSOLETE] print status of a panel's options
panels [OBSOLETE] -- [OBSOLETE] list of CoCoA panels
partitions -- partitions of an integer
permutations -- returns all permutations of the entries of a list
PerpIdealOfForm -- Ideal of derivations annihilating a form
pfaffian -- the Pfaffian of a skew-symmetric matrix
PkgName -- returns the name of a package
PlayCantStop -- First game in CoCoA
PlotPoints -- outputs the coordinates of the points to a file
PlotPointsOn -- outputs the coordinates of the points to a file
poincare [OBSOLESCENT] -- [OBSOLESCENT] Renamed HilbertSeries
PoincareMultiDeg [OBSOLETE] -- [OBSOLETE] Renamed HilbertSeriesMultiDeg
PoincareShifts [OBSOLETE] -- [OBSOLETE] Renamed HilbertSeriesShifts
PolyAlgebraHom -- homomorphism of polynomial algebras
PolyRingHom -- homomorphism of polynomial rings
PosetCharPoly -- characteristic polynomial of a poset from the relations of the poset
PosetDual -- dual of a poset from the relations of the poset
PosetJoin -- join between two elements of a poset from the relations of the poset
PosetMeet -- meet between two elements of a poset from the relations of the poset
PosetNRank -- rank of the node N from the relations of the graded poset
PosetPoincarePoly -- Poincare polynomial of a poset from the relations of the poset
PosetRank -- rank of the poset from its relations
power -- compute a power
PowerMod -- compute a modular power efficiently
PreImage [OBSOLESCENT] -- [OBSOLESCENT]
preimage0 -- preimage of a RINGELEM
PreprocessPts -- Reduce redundancy in a set of approximate points
PrevPrime, PrevProbPrime -- find the previous prime number
prim -- primitive part of a polynomial
PrimaryDecomposition -- primary decomposition of an ideal
PrimaryDecomposition0 [OBSOLETE] -- [OBSOLETE]
PrimaryDecompositionGTZ0 -- primary decomposition of a 0-dimensional ideal
PrimaryHilbertSeries -- primary
PrimaryPoincare [OBSOLESCENT] -- [OBSOLESCENT] renamed PrimaryHilbertSeries
PrimitiveRoot -- find a primitive root modulo a prime
primorial -- primorial function
print -- print the value of an expression
print on -- print to an output stream
PrintBettiDiagram -- print the diagram of the graded Betti numbers
PrintBettiMatrix -- print the matrix of the graded Betti numbers
PrintBettiNumbers -- print the (multi-)graded Betti numbers
println -- print the value of an expression
PrintRes -- print free resolution
PrintSectionalMatrix -- print sectional matrix
product -- the product of the elements of a list
protect -- protect a variable from being overwritten
PthRoot -- Compute p-th root

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Q

QQ -- the ring of rationals
QQEmbeddingHom -- returns the homomorphism QQ --> R
quit -- quit CoCoA
QuotientBasis -- vector space basis for zero-dimensional quotient rings
QuotientBasisSorted -- vector space basis for zero-dimensional quotient rings
QuotientingHom -- returns the projection homomorphism into a quotient ring
QZP -- change field for polynomials and ideals

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R

radical -- radical of an ideal
RadicalOfUnmixed -- radical of an unmixed ideal
random -- random integer
randomize [OBSOLETE] -- [OBSOLETE] randomize the coefficients of a given polynomial
randomized [OBSOLETE] -- [OBSOLETE] randomize the coefficients of a given polynomial
RandomLinearForm -- random linear form in polynomial ring
RandomNBitPrime -- Random prime with N bits
RandomPermutation -- random permutation (of indices)
RandomSmallPrime -- Random prime between 5 and N
RandomSparseNonSing01Mat -- random sparse non-singular (0,1) matrix
RandomSubset -- random subset
RandomSubsetIndices -- indices for random subset
RandomTuple -- random tuple
RandomTupleIndices -- indices for random tuples
RandomUnimodularMat -- random unimodular matrix
rank [OBSOLESCENT] -- [OBSOLESCENT] rank
RationalSolve -- Rational solutions for 0-dim polynomial system
RationalSolveHomog -- Rational solutions for 0-dim polynomial system
RatReconstructByContFrac -- rational reconstruction from modular image
RatReconstructByLattice -- rational reconstruction from modular image
RatReconstructPoly -- Rational reconstruction of polynomial coefficents
RatReconstructWithBounds -- deterministic rational reconstruction from modular image
ReadExpr [OBSOLESCENT] -- [OBSOLESCENT] renamed RingElem
RealRootRefine -- refine a real root of a univariate polynomial
RealRoots -- real roots of a univariate polynomial
RealRootsApprox -- approximations to the real roots of a univariate poly
record -- create a record
record field selector -- select a field of a record
ReducedGBasis -- reduced Groebner basis
ref -- passing function parameters by reference
RefineGCDFreeBasis [OBSOLETE] -- refine an integer GCD free basis
reg -- Castelnuovo-Mumford regularity of a module
RegularityIndex -- regularity index of a Hilbert function or series
RelNotes -- print the release notes
ReloadMan -- Reload CoCoAManual/CoCoAHelp.xml
remove -- remove an object in a list
repeat -- loop command
res -- free resolution
reseed -- reseed the pseudo-random number generator
Reset [OBSOLETE] -- [OBSOLETE] reset panels and random number seed to defaults
ResetPanels [OBSOLETE] -- [OBSOLETE] reset panels to their default values
resultant -- the resultant of two polynomials
return -- exit from a function
reverse, reversed -- reverse a list
RevLexMat -- matrix for rev lex term-ordering
rgin -- generic initial ideal wrt StdDegRevLex
RingElem -- convert an expression into a RINGELEM
RingElemList, RingElems -- convert expressions into a LIST of RINGELEM
RingEnv [OBSOLETE] -- [OBSOLETE] name of the ring environment
RingID -- identification for ring
RingOf -- the ring of the object
RingQQ -- the ring of rationals
RingQQt -- pre-defined polynomial rings
RingSet [OBSOLETE] -- [OBSOLETE] renamed RingsOf
RingsOf -- list of the rings of an object
RingZZ -- the ring of integers
rk -- rank of a matrix or module
RMap [OBSOLESCENT] -- [OBSOLESCENT] define ring homomorphism for function image
RootBound -- bound on roots of a polynomial over QQ
RootBoundTransform -- transform of a polynomial, helpful for checking RootBound
round -- round to integer
RowMat -- single row matrix
rref -- reduced row echelon form of a matrix

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S

SAGBI, SAGBIHomog -- SAGBI bases for subalgebra
SatSAGBI -- SAGBI bases for subalgebra
saturate -- saturation of ideals
ScalarProduct -- scalar product
ScientificStr -- convert integer/rational to a floating-point string
SectionalMatrix -- sectional matrix
seed [OBSOLETE] -- [OBSOLETE] replaced by reseed
SeparatorsOfPoints -- separators for affine points
SeparatorsOfProjectivePoints -- separators for projective points
SetCol -- set a list as a column into a matrix
SetEntry -- set an entry into a matrix
SetRow -- set a list as a row into a matrix
SetStackSize -- secret ;-)
SetVerbosityLevel -- set the verbosity level
shape -- extended list of types involved in an expression
sign -- the sign of a number
SimplestBinaryRatBetween -- find simplest binary rational in a closed interval
SimplestRatBetween -- find simplest rational in a closed interval
SimplexInfo -- Stanley-Reisner ideal, AlexanderDual complex, ideal of top simplices
SimplicialHomology -- simplicial homology of a top simplices list
singular value decomposition -- [OBSOLETE]
size [OBSOLETE] -- [OBSOLETE]
skip -- does nothing
SleepFor -- Make program sleep for a specified time
SmallestNonDivisor -- find smallest prime which does not divide an integer
SmoothFactor [OBSOLESCENT] -- [OBSOLESCENT] see FactorINT TrialDiv variant
SolomonTeraoIdeal -- Solomon-Terao ideal of an arrangement of hyperplanes wrt a poly
sort -- sort a list
SortBy -- sort a list
sorted -- sort a list
SortedBy -- sort a list
source -- read commands from a file or device
SourceRegion -- read commands from a region in a file
spaces -- return a string of spaces
sprint -- convert to a string
SprintTrunc -- convert to a string and truncate
SqFreeFactor -- compute a squarefree factorization
StableBBasis5 -- Stable Border Basis of ideal of points
StableIdeal -- stable ideal containing L
StagedTrees -- staged trees from Statistics
StandardInput -- an ISTREAM for reading the standard input
StandardOutput -- an OSTREAM for writing to the standard output
starting -- list functions starting with a given string
StdBasis -- Standard basis
StdDegLexMat -- matrix for std deg lex term-ordering
StdDegRevLexMat -- matrix for std deg rev lex term-ordering
StronglyStableIdeal -- strongly stable ideal containing L
SturmSeq -- Sturm sequence of a univariate polynomial
SubalgebraHom -- homomorphism into a subalgebra
SubalgebraMap [OBSOLETE] -- [OBSOLETE] algebra homomorphism representing a subalgebra
SubalgebraMinGens -- list of minimal generators as subalgebra
SubalgebraRepr [OBSOLESCENT] -- [OBSOLESCENT] representation of a polynomial as a subalgebra element
submat -- submatrix
submodule -- submodule generated by list
SubmoduleCols, SubmoduleRows -- convert a matrix into a module
SubmoduleOfMinGens -- submodule generated by minimal generators
subsets -- returns all sublists of a list
subst -- substitute values for indeterminates
substring -- substring of a string
sum -- the sum of the elements of a list
support -- the list of terms of a polynomial or moduleelem
swap -- swap two values
SwapCols -- swap two columns in a matrix
SwapRows -- swap two rows in a matrix
SwinnertonDyerPoly -- compute Swinnerton-Dyer polynomial with given roots
SylvesterMat -- the Sylvester matrix of two polynomials
SymbolRange -- range of symbols for the indeterminates of a PolyRing
SymmetricPolys -- list of symmetric polynomials
SystemCommand -- run a system command
syz -- syzygy modules
SyzOfGens -- syzygy module for a given set of generators

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T

tag -- returns the tag string of an object
tagged -- tag an object for pretty printing
tail -- remove the first element of a list
TensorMat [OBSOLESCENT] -- [OBSOLESCENT] renamed KroneckerProd
TgCone -- tangent cone
ThmProve [PROTOTYPE] -- [PROTOTYPE] ThmProve
TimeFrom -- time elapsed since a given moment
TimeOfDay -- the current time
TmpNBM [OBSOLETE] -- [OBSOLETE] renamed ApproxPointsNBM
TopLevel -- make a top-level variable accessible inside a function
TopLevelFunctions -- returns the functions available at top-level
toric -- saturate toric ideals
transposed -- the transposition of a matrix
try -- try command sequence, catch any errors
tuples -- N-tuples
TVecFromHF -- Type vector from Hilbert Function
TVecPoints -- points associated to type vector
TVecPrintRes -- resolution associated to type vector
TVecToHF -- Hilbert Function of a type vector
type -- the data type of an expression

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U

UnivariateIndetIndex -- the index of the indeterminate of a univariate polynomial
UniversalGBasis -- universal Groebner basis of the input ideal
unprotect -- remove protection from a variable
Unset [OBSOLETE] -- [OBSOLETE] set and unset panel options
untagged -- untag an object
use -- command for making a ring active

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V

valuation [OBSOLETE] -- [OBSOLETE]
VerbosityLevel -- verbosity level
VersionInfo -- version and info about CoCoA

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W

wdeg -- multi-degree of a polynomial
WeightsMatrix [OBSOLESCENT] -- [OBSOLESCENT] matrix of generalized weights for indeterminates
while -- loop command
WithoutNth -- removes the N-th component from a list
WLog [OBSOLETE] -- [OBSOLETE]

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X

XelMat -- matrices for std. term-orderings

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Z

zero -- zero of a ring
ZeroMat -- matrix filled with 0
ZPQ -- change field for polynomials and ideals
ZZ -- the ring of integers

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Part 2: CoCoA Tutorials and Programming Language

Part 2 - Chapter 1 - CoCoA Tutorials
2.1.1. Basic Tutorial for CoCoA-5
2.1.2. Tutorial: manual
2.1.3. Tutorial: Emacs UI (basic)
2.1.4. Tutorial: variables, assignment
2.1.5. Tutorial: arithmetic operators
2.1.6. Tutorial: printing
2.1.7. Tutorial: lists
2.1.8. Tutorial: polynomial rings, use command
2.1.9. Tutorial: polynomials
2.1.10. Tutorial: defining new functions
2.1.11. Tutorial: defining new functions (advanced)
2.1.12. Tutorial: homomorphisms
2.1.13. Tutorial: programming and debugging
2.1.14. Tutorial: feedback and reporting bugs
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Part 2 - Chapter 2 - Introduction to CoCoA Programming
2.2.1. An Overview of CoCoA Programming
2.2.2. All CoCoA commands
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Part 2 - Chapter 3 - Language Elements
2.3.1. Character Set and Special Symbols
2.3.2. Identifiers
2.3.3. Reserved Names
2.3.4. Comments
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Part 2 - Chapter 4 - Operators
2.4.1. CoCoA Operators: introduction
2.4.2. Algebraic Operators
2.4.3. Relational Operators
2.4.4. Selection Operators
2.4.5. Range Operator
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Part 2 - Chapter 5 - Evaluation and Assignment
2.5.1. Evaluation
2.5.2. Assignment
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Part 2 - Chapter 6 - Flow Control: Conditional Statements and Loops
2.6.1. Commands and Functions for Branching
2.6.2. Commands and Functions for Loops
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Part 2 - Chapter 7 - Verbosity and interrupt
2.7.1. Introduction to verbosity and interrupt
2.7.2. Commands and Functions implementing Verbosity
2.7.3. Commands and Functions implementing interruption
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Part 2 - Chapter 8 - Input/Output
2.8.1. Introduction to IO
2.8.2. Standard IO
2.8.3. File IO
2.8.4. String IO
2.8.5. Commands and Functions for IO
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Part 2 - Chapter 9 - CoCoA Packages
2.9.1. Introduction to Packages
2.9.2. First Example of a Package
2.9.3. Package essentials
2.9.4. Global Aliases
2.9.5. Sharing Your Package
2.9.6. Commands and Functions for Packages
2.9.7. Supported Packages
2.9.8. Galois Package
2.9.9. Integer Programming
2.9.10. Algebra of Invariants
2.9.11. Special Varieties
2.9.12. Statistics
2.9.13. Geometrical Theorem-Proving [PROTOTYPE]
2.9.14. Conductor
2.9.15. Matrix Normal Form
2.9.16. Control
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Part 2 - Chapter 10 - Linked libraries
2.10.1. CoCoALib
2.10.2. GMP
2.10.3. GSL
2.10.4. Frobby
2.10.5. MathSAT
2.10.6. Normaliz
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Part 2 - Chapter 11 - Migrating from CoCoA-4 and keeping up-to-date
2.11.1. Changes in the CoCoA language
2.11.2. Recent changes in the CoCoA-5 language
2.11.3. Obsolete and obsolescent functions
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Part 3: CoCoA datatypes

Part 3 - Chapter 1 - BOOL
3.1.1. Introduction to BOOL
3.1.2. Commands and Functions for BOOL
3.1.3. Commands and Functions returning BOOL
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Part 3 - Chapter 2 - INT
3.2.1. Introduction to INT
3.2.2. Commands and Functions for INT
3.2.3. Commands and Functions returning INT
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Part 3 - Chapter 3 - RAT
3.3.1. Introduction to RAT
3.3.2. Commands and Functions for RAT
3.3.3. Commands and Functions returning RAT
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Part 3 - Chapter 4 - STRING
3.4.1. String Literals
3.4.2. String Operations
3.4.3. Commands and Functions for STRING
3.4.4. Commands and Functions returning STRING
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Part 3 - Chapter 5 - LIST
3.5.1. Introduction to LIST
3.5.2. List Constructors
3.5.3. Commands and Functions for LIST
3.5.4. Commands and Functions returning LIST
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Part 3 - Chapter 6 - RECORD
3.6.1. Introduction to RECORD
3.6.2. Commands and Functions for RECORD
3.6.3. Commands and Functions returning RECORD
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Part 3 - Chapter 7 - FUNCTION
3.7.1. Introduction to FUNCTION
3.7.2. FUNCTIONs are first class objects
3.7.3. Commands and Functions for FUNCTION
3.7.4. Commands and Functions returning FUNCTION
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Part 3 - Chapter 8 - TYPE
3.8.1. Commands and Functions for TYPE
3.8.2. Commands and Functions returning TYPE
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Part 3 - Chapter 9 - RING
3.9.1. Introduction to RING
3.9.2. Polynomial Rings
3.9.3. Coefficient Rings
3.9.4. Indeterminates
3.9.5. Term Orderings
3.9.6. Module Orderings
3.9.7. Quotient Rings
3.9.8. Commands and Functions for RING
3.9.9. Commands and Functions returning RING
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Part 3 - Chapter 10 - RINGHOM
3.10.1. Introduction to RINGHOM
3.10.2. Composition of RINGHOM
3.10.3. Commands and Functions for RINGHOM
3.10.4. Commands and Functions returning RINGHOM
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Part 3 - Chapter 11 - RINGELEM
3.11.1. Introduction to RINGELEM
3.11.2. Evaluation of Polynomials
3.11.3. Commands and Functions for RINGELEM
3.11.4. Commands and Functions returning RINGELEM
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Part 3 - Chapter 12 - IDEAL
3.12.1. Commands and Functions for IDEAL
3.12.2. Commands and Functions returning IDEAL
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Part 3 - Chapter 13 - MAT
3.13.1. Introduction to MAT
3.13.2. Commands and Functions for MAT
3.13.3. Commands and Functions returning MAT
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Part 3 - Chapter 14 - MODULE
3.14.1. Commands and Functions for MODULE
3.14.2. Commands and Functions returning MODULE
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Part 3 - Chapter 15 - MODULEELEM
3.15.1. Introduction to MODULEELEM
3.15.2. Commands and Functions for MODULEELEM
3.15.3. Commands and Functions returning MODULEELEM
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Part 3 - Chapter 16 - Creating new types
3.16.1. Tagging an Object
3.16.2. Printing a Tagged Object
3.16.3. Commands and Functions for Tags
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