CoCoA-5.4.1x Manual

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 to AffHilbertSeries alias -- define aliases for package names aliases -- list of global aliases AllFactors -- All positive factors of a (small) positive integer AllReducedGroebnerBases [OBSOLETE] -- [OBSOLETE] Renamed to 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 to 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 to 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 CyclotomicFactorIndexes -- find indexes of 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 FirstCols, FirstRows -- submatrix of the first N cols or rows 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 to 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 to ElimHomogMat HSaturation -- saturation of ideals HVector -- the h-vector of a module or quotient object

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I

ID [OBSOLETE] -- [OBSOLETE] renamed to 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 [OBSOLETE] -- [OBSOLETE] renamed to 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 InvTotient -- Inverse totient function: preimages of Euler totient 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 to 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 to LinSolve LinSolve -- find a solution to a linear system LM -- the leading monomial of a polynomial or ModuleElem log [OBSOLESCENT] -- [OBSOLESCENT] renamed to 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 to 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 to 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 to 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 NewFreeModuleForSyz -- create a new FreeModule with shifts for syz 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 NewPolyRingWeights -- create a new PolyRing with weights 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 to HilbertSeries PoincareMultiDeg [OBSOLETE] -- [OBSOLETE] Renamed to HilbertSeriesMultiDeg PoincareShifts [OBSOLETE] -- [OBSOLETE] Renamed to 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 to 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 to 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] renamed: 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 StarRoot -- compute smallest root of an integer 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 module 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 to KroneckerProd TgCone -- tangent cone ThmProve [PROTOTYPE] -- [PROTOTYPE] ThmProve TimeFrom -- time elapsed since a given moment TimeOfDay -- the current time TmpNBM [OBSOLETE] -- [OBSOLETE] renamed to 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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