Alphabetical list of commands

ABCDEFGHIJKLMNOPQRSTUVWXZ

Special Characters

Shortcuts -- Special characters equivalent to commands

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A

Abs -- absolute value of a number Adjoint -- adjoint matrix AffHilbert -- the affine Hilbert function AffHilbertFn -- the affine Hilbert function AffHilbertSeries -- the affine Hilbert-Poincare series AffPoincare -- the affine Hilbert-Poincare series AlexanderDual_Frobby5 -- Alexander Dual of monomial ideals Alias -- define aliases for package names Alias In -- temporarily override global aliases Aliases -- list of global aliases AllIndetsCalled -- List of indeterminates of current ring with a given name AlmostQR -- QR decomposition of a matrix Append -- append an object to an existing list Ascii -- convert between characters and ascii code

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B

BBasis5 -- Border Basis of zero dimensional ideal BettiDiagram -- the diagram of the graded Betti numbers BettiMatrix -- the matrix of the graded Betti numbers Bin -- binomial coefficient BinExp -- binomial expansion Block -- group several commands into a single command BlockMatrix -- create a block matrix Break -- break out of a loop BringIn -- bring in objects from another ring

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C

Call -- apply a function to given arguments CartesianProduct, CartesianProductList -- Cartesian product of lists Cast -- type conversion Catch -- catch an error CFApprox -- continued fractions CFApproximants -- continued fractions Characteristic -- the characteristic of a ring CharPoly -- characteristic polynomial of a matrix Ciao -- quit CoCoA Clear -- clear the working memory or a ring-bound memory ClearDenom -- clear common denominator of a polynomial with rational coeffs Close -- close a device CloseLog -- close a log of a CoCoA session CocoaLimits -- limits on exponents and ring characteristics CocoaPackagePath -- returns the path to the CoCoA packages Coefficients -- list of coefficients of a polynomial or vector CoeffOfTerm -- coefficient of a term of a polynomial or vector ColMat -- single column matrix Colon -- ideal or module quotient ColumnVectors -- the list of column vectors of a matrix Comp -- the N-th component of a list Comps -- list of components of a vector Concat -- concatenate lists ConcatLists -- concatenate a list of lists Cond -- conditional expression ContFrac -- continued fractions denominators ContFracToRat -- convert continued fraction to rational Count -- count the objects in a list CurrentRing -- the current ring

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D

Dashes -- returns string of dashes Date -- the date DecimalStr -- convert rational number to decimal string Define -- define a function Deg -- the degree of a polynomial or vector DegLexMat -- matrices for std. term-orderings DegRevLexMat -- matrices for std. term-orderings Delete -- delete variables from the working memory Den -- denominator DensePoly -- the sum of all power-products of a given degree Depth -- Depth of a module Der -- the derivative of a rational function Describe -- information about an expression Destroy -- delete rings Det -- the determinant of a matrix DiagonalMat -- matrix with given diagonal Diff -- returns the difference between two lists Dim -- the dimension of a ring or quotient object Discriminant -- the discriminant of a polynomial Distrib -- the distribution of objects in a list Div -- quotient for integers DivAlg -- division algorithm

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E

E_ -- canonical vector Eigenvectors -- eigenvalues and eigenvectors of a matrix Elim -- eliminate variables EqSet -- checks if the set of elements in two lists are equal Equals -- returns a string of equal signs EquiIsoDec -- equidimensional isoradical decomposition Error -- return an error message Eval -- substitute numbers or polynomials for indeterminates EvalBinExp -- binomial expansion functions EvalHilbertFn -- evaluate the Hilbert function Ext -- presentation Ext modules as quotients of free modules

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F

Fact -- factorial function Factor -- factor a polynomial FactorMultiplicity -- multiplicity of a factor of an integer FGLM5 -- perform a FGLM Groebner Basis conversion Fields -- list the fields of a record First -- the first N elements of a list FirstNonZero -- the first non-zero entry in a vector FirstNonZeroPos -- the first non-zero entry in a vector Flatten -- flatten a list FloatApprox -- approx. of rational number of the form M*10^E FloatStr -- convert rational number to a float string For -- loop command Foreach -- loop command Format -- convert object to formatted string Fraction -- returns the quotient of its arguments Function -- return a function Functions -- list the functions of a package

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G

GB.Complete -- Complete an interactive Groebner-type calculation GB.GetBettiMatrix -- returns the Betti matrix computed so far GB.GetNthSyz -- returns the part of the Nth syzygy module computed so far GB.GetNthSyzShifts -- shifts of the Nth syzygy module computed so far GB.GetRes -- returns the resolution computed so far GB.GetResLen -- returns the length of the resolution computed so far GB.ResReport -- status of an interactive resolution calculation GB.Start_GBasis -- start interactive Groebner basis computation GB.Start_MinGens -- start interactive minimal generator calculation GB.Start_MinSyzMinGens -- start interactive calc., min. syzs. of min. gens. GB.Start_Res -- start interactive resolution computation GB.Start_Syz -- start interactive syzygy computation GB.Start_SyzMinGens -- start interactive calc. of syzygies of min. gens. GB.Stats -- status of an interactive Groebner-type calculation GB.Step -- take one step in an interactive Groebner-type calculation GB.Steps -- take steps in an interactive Groebner-type calculation GBasis -- calculate a Groebner basis GBasis5, and more -- calculations using the CoCoAServer GBasisTimeout -- compute a Groebner basis with a timeout GBM -- intersection of ideals for zero-dimensional schemes GCD -- greatest common divisor GCDFreeBasis -- determine (minimal) GCD free basis of a set of integers GenericPoints -- random projective points GenRepr -- representation in terms of generators Gens -- list of generators of an ideal Get -- read characters from a device GetEnv -- access shell variables GetErrMesg -- returns the message associated with an error Gin, Gin5 -- generic initial ideal GlobalMemory -- contents of global memory

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H

H.Browse -- browse the online help system H.Commands -- list commands according to type H.Man -- search online help system H.OutCommands -- print command descriptions to a file H.OutManual -- print the online manual to a file H.SetMore -- more-device for online help H.Syntax -- display the syntax of a command H.Tips -- quick tips for using online help H.Toc -- display the Table Of Contents of the online manual H.Tutorial -- run the CoCoA tutorial H.UnSetMore -- more-device for online help HColon -- ideal or module quotient Head -- the first element of a list Help -- extend online help for user-defined functions HGBM -- intersection of ideals for zero-dimensional schemes Hilbert -- the Hilbert-Poincare' function HilbertBasis -- Hilbert basis for a monoid HilbertFn -- the Hilbert function HilbertPoly -- the Hilbert polynomial HilbertSeries -- the Hilbert-Poincare series HilbertSeriesMultiDeg -- the Hilbert-Poincare series wrt a multigrading HilbertSeriesShifts -- the Hilbert-Poincare series HIntersection -- intersection of ideals HIntersectionList -- intersection of ideals Homogenized -- homogenize with respect to an indeterminate HSaturation -- saturation of ideals HVector -- the h-vector of a module or quotient object

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I

Ideal -- convert an expression into an ideal IdealAndSeparatorsOfPoints -- ideal and separators for affine points IdealAndSeparatorsOfProjectivePoints -- ideal and separators for points IdealOfPoints -- ideal of a set of affine points IdealOfProjectivePoints -- ideal of a set of projective points Identity -- the identity matrix If -- conditional statement ILogBase -- integer part of the logarithm Image -- apply ring homomorphism ImplicitPlot -- outputs the zero locus of a bivariate polynomial to a file ImplicitPlotOn -- outputs the zero locus of a bivariate polynomial to a file In -- create a list satisfying given conditions Indet -- individual indeterminates IndetInd -- the index of an indeterminate IndetIndex -- index of an indeterminate IndetName -- the name of an indeterminate Indets -- list of current indeterminates Insert -- insert an object in a list Interpolate -- interpolating polynomial Interreduce, Interreduced -- interreduce a list of polynomials or vectors Intersection -- intersect lists, ideals, or modules IntersectionList -- intersect lists, ideals, or modules Inverse -- multiplicative inverse IO.SprintTrunc -- convert to a string and truncate Iroot -- integer part of r-th root of an integer IrreducibleDecom_Frobby5 -- Irreducible decomposition of monomial ideals IsAntiSymmetric -- checks if a matrix is anti-symmetric IsDefined -- check if an expression has been defined IsDiagonal -- checks if a matrix is diagonal IsEven, IsOdd -- test whether an integer is even or odd IsHomog -- test whether given polynomials are homogeneous IsIn -- check if one object is contained in another IsLexSegment -- checks if an ideal is lex-segment IsNumber -- checks if the argument is a number IsPositiveGrading -- check if a matrix defines a positive grading IsPPrime -- checks if an integer is a probable prime IsPrime -- prime integer test Isqrt -- (truncated) square root of an integer IsServerReady -- checks if CoCoAServer is running IsStable -- checks if an ideal is stable IsStronglyStable -- checks if an ideal is strongly stable IsSubset -- checks if the elements of one list are a subset of another 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 IsZero -- test whether an object is zero

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J

Jacobian -- the Jacobian of a list of polynomials

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K

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L

Last -- the last N elements of a list Latex -- LaTeX formatting LC -- the leading coefficient of a polynomial or vector 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 Hilbert fn as I LinearSimplify -- find simplifying substitution for a univariate polynomial over Q LinKer, LinKerModP -- find the kernel of a matrix LinSol -- find a solution to a linear system List -- convert an expression into a list LM -- the leading monomial of a polynomial or vector Log -- the list of exponents of the leading term of a polynomial LogToTerm -- returns a monomial (power-product) with given exponents LPos -- the position of the leading power-product in a vector LPP -- the leading power-product of a polynomial or vector 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 Man -- search online help system MantissaAndExponent -- convert rational number to a float MapDown -- convert a constant polynomial to a number Mat -- convert an expression into a matrix MatConcatAntiDiag -- create a simple block matrix MatConcatDiag -- create a simple block matrix MatConcatHor -- create a simple block matrix MatConcatVer -- create a simple block matrix Max, Min -- a maximum or minimum element of a sequence or list MaximalStandardMonomials_Frobby5 -- Maximal standard monomials of monomial ideals MayerVietorisTreeN1 -- N-1st Betti multidegrees of monomial ideals using Mayer-Vietoris trees MDeg -- multi-degree of an polynomial Memory -- contents of local memory or ring-bound memory MinGens -- list minimal generators Minimalize -- remove redundant generators Minimalized -- remove redundant generators MinimalPresentation -- minimal presentation for a quotient Minors -- list of minor determinants of a matrix MinSyzMinGens -- minimal generators of syzygies of minimal generators Mod -- remainder for integers Mod2Rat -- reconstructing rationals from modular integers Module -- convert an expression into a module Monic -- divide polynomials by their leading coefficients Monomials -- the list of monomials of a polynomial or vector MonsInIdeal -- ideal generated by the monomials in an ideal More -- print a string, N lines at a time Multiplicity -- the multiplicity (degree) of a ring or quotient object

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N

NewId -- create a new identifier NewList -- create a new list NewMat -- create a new matrix NewVector -- create a new vector NextPPrime -- find the next largest probable prime number NextPrime -- find the next largest prime number NF -- normal form NFsAreZero -- test if normal forms are zero NonZero -- remove zeroes from a list Not, And, Or -- boolean operators NR -- normal reduction Num -- numerator NumCols -- number of columns in a matrix NumComps -- the number of components of a vector NumIndets -- number of indeterminates NumRows -- number of rows in a matrix

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O

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 -- status of a panel option Ord -- matrix defining a term-ordering

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P

Packages -- list of loaded packages Panel -- print status of a panel's options Panels -- list of CoCoA panels Partitions -- partitions of an integer Permutations -- returns all permutations of the entries of a list Pfaffian -- the Pfaffian of a skew-symmetric matrix PkgName -- returns the name of a package PlotPoints -- outputs the coordinates of the points to a file PlotPointsOn -- outputs the coordinates of the points to a file Poincare -- the Hilbert-Poincare series PoincareMultiDeg -- the Hilbert-Poincare series wrt a multigrading PoincareShifts -- the Hilbert-Poincare series Poly -- convert an expression into a polynomial PositiveGrading4 -- extend a positive grading (for CoCoA-4 multigradings limitation) PowerMod -- compute a modular power efficiently PreprocessPts5 -- Reduce redundancy in a set of approximate points PrimaryDecomposition -- primary decomposition of an ideal Print -- print the value of an expression Print On -- print to an output device PrintLn -- print the value of an expression Product -- the product of the elements of a list

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Q

Quit -- quit CoCoA QuotientBasis -- vector space basis for zero-dimensional quotient rings QZP -- change field for polynomials and ideals

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R

Radical -- radical of an ideal RadicalOfUnmixed -- radical of an unmixed ideal Rand -- random integer Randomize -- randomize the coefficients of a given polynomial Randomized -- randomize the coefficients of a given polynomial Rank -- rank of a module RealRootRefine -- refine a real root of a univariate polynomial RealRoots -- computes the real roots of a univariate polynomial RealRootsApprox -- computes approximations to the real roots of a univariate polynomial Record -- create a record ReducedGBasis -- compute reduced Groebner basis RefineGCDFreeBasis -- refine an integer GCD free basis RegularityIndex -- regularity index of a Hilbert function or series RelNotes -- print the release notes Remove -- remove an object in a list Repeat -- loop command Res -- free resolution Reset -- reset panels and random number seed to defaults ResetPanels -- reset panels to their default values Resultant -- the resultant of two polynomials Return -- exit from a structured command Reverse, Reversed -- reverse a list RevLexMat -- matrices for std. term-orderings Ring -- returns the ring with a given name RingEnv -- name of the current ring RingEnvs -- names of all defined rings RingEnvSet -- set of names of the ring environments RMap -- define ring homomorphism RootBound -- bound on roots of a polynomial over Q RowMat -- single row matrix

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S

Saturation -- saturation of ideals ScalarProduct -- scalar product Seed -- seed for Rand SeparatorsOfPoints -- separators for affine points SeparatorsOfProjectivePoints -- separators for projective points Set, Unset -- set and unset panel options Shape -- extended list of types involved in an expression Sign -- the sign of a number Size -- the amount of memory used by an object Skip -- does nothing SmoothFactor -- find small prime factors of an integer Sort -- sort a list SortBy -- sort a list Sorted -- sort a list SortedBy -- sort a list Source -- read commands from a file or device Spaces -- return a string of spaces Sprint -- convert to a string StableBBasis5 -- Stable Border Basis of ideal of points StableBBasisNBM5 -- Numerical Border Basis of ideal of points StableIdeal -- stable ideal containing L StarPrint -- print polynomial with *'s for multiplications Starting -- list functions starting with a given string StronglyStableIdeal -- strongly stable ideal containing L Submat -- submatrix Subsets -- returns all sublists of a list Subst -- substitute values for indeterminates Sum -- the sum of the elements of a list Support -- the list of terms of a polynomial or vector Sylvester -- the Sylvester matrix of two polynomials Syz -- syzygy modules SyzMinGens -- syzygy module for a set of minimal generators 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 -- returns the tensor product of two matrices Toric -- saturate toric ideals Transposed -- the transposition of a matrix Tuples -- N-tuples Type -- the data type of an expression TypeOfCoeffs -- type of the coefficients of the current ring Types -- lists all data types

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U

UnivariateIndetIndex -- the index of the indeterminate of a univariate polynomial Unset -- set and unset panel options Untagged -- untag an object Use -- command for making a ring active Using -- perform commands in non-active ring

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V

Var -- function calls by reference, other complex referencing Vector -- create a vector

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W

WeightsList -- first row of the weights matrix WeightsMatrix -- matrix of generalized weights for indeterminates While -- loop command WithoutNth -- removes the N-th component from a list WLog -- weighted list of exponents

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X

XelMat -- matrices for std. term-orderings

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Z

ZPQ -- change field for polynomials and ideals

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