Macaulay2Doc : Table of Contents

• Macaulay2Doc -- Macaulay2 documentation

  • Copyright and license

    • Singular-Factory
    • frobby
    • MPIR
    • GNU MP
    • MPFR
    • MPFI
    • GC garbage collector
    • LAPACK
    • BLAS
    • NTL library
    • FLINT
    • givaro
    • fflas-ffpack
    • MPSolve -- a library for finding roots of univariate polynomials
    • Boost -- a collection of C++ source libraries
    • 4ti2
    • gfan
    • normaliz
    • csdp
    • nauty
    • cdd+
    • lrslib
    • topcom
    • cohomCalg
  • Acknowledgements
  • The authors

    • Daniel R. Grayson
    • Michael E. Stillman
    • David Eisenbud
  • Other sources of information about Macaulay2
  • getting started

    • checking your Macaulay2 installation -- and bookmarking the Macaulay2 documentation
    • finding the Macaulay2 files
    • teaching M2 how to find its shared libraries
    • teaching your shell how to find M2
    • moving or copying the Macaulay2 files
    • running Macaulay2 in a terminal window
    • using Macaulay2 with emacs

      • setting up the Macaulay2 emacs interface
      • teaching emacs how to find M2-init.el
      • teaching emacs how to find M2
      • running Macaulay2 in emacs
      • using Macaulay2 with emacs after it has been set up
      • editing Macaulay2 code with emacs
    • using Macaulay2 with TeXmacs
    • a first Macaulay2 session
    • reading the documentation

      • about -- search the documentation
      • apropos -- symbols matching a pattern
      • help -- view documentation nodes
      • viewHelp -- view online documentation in a web browser
      • infoHelp -- view documentation in Info format
      • ? Symbol -- view brief documentation of a symbol
      • examples -- list the examples in documentation
    • getting help or reporting bugs
  • rings

    • basic rings of numbers
    • integers modulo a prime
    • finite fields
    • polynomial rings
    • monoid -- make or retrieve a monoid

      • monoid(...,Variables=>...) -- specify the names of the indeterminates
      • monoid(...,Local=>...) -- specify local or global monomial order
      • monoid(...,Inverses=>...) -- allow negative exponents in monomials
      • monoid(...,Weights=>...) -- specify weights of the variables
      • monoid(...,Degrees=>...) -- specify the degrees of the variables
      • monoid(...,DegreeLift=>...) -- specify maps between degree groups
      • monoid(...,Heft=>...) -- specify a heft vector
      • monoid(...,Join=>...) -- specify how to handle degrees in the coefficient ring
      • monoid(...,MonomialOrder=>...) -- specify the monomial ordering
      • monoid(...,MonomialSize=>...) -- specify the bit-length of monomial exponents in the ring
      • monoid(...,SkewCommutative=>...) -- specify Skew commuting variables in the ring
      • monoid(...,WeylAlgebra=>...) -- specify differential operators in the ring
    • monomial orderings

      • examples of specifying alternate monomial orders
      • monomial orders for free modules

        • Schreyer orders -- induced monomial order on a free module
      • packing monomials for efficiency
      • GRevLex -- graded reverse lexicographical monomial order.
      • Lex -- lexicographical monomial order.
      • GLex -- graded lexicographic ordering
      • Weights -- assigning weights to the variables
      • Eliminate -- elimination order
      • GroupLex -- defines a ring where some variables are inverted
      • GroupRevLex
      • definition of product (block) orders
      • RevLex -- reverse lexicographic ordering
      • NCLex -- Non-commutative lexicographical order.
      • MonomialOrder -- monomial ordering
      • obtaining the monomial order of a ring
    • graded and multigraded polynomial rings

      • heft vectors

        • heft -- heft vector of ring or monoid
        • findHeft -- find a heft vector for a list of degrees
    • quotient rings
    • manipulating polynomials
    • factoring polynomials
    • fraction fields
    • finite field extensions
    • exterior algebras
    • symmetric algebras
    • tensor products of rings
    • Weyl algebras
    • associative algebras
  • ideals

    • creating an ideal
    • ideals to and from matrices
    • ideals to and from modules
    • sums, products, and powers of ideals
    • equality and containment
    • extracting generators of an ideal
    • dimension, codimension, and degree
    • intersection of ideals
  • matrices

    • inputting a matrix
    • projections, inclusions, and permutations
    • random and generic matrices
    • extracting information about a matrix
    • basic arithmetic of matrices
    • concatenating matrices
    • differentiation
    • rank of a matrix
    • determinants and minors
    • Pfaffians
    • exterior power of a matrix
    • format and display of matrices in Macaulay2
    • importing and exporting matrices
  • substitution and maps between rings

    • substitute -- substituting values for variables
    • working with multiple rings
    • basic construction, source and target of a ring map
    • evaluation and composition of ring maps
    • kernel and coimage of a ring map
    • graphIdeal(RingMap) -- the ideal of the graph of the regular map corresponding to a ring map
    • graphRing(RingMap) -- the coordinate ring of the graph of the regular map corresponding to a ring map
  • modules

    • free modules
    • matrices to and from modules -- including kernel, cokernel and image
    • submodules and quotients
    • subquotient modules -- the way Macaulay2 represents modules
    • module homomorphisms
    • right modules or left modules?
    • Hilbert functions and free resolutions -- including degree and betti numbers

      • isHomogeneous -- whether something is homogeneous (graded)
      • codim(Module) -- codimension of the support of a module
      • degree(Module)
      • genera(CoherentSheaf) -- list of the successive linear sectional arithmetic genera
      • hilbertSeries(Module) -- compute the Hilbert series of the module
      • poincare -- assemble degrees of a ring, module, or ideal into a polynomial
      • poincareN -- assemble degrees into polynomial
      • reduceHilbert -- reduce a Hilbert series expression
      • betti -- display or modify a Betti diagram

        • BettiTally -- the class of all Betti tallies
        • MultigradedBettiTally -- the class of all multigraded Betti tallies
        • multigraded -- convert a Betti tally into a multigraded Betti tally
        • betti(BettiTally) -- view and set the weight vector of a Betti diagram
        • betti(...,Minimize=>...) -- minimal betti numbers of a non-minimal free resolution
        • minimalBetti -- minimal betti numbers of (the minimal free resolution of) a homogeneous ideal or module
    • basis -- basis or generating set of all or part of a ring, ideal or module
    • exterior power of a module
  • chain complexes

    • free resolutions of modules

      • computing resolutions
    • extracting information from chain complexes
    • making chain complexes by hand
    • manipulating chain complexes
    • maps between chain complexes
  • varieties

    • coherent sheaves
    • Variety -- the class of all algebraic varieties
  • Gröbner bases

    • what is a Groebner basis?
    • Groebner basis examples and applications

      • simple Groebner basis computations over various rings
    • computing Groebner bases
  • normal forms
  • elimination of variables
  • combinatorial functions
  • The Macaulay2 language

    • variables

      • valid names
      • assigning values
      • local variables in a file
      • viewing the symbols defined so far
      • subscripted variables
      • numbered variables
    • numeric types
    • strings and nets -- an overview of strings and nets in Macaulay2

      • regular expressions -- syntax for regular expressions

        • match -- regular expression matching
        • regex -- evaluate a regular expression search
        • separate -- split a string into substrings using a regular expression
        • select(String,String,String) -- select and reformat substrings matching a regular expression
        • replace(String,String,String) -- replace substrings matching a regular expression
        • regexQuote -- escape special characters in regular expressions
      • concatenate -- join strings
      • format -- format a string or a real number
      • lines -- split a string into lines
      • ascii -- ASCII character conversion
      • utf8 -- encode and decode unicode utf-8-encoded strings
      • substring -- extract part of a string
      • String _ ZZ -- get element from string
      • String _ Sequence -- substring
      • horizontalJoin -- join nets or strings horizontally
      • pad -- pad a string or net with spaces
      • stack -- join nets or string vertically
      • unstack -- list the rows of a net
      • String -- the class of all strings

        • net(String) -- convert a string to a net
        • toString -- convert to a string
        • toExternalString -- convert to a readable string
        • /// -- delineate a string with slashes
        • " -- delineate a string with quotation marks
      • Net -- the class of all nets and strings

        • net -- format for printing, as a net
        • width(Net) -- width of a net
        • height -- height of a net
        • depth -- depth of a net
    • lists and sequences -- a detailed overview of lists and sequences in Macaulay2
    • hash tables

      • copy -- copy an object
      • hashTable -- make a hash table
      • keys -- keys used in a hash table, dictionary, or database
      • values -- values in a hash table
      • pairs -- list the pairs in a hash table, dictionary, or basic list
      • isMutable -- whether something may be modified
      • remove -- remove an entry from a mutable hash table, list, or database
      • applyKeys -- apply a function to each key in a hash table
      • applyValues -- apply a function to each value in a hash table
      • applyPairs -- apply a function to each pair in a hash table
      • merge -- merge two hash tables
      • combine -- combine hash tables
      • hashing
      • HashTable -- the class of all hash tables
      • MutableHashTable -- the class of all mutable hash tables
    • operators

      • = -- assignment
      • := -- assignment of method or new local variable
      • <- -- assignment with left side evaluated
      • == -- equality
      • != -- inequality
      • Thing === Thing -- strict equality
      • =!= -- strict inequality
      • < -- less than
      • <= -- less than or equal
      • > -- greater than
      • >= -- greater than or equal
      • ? -- comparison operator
      • and -- conjunction
      • or -- disjunction
      • not -- negation
      • SPACE -- blank operator; often used for function application, making polynomial rings
      • -> -- make a function
      • @@ -- a binary operator
      • \\ -- a binary operator
      • \ -- a binary operator
      • _ -- a binary operator, used for subscripting and access to elements
      • . -- access to elements whose key is a symbol
      • .? -- check for presence of elements whose key is a symbol
      • # -- length or access to elements
      • #? -- check existence of value in a list, hash table, database, or string
      • ! -- factorial
      • + -- a unary or binary operator, usually used for addition
      • - -- a unary or binary operator, usually used for negation or subtraction
      • * -- a binary operator, usually used for multiplication
      • / -- a binary operator, usually used for division
      • // -- a binary operator, usually used for quotient
      • % -- a binary operator, usually used for remainder and reduction
      • ^ -- a binary operator, usually used for powers
      • & -- a binary operator
      • ++ -- a binary operator, usually used for direct sum
      • ** -- a binary operator, usually used for tensor product or Cartesian product
      • ^** -- a binary operator, usually used for tensor or Cartesian power
      • ~ -- a unary postfix operator
      • (*) -- a unary postfix operator, used for indicating a graded object
      • : -- a binary operator, uses include repetition; ideal quotients
      • .. -- a binary operator, used for sequences of consecutive items
      • ..< -- a binary operator, used for sequences of consecutive items, not including the endpoint
      • << -- a binary operator (file output, ...)
      • >> -- a binary operator, uses include bit shifting, or attaching optional inputs to functions
      • | -- a binary operator, often used for horizontal concatenation
      • || -- a binary operator, often used for vertical concatenation
      • @ -- a binary operator
      • ^^ -- logical exclusive-or
      • |- -- a binary operator
      • <==> -- a binary operator
      • ===> -- a binary operator
      • ==> -- a binary operator
      • <=== -- a unary and binary operator
      • <== -- a unary and binary operator
      • => -- produce an Option
      • , -- the comma, used for separating entries in a list or sequence
      • ; -- expression separator
      • precedence of operators
      • operatorAttributes -- a hashtable with information about Macaulay2 operators
    • conditional execution -- if-then-else statements
    • while -- while loops
    • for -- for loops
    • mapping over hash tables -- apply a function to each element of a hash table
    • error handling -- signalling and trapping errors

      • error -- deliver error message
      • try -- catch an error
    • catch -- catch a thrown exception
    • using functions
    • using functions with optional inputs
    • making functions
    • local variables in a function
    • making functions with a variable number of arguments
    • making functions with multiple return values
    • making new functions with optional arguments
    • using hooks

      • hooks -- list hooks attached to a key
      • addHook -- add a hook function to an object for later processing
      • runHooks -- run the hook functions stored in an object
    • code -- display source code
    • what a class is

      • class -- class of an object
      • parent -- parent type of an object
      • instance -- whether something has a certain type
      • ancestor(Type,Type) -- whether one type is an ancestor of another
    • installing methods
    • binary methods
    • inheritance
    • making new classes
    • new -- new objects and new types

      • newClass -- set the class and parent of an object
    • printing and formatting for new classes
    • making a new method function

      • method -- make a new method function

        • specifying typical values
        • MethodFunction -- a type of method function
        • MethodFunctionSingle -- a type of method function
        • MethodFunctionBinary -- a type of method function
        • MethodFunctionWithOptions -- a type of method function
    • methods -- list methods
    • printing to the screen
    • reading files
    • getting input from the user
    • creating and writing files
    • saving polynomials and matrices in files
    • two dimensional formatting
    • file manipulation -- Unix file manipulation functions

      • fileExists -- whether a file exists
      • isRegularFile -- whether a file is a regular file
      • isDirectory -- whether a file is a directory
      • fileTime -- get or set file modified time
      • copyFile(String,String)
      • moveFile(String,String)
      • removeFile -- remove a file
      • findFiles -- find files recursively
      • symlinkFile -- make a symbolic link to a file
      • linkFile(String,String) -- make a new link to a file
      • readlink -- readlink fn
      • mkdir
      • makeDirectory(String) -- make a directory
      • copyDirectory(String,String)
      • removeDirectory -- remove a directory
      • symlinkDirectory(String,String) -- make symbolic links for all files in a directory tree
      • readDirectory -- read the contents of a directory
      • realpath -- convert a filename to one passing through no symbolic links
      • temporaryFileName -- make a temporary file name
    • communicating with programs
    • using sockets
    • packages -- Macaulay2 packages

      • Package -- the class of all packages
      • loadedPackages -- the list of loaded packages
      • package -- get containing package
      • loadPackage -- load a package
      • readPackage -- read the package preamble
      • needsPackage -- load a package if not already loaded
      • installPackage -- load and install a package and its documentation
      • installedPackages
      • uninstallPackage -- uninstall a package and remove its documentation
      • uninstallAllPackages
      • importFrom -- import symbols to the current private dictionary
      • use(Package) -- activate a previously dismissed package
      • check -- perform tests of a package
      • debug -- open the private dictionary of a package
      • dismiss -- dismiss a package
      • options(Package) -- get options
      • makePackageIndex -- create an index of installed packages
      • PackageDictionary -- the dictionary for names of packages
    • creating a package

      • an example of a package
      • currentPackage -- the current package
      • newPackage -- the preamble of a package
      • export -- export symbols from a package
      • exportFrom -- export symbols from a package's private dictionary
      • exportMutable -- export mutable symbols from a package
      • beginDocumentation -- start documentation section of a package
      • TEST -- add a test for a package
      • endPackage -- end a new package
    • writing documentation

      • hypertext list format
      • document -- create a documentation node

        • package documentation template
        • function documentation template
        • optional argument documentation template
        • overview documentation template
        • Key -- key of a documentation node
        • Headline -- headline of a documentation node
        • BaseFunction -- the basename of a function in a documentation node
        • Usage -- ways to use a function in a documentation node
        • Inputs -- inputs for a function in a documentation node
        • Outputs -- outputs for a function in a documentation node
        • Consequences -- side-effects of a function in a documentation node
        • Acknowledgement -- acknowledgements listed in a documentation node
        • Contributors -- non-author contributors listed in a documentation node
        • References -- references listed in a documentation node
        • Caveat -- warnings or limitations listed in a documentation node
        • SeeAlso -- cross-references to other documentation nodes
        • Subnodes -- a menu of documentation nodes
        • SourceCode -- sources displayed in a documentation node
        • ExampleFiles -- data files corresponding to a documentation node
      • EXAMPLE -- hypertext EXAMPLE item
      • SYNOPSIS -- a standardized synopsis for use in documentation nodes
      • documentation keys

        • DocumentTag -- the class of all document tags
    • parallel programming with threads and tasks

      • createTask -- create a task
      • addCancelTask(Task,Task) -- specify that the completion of one task triggers the cancellation of another
      • addDependencyTask(Task,Task) -- schedule a task, but ensure that it will not run until another task finishes
      • addStartTask(Task,Task) -- schedule a task upon completion of another
      • isReady(Task) -- whether a task is finished
      • schedule -- schedule a task for execution
      • taskResult(Task) -- retrieve the value returned by a task
      • cancelTask(Task) -- stop a task
      • isCanceled(Task) -- whether a task has been canceled
      • setIOExclusive -- exclusive I/O for the current thread
      • setIOSynchronized -- synchronized I/O for threads
      • setIOUnSynchronized -- unsynchronized I/O for threads
      • allowableThreads -- the current maximum number of simultaneously running tasks
      • threadVariable -- create a symbol whose value in one thread is not shared with others
      • maxAllowableThreads -- the maximum possible number of simultaneously running tasks
      • Task -- the class of all tasks
    • system facilities

      • autoload -- arrange for a function to be loaded automatically
      • initialization file
      • input -- read Macaulay2 commands and echo
      • load -- read Macaulay2 commands
      • needs -- read Macaulay2 commands if necessary
      • end -- stop loading a file
      • clearEcho -- turn off echoing
      • setEcho -- turn on echoing
      • top level loop

        • capture -- evaluate Macaulay2 code and capture the output
      • restart -- restart Macaulay2
      • addStartFunction -- add a startup function
      • addEndFunction -- add an ending function
      • alarm -- set an alarm
      • currentDirectory -- current working directory
      • exec -- execute another program
      • exit -- exit the program
      • fork -- fork the process
      • getenv -- get value of environment variable
      • processID -- the process identifier
      • path -- list of directories to look in
      • quit -- quit the program
      • run -- run an external command
      • sleep -- sleep for a while
      • nanosleep -- sleep for a given number of nanoseconds
      • Time -- the class of all timing results
      • time -- time a computation
      • timing -- time a computation
      • elapsedTime -- time a computation using time elapsed
      • elapsedTiming -- time a computation using time elapsed
      • wait -- wait for child process
      • commandLine -- the command line arguments
      • environment -- the environment variables
      • version -- information about this version of the program
      • collectGarbage -- collect the garbage in memory
      • registerFinalizer -- register a string that will be displayed when an object is garbage collected
    • debugging
  • packages provided with Macaulay2
  • authors of Macaulay2 packages
  • changes to Macaulay2, by version

    • changes made for the next release
    • changes, 1.22
    • changes, 1.21
    • changes, 1.20
    • changes, 1.19.1
    • changes, 1.19
    • changes, 1.18
    • changes, 1.17
    • changes, 1.16
    • changes, 1.15
    • changes, 1.14
    • changes, 1.13
    • changes, 1.12
    • changes, 1.11
    • changes, 1.10
    • changes, 1.9.2
    • changes, 1.9.1
    • changes, 1.9
    • changes, 1.8.2
    • changes, 1.8.1
    • changes, 1.8
    • changes, 1.7
    • changes, 1.6
    • changes, 1.5
    • changes, 1.4
    • changes, 1.3.1
    • changes, 1.3
    • changes, 1.2
    • changes, 1.0 and 1.1
    • list of obsolete functions
  • mathematical examples

    • Tutorial: Elementary uses of Groebner bases
    • Tutorial: Canonical Embeddings of Plane Curves and Gonality
    • Tutorial: Fano varieties
    • Tutorial: Divisors
  • basic commutative algebra

    • Elementary uses of Groebner bases I. Math 634 Fall 2005
    • modules in Macaulay2
    • M2SingularBook -- Macaulay2 examples for the Singular book

      • Singular Book 1.1.8 -- computation in fields
      • Singular Book 1.1.9 -- computation in polynomial rings
      • Singular Book 1.1.10 -- methods for creating ring maps
      • Singular Book 1.2.3 -- leading data
      • Singular Book 1.2.13 -- monomial orderings
      • Singular Book 1.3.3 -- properties of ring maps
      • Singular Book 1.3.13 -- computation in quotient rings
      • Singular Book 1.3.15 -- computing with radicals
      • Singular Book 1.4.9 -- global versus local rings
      • Singular Book 1.5.10 -- realization of rings
      • Singular Book 1.6.13 -- normal form
      • Singular Book 1.7.10 -- standard bases
      • Singular Book 1.7.12 -- highest corner
      • Singular Book 1.8.1 -- ideal membership
      • Singular Book 1.8.2 -- linear combination of ideal members
      • Singular Book 1.8.4 -- elimination of variables
      • Singular Book 1.8.6 -- Zariski closure of the image
      • Singular Book 1.8.7 -- solving equations
      • Singular Book 1.8.9 -- radical membership
      • Singular Book 1.8.11 -- intersection of ideals
      • Singular Book 1.8.13 -- quotient of ideals
      • Singular Book 1.8.15 -- saturation
      • Singular Book 1.8.18 -- kernel of a ring map
      • Singular Book 1.8.19 -- algebraic dependence
      • Singular Book 1.8.20 -- subalgebra membership
      • Singular Book 2.1.6 -- matrix operations
      • Singular Book 2.1.7 -- maps induced by Hom
      • Singular Book 2.1.10 -- Submodules of A^n
      • Singular Book 2.1.13 -- kernel, image and cokernel of a module homomorphism
      • Singular Book 2.1.20 -- sum, intersection, module quotient
      • Singular Book 2.1.24 -- submodules, presentation of a module
      • Singular Book 2.1.26 -- computation of Hom
      • Singular Book 2.1.34 -- minimal presentations, prune
      • Singular Book 2.2.15 -- graded rings and modules
      • Singular Book 2.3.10 -- normal form
      • Singular Book 2.3.12 -- standard bases
      • Singular Book 2.4.12 -- resolution and Betti numbers
      • Singular Book 2.4.15 -- homogeneous resolution and graded Betti numbers
      • Singular Book 2.5.5 -- syzygies
      • Singular Book 2.5.18 -- Schreyer resolution
      • Singular Book 2.6.3 -- diagonal form
      • Singular Book 2.6.11 -- cyclic decomposition
      • Singular Book 2.6.15 -- Jordan normal form
      • Singular Book 2.7.5 -- tensor product of maps
      • Singular Book 2.7.9 -- tensor product of modules
      • Singular Book 2.7.14 -- tensor product of rings
      • Singular Book 2.8.1 -- module membership
      • Singular Book 2.8.3 -- elimination of module components
      • Singular Book 2.8.5 -- intersection of modules
      • Singular Book 2.8.6 -- quotient of modules
      • Singular Book 2.8.7 -- radical, zerodivisors of modules
      • Singular Book 2.8.8 -- annihilator and Fitting ideal
      • Singular Book 2.8.9 -- kernel of a module homomorphism
      • Singular Book 2.8.10 -- solving linear equations
  • replacements for commands and scripts from Macaulay
  • how Macaulay2 finds its files

    • currentLayout -- relative locations of Macaulay2 files
    • prefixDirectory -- the prefix directory
  • Type -- the class of all types
  • Function -- the class of all functions
• # BasicList -- length or cardinality
• - List -- negation of a vector
• - Vector -- negation of a Vector
• - VirtualTally -- negation of a VirtualTally
• -- -- comment
• > ZZ -- greater than
• >= ZZ -- greater than or equal
• ^* -- a unary postfix operator, used for indicating pullback maps
• _* -- a unary postfix operator, used for indicating pushforward maps
• abs -- absolute value function
• accumulate -- apply a binary operator repeatedly
• acos -- arccosine
• acosh -- inverse hyperbolic cosine
• acot -- compute the inverse cotangent
• acoth -- compute the hyperbolic arctangent of a number
• AdditionalPaths -- list of non-standard paths to search for a program
• Adjacent -- the class of all adjacent expression pairs
• adjoint'(Matrix,Module,Module) -- an adjoint map
• adjoint(Matrix,Module,Module) -- an adjoint map
• AffineVariety -- the class of all affine varieties
• AffineVariety ** Ring -- a binary operator, usually used for tensor product or Cartesian product
• AfterEval -- top level method applied after evaluation
• AfterNoPrint -- top level method applied after not printing
• AfterPrint -- top level method applied after printing
• agm -- arithmetic-geometric mean
• all -- whether all elements satisfy a specified condition
• all(BasicList,BasicList,Function) -- whether all corresponding elements of a pair of lists satisfy a condition
• all(BasicList,Function) -- whether all elements of a list satisfy a specified condition
• all(HashTable,Function) -- whether all key/value pairs in a hash table satisfy a specified condition
• ambient -- ambient free module of a subquotient, or ambient ring
• ambient(GaloisField) -- corresponding quotient ring
• ambient(Matrix)
• ambient(Module) -- ambient free module
• ambient(Ring) -- ambient polynomial ring
• ancestors -- the ancestors of something
• AngleBarList -- the class of lists delimited by <| ... |>
• antipode -- antipode for skew commuting polynomial rings
• any -- whether any elements satisfy a specified condition
• any(BasicList,BasicList,Function) -- whether any corresponding elements of a pair of lists satisfy a condition
• any(BasicList,Function) -- whether any elements of a list satisfy a specified condition
• any(HashTable,Function) -- whether all key/value pairs in a hash table satisfy a specified condition
• append -- add an element to the end of a list
• applicationDirectory -- the path to the user's application directory
• applicationDirectorySuffix -- suffix that determines the user's application directory
• apply -- apply a function to each element
• apply(BasicList,BasicList,Function) -- apply a function to pairs of elements, one from each list
• apply(BasicList,Function) -- apply a function to each element of a list
• apply(Thing,Function) -- apply a function to an object with an iterator
• apply(ZZ,Function) -- apply a function to {0,..., n-1}
• applyTable -- apply a function to each element of a table
• argument -- specify the function in a scripted functor for an argument
• Ascending -- specify ascending order
• asin -- arcsine
• asinh -- inverse hyperbolic sine
• assert -- check whether something is true, raise an error if not
• AssociativeExpression -- the class of associative expressions
• atan -- compute the arctangent of a number
• atan2 -- compute an angle of a certain triangle
• atanh -- compute the hyperbolic arctangent of a number
• atEndOfFile(File) -- test for end of file
• backtrace -- whether a backtrace is displayed following an error message
• Bag -- the class of all bags
• Bareiss
• baseFilename -- the base part of a filename or path
• baseName -- the base name of a generator
• baseRing -- produce the ring from which a ring was formed
• baseRings -- store the list of base rings of a ring
• BasicList -- the class of all basic lists
• BasicList # ZZ -- get value from list, hash table, database, or string
• BeforePrint -- top level method applied before printing results
• benchmark -- accurate timing of execution
• BesselJ -- Bessel function of the first kind
• BesselY -- Bessel function of the second kind
• Beta -- Beta function
• between -- insert something between elements of a list
• between(Thing,VisibleList) -- insert something between elements of a list
• BinaryOperation -- the class of all binary operation expressions
• binomial -- binomial coefficient
• blockMatrixForm -- global flag for block printing
• Boolean -- the class of Boolean values
• borel(Matrix) -- make a Borel fixed submodule
• break -- break from a loop
• cache -- a key under which to store cache tables
• CacheFunction -- the class of cache functions
• CacheTable -- hash tables for caching
• cacheValue -- cache values of functions in their arguments
• CatalanConstant -- Catalan's constant
• CC -- the class of all complex numbers
• CC' -- the parent class of all rings of complex numbers
• ceiling -- ceiling function
• ceiling(Number) -- ceiling function
• centerString -- center a string or net
• ChainComplex -- the class of all chain complexes
• chainComplex -- make a chain complex
• ChainComplex ** ChainComplex -- tensor product
• ChainComplex ** ChainComplexMap -- tensor product
• ChainComplex ** GradedModule -- tensor product
• ChainComplex ** Ring -- a binary operator, usually used for tensor product or Cartesian product
• ChainComplex ++ ChainComplex -- direct sum
• ChainComplex ^ ZZ -- access member, cohomological degree
• ChainComplex _ ZZ -- component
• ChainComplex _ ZZ = Thing -- install component of chain complex
• ChainComplex Array -- degree shift
• chainComplex(GradedModule) -- make a chain complex from a graded module
• chainComplex(List) -- make a chain complex
• chainComplex(Matrix) -- make a small chain complex
• chainComplex(Ring) -- make an empty chain complex over a ring
• ChainComplexMap -- the class of all maps between chain complexes
• ChainComplexMap ** ChainComplex -- tensor product
• ChainComplexMap ** ChainComplexMap -- tensor product
• ChainComplexMap ^ ZZ -- iterated composition
• ChainComplexMap _ ZZ -- component map
• ChainComplexMap _ ZZ = Thing -- install component of chain complex map
• ChainComplexMap Array -- degree shift
• changeBase -- change integer base
• char -- computes the characteristic of the ring or field
• characters -- get characters from a string
• chi
• clean -- Set to zero elements that are approximately zero
• clearAll -- forget everything
• clearOutput -- forget output values
• close -- close a file
• closeIn -- close an input file
• closeOut -- close an output file
• codim -- compute the codimension
• codim(CoherentSheaf) -- codimension of the support of a coherent sheaf on a projective variety
• codim(Ideal) -- compute the codimension
• codim(Ideal,Generic=>...)
• codim(MonomialIdeal) -- compute the codimension
• codim(ProjectiveVariety) -- codimension of the projective variety
• codim(QuotientRing) -- compute the codimension
• coefficient -- coefficient of a monomial
• coefficientRing -- get the coefficient ring
• coefficients -- monomials and their coefficients
• Cofactor
• CoherentSheaf -- the class of all coherent sheaves
• CoherentSheaf (*) -- sum of twists
• CoherentSheaf ** CoherentSheaf -- tensor produce of coherent sheaves
• CoherentSheaf ++ CoherentSheaf -- direct sum of coherent sheaves
• CoherentSheaf / CoherentSheaf -- quotient of coherent sheaves
• CoherentSheaf ^ ZZ -- direct sum
• CoherentSheaf ^** ZZ -- tensor power
• CoherentSheaf ZZ -- canonical twist of a coherent sheaf
• cohomology -- general cohomology functor
• coimage -- coimage of a map
• cokernel -- cokernel of a map of modules, graded modules, or chaincomplexes
• columnAdd -- add a multiple of one column to another
• columnate -- arrange strings in columns
• columnMult -- multiply a column by a ring element
• columnPermute -- permute some columns
• columnRankProfile -- find the column rank profile of a mutable matrix
• columnSwap -- interchange columns
• Command -- the class of all commands
• commandInterpreter -- the top level command interpreter
• commonest -- the most common elements of a list or tally
• commonRing -- find smallest containing ring
• comodule -- submodule to quotient module
• compactMatrixForm -- global flag for compact printing
• CompiledFunction -- the class of all compiled functions
• CompiledFunctionBody -- the class of all compiled function bodies
• CompiledFunctionClosure -- the class of all compiled function closures
• Complement -- a Strategy option value
• complement(Matrix) -- find the minimal generators for cokernel of a matrix (low level form)
• complete
• complete(ChainComplex) -- complete the internal parts
• ComplexField -- the class of all complex fields
• components -- list the components of a direct sum
• components(ChainComplex) -- list the components of a direct sum
• compose(Module,Module,Module) -- composition as a pairing on Hom-modules
• compositions -- list the compositions of an integer
• compress -- extract nonzero columns from a matrix
• computing syzygies
• cone -- mapping cone of a chain map
• cone(ChainComplexMap) -- mapping cone of a chain map
• conjugate -- complex conjugate
• conjugate(CC) -- complex conjugate
• conjugate(Partition) -- conjugate a partition
• connectionCount -- the number of connections
• Constant
• constructing maps between modules
• content -- the content of a polynomial
• continue -- continue with the next iteration of a loop
• contract -- contract one matrix by another
• contract'(Matrix,Matrix) -- contract a matrix by a matrix, the dual notion
• contract(Matrix,Matrix) -- contract a matrix by a matrix
• COPYING-GPL-2 -- GNU General Public License, version 2
• COPYING-GPL-3 -- GNU General Public License, version 3
• copyright -- a string containing the copyright notice for Macaulay2
• Core -- the core part of Macaulay2
• cos -- compute the cosine
• cosh -- compute the hyperbolic cosine
• cot -- cotangent
• cotangentSheaf -- cotangent sheaf of a projective variety
• cotangentSheaf(ProjectiveVariety) -- cotangent sheaf of a projective variety
• cotangentSheaf(ZZ,ProjectiveVariety) -- exterior powers of the cotangent sheaf of a projective variety
• coth -- hyperbolic cotangent
• cover -- get the covering free module
• cover(Matrix) -- get the covering free module
• cover(Module) -- get the covering free module
• coverMap(Module) -- the surjective map from a free module to a module corresponding to the generators
• cpuTime -- seconds of cpu time used since Macaulay2 began
• csc -- cosecant
• csch -- hyperbolic cosecant
• current -- the pseudocode that produced an error
• currentColumnNumber -- current column number of the current input file
• currentFileDirectory -- the directory containing the current input file
• currentFileName -- the current source file
• currentPosition -- the current position in the current source file
• currentRowNumber -- current row number of the current input file
• currentTime -- get the current time
• Database -- the class of all database files
• dd -- differential in a chain complex
• debugError -- a function to debug
• debuggingMode -- whether to enter the debugger when an error occurs
• debugLevel -- current level debugging
• decompose
• deepSplice -- remove all levels of subsequences from a sequence or list
• default -- default member of a family
• defaultPrecision -- default precision of numbers
• Degree -- specify the degree of a map
• degree
• degree(Ideal)
• degree(ProjectiveHilbertPolynomial)
• degree(ProjectiveVariety)
• degree(Ring)
• degree(RingElement)
• degree(RingElement,RingElement) -- degree with respect to a variable
• degreeLength -- the length of the degree vector
• DegreeOrder -- sort primarily by degree
• degrees -- degrees of generators
• degrees(Matrix) -- degrees of target and source
• degrees(Ring) -- degrees of generators
• degreesRing -- the ring or monoid of degrees

  • degreesRing(List) -- the ring or monoid of degrees
• delete -- delete some elements of a list
• demark -- insert a string between elements of a list of strings
• denominator -- denominator of a fraction
• Descending -- specify descending order
• Descent
• Describe -- the class of the output of describe
• describe -- real description
• determinant -- determinant of a matrix
• determinant(...,Strategy=>...) -- choose between Bareiss, Cofactor and Dynamic algorithms
• Developer's Corner

  • engine
• diagonalMatrix -- make a diagonal matrix
• diagonalMatrix(Matrix) -- make a diagonal matrix from entries of a matrix
• diagonalMatrix(Ring,ZZ,ZZ,List) -- make a diagonal matrix from a list
• diameter -- diameter of an interval
• Dictionary -- the class of all dictionaries
• dictionary -- determine the dictionary to which a symbol belongs
• dictionaryPath
• diff -- differentiate or take difference

  • diff(RingElement,RingElement) -- differentiation
  • diff(RingElement,Matrix) -- differentiate each entry of a matrix
  • diff(Matrix,RingElement) -- differentiation
  • diff(Matrix,Matrix) -- differentiate a matrix by a matrix
  • diff(ProjectiveHilbertPolynomial) -- differentiate or take difference
  • diff(ProjectiveHilbertPolynomial,ZZ) -- differentiate or take difference
• diff and contract
• diff'(Matrix,Matrix) -- differentiate a matrix by a matrix, the dual notion
• difference -- difference
• Digamma -- Digamma function
• dim -- compute the Krull dimension
• dim(AffineVariety) -- dimension of the affine variety
• dim(Ideal) -- compute the Krull dimension
• dim(Module) -- compute the Krull dimension
• dim(ProjectiveHilbertPolynomial) -- the degree of the Hilbert polynomial
• dim(ProjectiveVariety) -- dimension of the projective variety
• dim(Ring) -- compute the Krull dimension
• directSum -- direct sum of modules or maps
• disassemble -- disassemble pseudocode or a function
• Divide -- the class of all divide expressions
• divideByVariable -- divide all columns by a (power of a) variable
• division in polynomial rings with monomials less than 1
• do -- loop control
• drop -- drop some elements from a list or sequence.
• dual -- dual module or map
• dual(ChainComplex) -- dual
• dual(ChainComplexMap) -- dual of a chain complex
• dual(CoherentSheaf) -- dual coherent sheaf
• dual(Matrix) -- dual of a map
• dual(Module) -- dual module
• dual(MonomialIdeal) -- the Alexander dual of a monomial ideal
• dual(MonomialIdeal,List) -- the Alexander dual
• dual(MonomialIdeal,RingElement) -- the Alexander dual
• dual(MonomialIdeal,Strategy=>...)
• Dynamic
• eagonNorthcott(Matrix) -- Eagon-Northcott complex of a matrix of linear forms
• echoOff -- turn off echoing
• echoOn -- turn on echoing
• edit -- edit source code
• eigenvalues -- find eigenvalues of a matrix
• eigenvalues(...,Hermitian=>...) -- Hermitian=>true means assume the matrix is symmetric or Hermitian
• eigenvectors -- find eigenvectors of a matrix over RR or CC
• eigenvectors(...,Hermitian=>...) -- Hermitian=>true means assume the matrix is symmetric or Hermitian
• eint -- exponential integral
• elements -- list of elements
• else -- condition testing
• End -- module of endomorphisms
• endl -- end an output line
• Engine -- specify whether a ring is handled by the engine
• engineDebugLevel -- current engine debugging level
• EngineRing -- the class of rings handled by the engine
• entries -- lists the entries of a matrix
• entries(Vector) -- lists the entries of a vector
• equality and containment of modules
• Equation -- the class of all equation expressions
• erase -- remove a global symbol
• erf -- error function
• erfc -- complementary error function
• errorDepth -- set the error printing depth
• euler -- Euler characteristic
• euler(CoherentSheaf) -- Euler characteristic of coherent sheaf
• euler(Ideal) -- Euler characteristic
• euler(Module) -- Euler characteristic
• euler(ProjectiveHilbertPolynomial) -- constant term of the Hilbert polynomial
• euler(ProjectiveVariety) -- topological Euler characteristic of a (smooth) projective variety
• euler(Ring) -- Euler characteristic
• EulerConstant -- the Euler-Mascheroni constant
• eulers -- list the sectional Euler characteristics
• eulers(CoherentSheaf) -- list the sectional Euler characteristics
• eulers(Ideal) -- list the sectional Euler characteristics
• eulers(Ring) -- list the sectional Euler characteristics
• even -- tell whether an integer is even
• exp -- exponential function
• exp(RingElement) -- exponential function
• expm1 -- exponential minus 1
• exponents -- the exponents of a polynomial
• Expression -- the class of all expressions
• expression -- convert to an expression
• Ext -- compute an Ext module
• Ext(Module,Module) -- total Ext module
• Ext^ZZ(CoherentSheaf,CoherentSheaf) -- global Ext
• Ext^ZZ(CoherentSheaf,SumOfTwists) -- global Ext
• Ext^ZZ(Matrix,Module) -- map between Ext modules
• Ext^ZZ(Module,Matrix) -- map between Ext modules
• Ext^ZZ(Module,Module) -- Ext module
• extend -- extend a module map to a chain map, if possible
• extend(ChainComplex,ChainComplex,Matrix) -- extend a module map to a chain map, if possible
• exteriorPower -- exterior power
• exteriorPower(...,Strategy=>...) -- choose between Bareiss, Cofactor and Dynamic algorithms
• exteriorPower(ZZ,CoherentSheaf) -- exterior power
• exteriorPower(ZZ,Matrix) -- exterior power of a matrix
• exteriorPower(ZZ,Module) -- exterior power of a module
• extracting elements
• factor -- factor a ring element or a ZZ-module
• factor(Module) -- factor a ZZ-module
• factor(RingElement) -- factor a ring element
• false
• Fano -- Fano scheme
• Fano(ZZ,Ideal) -- Fano scheme
• Fano(ZZ,Ideal,Ring) -- Fano scheme
• FastNonminimal -- compute a non-minimal graded free resolution
• File -- the class of all files
• File << Thing -- print to a file
• fileDictionaries -- local dictionaries for loaded files
• fileExecutable
• fileExitHooks -- a list of hooks (functions) to execute when the current file has been loaded
• fileLength -- the length of a file
• fileMode -- set or get file mode
• fileMode(File) -- get file mode
• fileMode(String) -- get file mode
• fileMode(ZZ,File) -- set file mode
• fileMode(ZZ,String) -- set file mode
• FilePosition -- the class of all file positions
• fileReadable
• fileWritable
• fillMatrix -- fill a mutable matrix with random numbers
• finding your files under cygwin
• FindOne -- find a primitive element
• findProgram -- load external program
• findSynonyms(Symbol) -- find synonyms of symbols
• fine control of a Groebner basis computation
• first -- first element of a list
• firstkey -- get the first key
• fittingIdeal -- Fitting ideal of a module
• flagLookup -- flag a symbol
• FlatMonoid
• flatten -- flatten a nested list or a matrix
• flattenRing -- write a ring as a (quotient of a) polynomial ring
• flattenRing(...,CoefficientRing=>...) -- optionally specify the desired coefficient ring of the flattened ring
• flattenRing(...,Result=>...) -- optionally specify which output(s) to return (see Description for details)
• flip(Module,Module) -- matrix of commutativity of tensor product
• floor -- floor function
• floor(Number) -- floor function
• flush -- flush output to file
• fold -- apply a binary operator repeatedly
• forceGB -- declare that the columns of a matrix are a Gröbner basis
• forceGB(...,ChangeMatrix=>...) -- inform Macaulay2 about the change of basis matrix from GB to generators
• forceGB(...,MinimalMatrix=>...) -- specify the minimal generator matrix
• forceGB(...,SyzygyMatrix=>...) -- specify the syzygy matrix
• formation -- recover the methods used to make a module
• frac -- construct a fraction field
• fraction
• FractionField -- the class of all fraction fields
• frames -- get the frames associated to a closure
• from -- loop control
• Function @@ Function -- composition of functions
• Function _ Thing -- attach the first argument to a function of two or more arguments
• Function Thing -- function application
• FunctionApplication -- the class of all function application expressions
• FunctionBody -- the class of function bodies
• functionBody -- get the body of a function
• FunctionClosure -- the class of all function closures
• GaloisField -- the class of all Galois fields
• Gamma -- Gamma function
• gb -- compute a Gröbner basis
• gbRemove -- remove Gröbner basis
• gbSnapshot -- the Gröbner basis matrix as so far computed
• gbTrace -- provide tracing output during various computations in the engine.
• gcd -- greatest common divisor
• gcdCoefficients -- gcd with coefficients
• GCstats -- information about the status of the garbage collector
• genera -- list of the successive linear sectional arithmetic genera
• genera(Ideal) -- list of the successive linear sectional arithmetic genera
• genera(ProjectiveVariety) -- list of the successive linear sectional arithmetic genera
• genera(Ring) -- list of the successive linear sectional arithmetic genera
• GeneralOrderedMonoid -- the class of all ordered free commutative monoids
• generateAssertions -- generate assert statements from experimental input
• generator -- provide a single generator
• generators -- provide matrix or list of generators
• generators of ideals and modules
• generators(GroebnerBasis) -- the generator matrix of a Gröbner basis
• generators(Ideal) -- the generator matrix of an ideal
• generators(Module) -- the generator matrix of a module
• generators(Monoid) -- list of generators
• generators(Ring) -- the list of generators of a ring
• genericMatrix -- make a generic matrix of variables
• genericSkewMatrix -- make a generic skew symmetric matrix of variables
• genericSymmetricMatrix -- make a generic symmetric matrix
• genus -- arithmetic genus
• genus(CoherentSheaf) -- arithmetic genus
• genus(ProjectiveVariety) -- arithmetic genus
• genus(Ring) -- arithmetic genus
• get -- get the contents of a file
• getc -- get a byte
• getChangeMatrix -- get the change of basis matrix
• getGlobalSymbol -- create a global symbol in a global dictionary
• getNetFile -- get the sequence of completed lines (nets) from a net file
• getNonUnit -- retrieve a previously discovered non-unit
• getPrimeWithRootOfUnity -- find a prime p with a primitive n-th root of unity r in ZZ/p
• getSymbol -- make a global user symbol from a string
• getWWW -- get a web page
• GF -- make a finite field
• global -- get a global symbol
• globalAssign -- global assignment, in function form
• globalAssignFunction -- the standard method for the global assignment hook
• GlobalAssignHook -- hook for assignment to global variables
• globalAssignment -- install standard global assignment method
• globalAssignmentHooks -- assignment hooks for global symbols
• GlobalDictionary -- the class of all global dictionaries
• globalReleaseFunction -- the standard method for the global variable release hook
• GlobalReleaseHook
• graded modules
• GradedModule -- the class of all graded modules
• gradedModule -- make a graded module
• GradedModule ** ChainComplex -- tensor product
• GradedModule ** GradedModule -- a binary operator, usually used for tensor product or Cartesian product
• GradedModule ** Module -- a binary operator, usually used for tensor product or Cartesian product
• GradedModule Array -- degree shift
• GradedModuleMap -- the class of all maps between graded modules
• gradedModuleMap -- make a map of graded modules
• GradedModuleMap | GradedModuleMap -- a binary operator, often used for horizontal concatenation
• GradedModuleMap || GradedModuleMap -- a binary operator, often used for vertical concatenation
• Grassmannian(ZZ,ZZ) -- the Grassmannian of linear subspaces of a vector space
• GroebnerBasis -- the class of all Gröbner bases
• groebnerBasis -- Gröbner basis, as a matrix
• GroebnerBasisOptions
• groupID -- the process group identifier
• handleInterrupts
• handling hypertext
• HardDegreeLimit -- compute only up to this degree
• hash -- hash code of an object
• HeaderType -- a class of lists with abbreviated constructors
• height(File) -- get window height
• HH -- general homology and cohomology functor
• hh -- Hodge numbers of a smooth projective variety
• HH ChainComplex -- homology of a chain complex
• HH ChainComplexMap -- homology of a chain complex map
• HH^ZZ ChainComplex -- cohomology of a chain complex
• HH^ZZ ChainComplexMap -- cohomology of a chain complex map
• HH^ZZ Module -- local cohomology of a module
• HH^ZZ SheafOfRings -- cohomology of a sheaf of rings on a projective variety
• HH^ZZ SumOfTwists -- coherent sheaf cohomology module
• HH^ZZ(ProjectiveVariety,CoherentSheaf) -- cohomology of a coherent sheaf on a projective variety
• HH_ZZ ChainComplex -- homology of a chain complex
• HH_ZZ ChainComplexMap -- homology of a chain complex map
• hilbertFunction -- the Hilbert function
• hilbertPolynomial -- compute the Hilbert polynomial
• hilbertPolynomial(...,Projective=>...) -- choose how to display the Hilbert polynomial
• hilbertPolynomial(CoherentSheaf) -- compute the Hilbert polynomial of the coherent sheaf
• hilbertPolynomial(Ideal) -- compute the Hilbert polynomial of the quotient of the ambient ring by the ideal
• hilbertPolynomial(Module) -- compute the Hilbert polynomial of the module
• hilbertPolynomial(ProjectiveVariety) -- compute the Hilbert polynomial of the projective variety
• hilbertPolynomial(Ring) -- compute the Hilbert polynomial of the ring
• hilbertSeries -- compute the Hilbert series
• hilbertSeries(...,Order=>...) -- display the truncated power series expansion
• hilbertSeries(...,Reduce=>...) -- reduce the Hilbert series
• hilbertSeries(Ideal) -- compute the Hilbert series of the quotient of the ambient ring by the ideal
• hilbertSeries(PolynomialRing) -- compute the Hilbert series of a ring
• hilbertSeries(ProjectiveHilbertPolynomial) -- compute the Hilbert series of a projective Hilbert polynomial
• hilbertSeries(ProjectiveVariety) -- compute the Hilbert series of a projective variety
• hold -- hold something in a holder expression
• Holder -- the class of all holder expressions
• Hom -- module of homomorphisms
• Hom(ChainComplexMap,Module) -- induced map on Hom
• Hom(CoherentSheaf,CoherentSheaf) -- global Hom
• Hom(Matrix,Matrix) -- induced map on Hom
• Hom(Matrix,Module) -- induced map on Hom
• Hom(Module,ChainComplex)
• Hom(Module,ChainComplexMap) -- induced map on Hom
• Hom(Module,Matrix) -- induced map on Hom
• Hom(Module,Module) -- module of homomorphisms
• homeDirectory -- the home directory of the user
• Homogeneous -- a Strategy option value
• homogenize -- homogenize with respect to a variable
• homology -- general homology functor
• homology(Matrix,Matrix) -- homology of a pair of maps
• homomorphism -- get the homomorphism from element of Hom
• homomorphism' -- get the element of Hom from a homomorphism
• homomorphisms (maps) between modules -- including elements of modules
• html -- convert to HTML format
• httpHeaders(String) -- prepend http headers to a string
• hypertext -- prepare hypertext for display
• id -- identity map
• Ideal -- the class of all ideals
• ideal -- make an ideal
• Ideal * Ideal -- product of ideals
• Ideal + Ideal -- sum of ideals
• Ideal / Function -- apply a function to generators of an ideal
• Ideal / Ideal -- quotient module
• Ideal ^ Array -- bracket power of an ideal
• Ideal ^ ZZ -- power of an ideal
• Ideal _* -- get the list of generators of an ideal
• ideal(List) -- make an ideal
• ideal(Matrix) -- make an ideal
• ideal(Module) -- converts a module to an ideal
• ideal(MonomialIdeal) -- converts a monomial ideal to an ideal
• ideal(Ring) -- returns the defining ideal
• ideal(RingElement) -- make an ideal
• ideal(Variety) -- returns the defining ideal
• identity -- the identity function
• if -- condition testing
• ii -- the square root of -1
• image -- image of a map
• imaginaryPart -- imaginary part
• ImmutableType -- the class of immutable types
• in -- a keyword used in for-loops
• incomparable -- a result indicating incomparability
• independentSets -- some size-maximal independent subsets of variables modulo an ideal
• indeterminate -- an indeterminate number
• IndeterminateNumber -- the class of all indeterminate numbers
• index -- numeric index of a ring variable
• indexComponents -- specify keys for components of a direct sum
• IndexedVariable -- the class of all indexed variables
• IndexedVariable .. IndexedVariable -- sequences of consecutive indexed variables
• IndexedVariable ..< IndexedVariable -- sequences of consecutive indexed variables
• IndexedVariable _ Ring -- get a ring variable by name
• IndexedVariableTable
• indices -- indices of a polynomial; also components for a direct sum
• indices(RingElement) -- indices of variables occurring in a polynomial
• inducedMap -- compute an induced map
• inducedMap(...,Degree=>...) -- specify the degree of a map
• inducedMap(...,Verify=>...) -- verify that a map is well-defined
• inducedMap(Module,Module) -- compute the map induced by the identity
• inducedMap(Module,Module,Matrix) -- compute the induced map
• inducesWellDefinedMap -- whether a map is well defined
• InexactField -- the class of inexact fields
• InexactFieldFamily -- the class of all families of inexact fields
• InexactNumber
• InexactNumber'
• InfiniteNumber -- the class of all infinite numbers
• infinity -- infinity
• info -- convert hypertext to info format
• information about a map of modules
• Inhomogeneous -- a Strategy option value
• initial help
• insert -- copy a list, inserting an element
• installAssignmentMethod -- install methods assignment to the value of an operator
• installMethod -- install methods
• instances(Type)
• integrate -- numerical integration
• interpreterDepth -- nesting depth of the interpreter
• intersect -- compute an intersection
• intersect(Ideal,Ideal) -- compute an intersection of a sequence of ideals or modules
• intersect(RRi) -- computes intersection of input intervals
• intersection -- compute an intersection
• intersection(RRi,RRi) -- computes intersection of input intervals
• interval -- construct an interval
• inverse -- compute the inverse
• inverse(Matrix) -- compute the inverse
• inverseErf -- inverse error function
• InverseMethod -- compute reciprocals
• inversePermutation -- inverse permutation
• inverseRegularizedBeta -- inverse of the regularized beta function
• inverseRegularizedGamma -- inverse of the upper regularized gamma function
• Invoking the program
• irreducibleCharacteristicSeries -- irreducible characteristic series of an ideal
• isAffineRing -- whether something is an affine ring
• isANumber -- whether a number is not not-a-number
• isBorel -- whether an ideal is fixed by upper triangular changes of coordinates
• isCommutative -- whether a ring is commutative
• isConstant -- whether a ring element is constant
• isDirectSum -- whether something is a direct sum
• isEmpty
• isEmpty(RRi) -- empty test for an interval
• isField -- whether something is a field
• isFinite -- whether a number is finite
• isFinitePrimeField -- whether a ring is a finite prime field
• isFreeModule -- whether something is a free module
• isGlobalSymbol -- whether a global symbol with a given name exists
• isIdeal -- whether something is an ideal
• isInfinite -- whether a number is infinite
• isInjective -- whether a map is injective
• isInputFile(File) -- whether a file is open for input
• isIsomorphism -- whether a map is an isomorphism
• isListener(File) -- whether a file is open for listening
• isMember -- test membership in a list or set
• isMember(QQ,RRi) -- membership test in an interval
• isModule -- whether something is a module
• isMonomialIdeal -- whether something is a monomial ideal
• isOpen -- whether a file or database is open
• isOutputFile(File) -- whether a file is open for output
• isPolynomialRing -- whether something is a polynomial ring
• isPrime -- whether a integer or polynomial is prime
• isPrimitive -- whether an element is a primitive element of a finite field
• isPseudoprime(ZZ) -- whether an integer is probably prime
• isQuotientModule -- whether something is evidently a quotient of a free module
• isQuotientOf -- whether one thing is a quotient of another
• isQuotientOf(Ring,Ring) -- whether one ring is a quotient of another
• isQuotientOf(Type,Ring) -- whether one ring is a quotient of a ring of a given type
• isQuotientRing -- whether something is a quotient ring
• isReady(File) -- whether a file has data available for reading
• isReal -- whether a number is real
• isRing -- whether something is a ring
• isSkewCommutative -- whether a ring has skew commuting variables
• isSorted -- whether a list is sorted
• isSquareFree -- whether something is square free monomial ideal
• isSubmodule -- whether a module is evidently a submodule of a free module
• isSubquotient(Module,Module) -- check whether a module is a subquotient of another
• isSubset -- whether one object is a subset of another
• isSubset(Ideal,Ideal) -- whether one object is a subset of another
• isSubset(Module,Module) -- whether one object is a subset of another
• isSubset(RRi,RRi) -- subset test for intervals
• isSubset(Set,Set) -- whether one object is a subset of another
• isSurjective -- whether a map is surjective
• isTable -- whether something is a list of lists of equal length
• isUnit -- whether a ring element is a unit
• isWellDefined -- whether a map is well defined
• Iterator -- class for iterators
• iterator -- get an iterator
• Jacobian
• jacobian -- the Jacobian matrix of partial derivatives
• jacobian(Ideal) -- the Jacobian matrix of the generators of an ideal
• jacobian(Matrix) -- the matrix of partial derivatives of polynomials in a matrix
• jacobian(Ring) -- the Jacobian matrix of the polynomials defining a quotient ring
• join -- join lists, sequences, and iterable objects
• Jupyter -- top level printing method used in the Jupyter kernel
• KeepFiles -- whether to keep temporary files
• kernel -- kernel of a ringmap, matrix, or chain complex
• kernel(...,SubringLimit=>...)
• kernel(ChainComplexMap) -- kernel of a chain complex map
• kernel(Matrix) -- kernel of a matrix
• kernel(RingMap) -- kernel of a ringmap
• Keyword -- the class of all keywords
• kill -- kill a process
• kill(ZZ) -- kill a process
• koszul -- Koszul complex or specific matrix in the Koszul complex
• koszul(Matrix) -- the Koszul complex
• koszul(ZZ,Matrix) -- a differential in a Koszul complex
• last -- last element of a list
• lcm -- least common multiple
• lcm(MonomialIdeal) -- least common multiple of all minimal generators
• leadCoefficient -- the coefficient of the leading term
• leadComponent -- the leading component(s) of a vector or matrix
• leadMonomial -- the leading monomial of a ring element
• leadTerm -- get the greatest term
• leadTerm(Ideal) -- get the ideal of greatest terms
• leadTerm(Matrix) -- get the greatest term of each column
• leadTerm(RingElement) -- get the greatest term
• leadTerm(ZZ,Ideal) -- get the ideal of lead polynomials
• leadTerm(ZZ,Matrix) -- get the matrix of lead polynomials of each column
• leadTerm(ZZ,RingElement) -- get the lead polynomials using part of the monomial order
• left -- left endpoint of an interval
• left shift
• length -- length of an object
• length(ChainComplex) -- length of a chain complex or graded module
• length(Dictionary) -- length of a dictionary
• length(GradedModule) -- length of a graded module
• length(Module) -- Computes the length of a module
• length(String) -- length of a string
• length(VisibleList) -- length of a visible list
• LengthLimit -- stop when the resolution reaches this length
• lift -- lift to another ring
• liftable -- whether lifting to another ring is possible
• limitFiles
• limitProcesses
• lineNumber -- current line number
• list -- loop control
• List ** List -- Cartesian product of two lists
• List + List -- sum of two vectors
• List - List -- difference of two vectors
• List .. List -- rectangular sequences of consecutive lists and sequences
• List ..< List -- rectangular sequences of consecutive lists and sequences
• List / Thing -- vector division
• List << List -- component-wise comparison of lists
• List | List -- join lists, sequences or arrays
• listForm -- convert to list form
• listLocalSymbols -- display of local symbols and their values
• listSymbols -- compact display of symbols and their values
• listUserSymbols -- display the user's symbols
• lngamma -- logarithm of the Gamma function
• loadDepth -- the load depth
• loadedFiles
• local -- get a local symbol
• localDictionaries -- get local dictionaries
• LocalDictionary -- the class of all local dictionaries
• locate -- locate source code
• log -- logarithm function
• log1p -- logarithm of 1+x
• LongPolynomial -- a Strategy option value
• lookup -- look up methods
• lookupCount -- reference count for a symbol
• LowerBound -- the class of lower bound objects
• LUdecomposition -- LU decomposition
• making modules from matrices
• manipulating modules
• Manipulator -- the class of all file manipulators
• map -- make a map

  • map(Module,Module,Function) -- create a matrix by specifying a function that gives each entry
  • map(Module,Module,List) -- create a matrix by giving a sparse or dense list of entries
  • map(Module,Module,RingElement) -- construct the map induced by multiplication by a ring element on the generators
  • map(Module,Module,Matrix) -- create the matrix induced on generators by a given matrix
  • map(Module,ZZ,Function) -- create a matrix from a free module by specifying a function that gives each entry
  • map(Module,Nothing,List) -- create a matrix by giving a doubly nested list of ring elements
  • map(Module,ZZ,List) -- create a matrix by giving a sparse or dense list of entries
  • map(Module,Nothing,Matrix) -- recast a matrix to have a new target, and a free module as source
  • map(Matrix) -- make a matrix with a different degree
  • map(Module,Module,RingMap,Matrix) -- homomorphism of modules over different rings
  • map(Ring,Ring) -- make a ring map, using the names of the variables
  • map(Ring,Ring,List) -- make a ring map
  • map(Ring,Ring,Matrix) -- make a ring map
  • map(ChainComplex,ChainComplex,Function) -- make a map of chain complexes
• map(Module) -- identity map
• map(Ring,Matrix) -- make a ring map
• MapExpression -- the class of map expressions
• maps between modules
• markedGB(Matrix,Matrix) -- make a marked Gröbner basis
• mathML -- convert to MathML format
• Matrix -- the class of all matrices
• matrix -- make a matrix
• Matrix % GroebnerBasis -- calculate the normal form of ring elements and matrices using a (partially computed) Gröbner basis
• Matrix * Matrix -- matrix multiplication
• Matrix ** Matrix -- tensor product
• Matrix ** Module -- tensor product
• Matrix ** Ring -- tensor product
• Matrix ** RingElement -- a binary operator, usually used for tensor product or Cartesian product
• Matrix ++ Matrix -- direct sum of maps
• Matrix // Matrix -- factor a map through another
• Matrix ^ ZZ -- power
• Matrix _ Sequence -- get entry of matrix
• Matrix | Matrix -- join matrices horizontally
• Matrix || Matrix -- join matrices vertically
• matrix(List) -- create a matrix from a doubly-nested list of ring elements or matrices
• matrix(Matrix) -- the matrix between generators
• matrix(MutableMatrix) -- make a matrix from a mutable one
• matrix(Ring,List) -- create a matrix from a doubly nested list of ring elements or matrices
• matrix(RingElement) -- make a matrix from a ring element
• matrix(RingMap) -- the matrix associated to a ring map
• matrix(Vector) -- make a matrix
• MatrixExpression -- the class of all matrix expressions
• max -- yields the maximum element in a list or sequence
• max(GradedModule) -- yields the maximum element in a list or sequence
• maxExponent
• maxPosition -- position of the largest element
• memoize -- record results of function evaluation for future use
• mergePairs -- merge sorted lists of pairs
• methodOptions(Function) -- recover the options used when a method function was created
• methods for normal forms and remainder -- normal form of ring elements and matrices
• midpoint -- midpoint of an interval
• min -- yields the minimum element in a list or sequence
• min(GradedModule) -- yields the minimum element in a list or sequence
• minExponent
• mingens -- minimal generator matrix
• mingens(GroebnerBasis) -- (partially constructed) minimal generator matrix
• mingens(Module) -- minimal generator matrix
• mingle -- mingle elements of several lists
• minimal presentations and generators
• MinimalGenerators -- whether to compute minimal generators and return a trimmed set of generators
• minimalPresentation -- compute a minimal presentation
• minimalPresentation(Ideal) -- compute a minimal presentation of the quotient ring defined by an ideal
• minimalPresentation(Matrix) -- minimally present source and target of a matrix
• minimalPresentation(Module) -- minimal presentation of a module
• minimalPresentation(Ring) -- compute a minimal presentation of a quotient ring
• minimizeFilename -- minimize a file name
• MinimumVersion -- the minimum required version of a program
• minors(...,Strategy=>...) -- choose between Bareiss, Cofactor and Dynamic algorithms
• minors(ZZ,Matrix) -- ideal generated by minors
• minPosition -- position of the smallest element
• Minus -- the class of all minus expressions
• minus -- additive inverse
• mod -- reduce modulo an integer
• Module -- the class of all modules
• module -- make or get a module
• Module ** Module -- tensor product
• Module ** Ring -- tensor product
• Module + Module -- sum of submodules
• Module ++ Module -- direct sum of modules
• Module / Module -- quotient module
• Module ^ Array -- projection onto summand
• Module ^ List -- projection onto summand
• Module ^ ZZ -- direct sum
• Module ^** ZZ -- tensor power
• Module _ Array -- inclusion from summand
• Module _ List -- map from free module to some generators
• Module Array -- make a chain complex from a module
• module(CoherentSheaf) -- get the module defining a coherent sheaf
• module(Ring) -- make or get a module
• module(SheafOfRings) -- make or get a module
• module(Vector) -- the module of a vector
• modulo(Matrix,Matrix) -- find the pre-image (pullback) of image of a map (low level version)
• Monoid -- the class of all monoids
• monoid(Ring) -- make or retrieve a monoid
• MonoidElement -- the class of all monoid elements
• monomialCurveIdeal -- make the ideal of a monomial curve
• MonomialIdeal -- the class of all monomial ideals handled by the engine
• monomialIdeal -- make a monomial ideal
• MonomialIdeal - MonomialIdeal -- monomial ideal difference
• monomialIdeal(Ideal) -- monomial ideal of lead monomials of a Gröbner basis
• monomialIdeal(Matrix) -- monomial ideal of lead monomials
• Monomials -- whether to display the number of monomial
• monomials -- matrix of monomials in a ring element or matrix
• monomials(...,Variables=>...) -- specify variables
• monomialSubideal -- find the largest monomial ideal in an ideal
• multidegree -- multidegree
• mutable matrices
• mutableIdentity(Ring,ZZ) -- make a mutable identity matrix
• MutableList -- the class of all mutable lists
• MutableMatrix -- the class of all mutable matrices
• mutableMatrix -- make a mutable matrix
• MutableMatrix _ Sequence = Thing -- assignment to an element of a mutable matrix
• mutableMatrix(Ring,ZZ,ZZ) -- make a mutable matrix filled with zeroes
• Net ^ ZZ -- raise a net or string
• Net | Net -- join strings or nets
• Net || Net -- join nets or strings vertically
• NetFile -- the class of all net files
• netList(VisibleList) -- a table of boxes
• new ChainComplex -- make a new chain complex from scratch
• new HashTable from List -- make a hash table from a list
• newCoordinateSystem -- change variables
• NewFromMethod
• newline -- the new line character sequence
• NewMethod
• newNetFile -- create a new net file
• NewOfFromMethod
• NewOfMethod
• newRing -- make a copy of a ring, with some features changed
• next -- get the next object from an iterator
• nextkey -- the next key in a database
• nextPrime -- compute the smallest prime greater than or equal to a given number
• NoPrint -- top level method for non-printing results
• norm
• not documented yet
• Nothing -- the empty class
• notify -- whether to notify the user when a file is loaded
• notImplemented -- print an 'not implemented' error message
• null -- the unique member of the empty class
• nullaryMethods
• nullhomotopy -- make a null homotopy
• nullSpace -- find the null space of a mutable matrix
• Number -- the class of all numbers
• number -- count how many elements of a list satisfy a condition
• NumberedVerticalList -- a type of visible self-initializing list that prints vertically
• numColumns(Matrix) -- number of columns in a matrix or mutable matrix
• numerator -- numerator of a fraction
• numeric -- convert to floating point
• numericInterval -- converts to an interval
• numgens -- the number of generators
• numgens(CoherentSheaf) -- the number of generators of the underlying module
• numgens(Ideal) -- number of generators of an ideal
• numgens(Module) -- number of generators of a module
• numgens(Monoid) -- number of generators of a monoid
• numgens(Ring) -- number of generators of a polynomial ring
• numRows(Matrix) -- number of rows in a matrix or mutable matrix
• odd -- tell whether an integer is odd
• of -- a keyword
• ofClass -- English phrases for types
• on -- trace a function each time it's run
• OneExpression -- the class all one expressions
• oo -- the last output value
• OO _ Variety -- the structure sheaf
• ooo -- the next to the last output value
• oooo -- the third to the last output value
• openDatabase -- open a database file
• openDatabaseOut -- open a database file for writing
• openFiles -- list the open files
• openIn(String) -- open an input file
• openInOut -- open an input output file
• openListener(String) -- open a port for listening
• openOut(String) -- open an output file
• openOutAppend(String) -- open an output file for appending
• Option -- the class of all key-value pairs K => V
• options -- get options
• options(Command) -- get optional arguments and default values of functions and methods
• options(Monoid) -- get values used for optional arguments
• OptionTable -- the class of hash tables for optional arguments
• OptionTable ++ OptionTable -- a binary operator, usually used for direct sum
• OptionTable >> Function -- attaching options to a function
• Order -- specify the order of a Hilbert series required
• order -- a key used internally
• OrderedMonoid -- the class of all ordered monoids
• OutputDictionary -- the dictionary for output values
• override -- override default values for optional arguments
• pack -- pack elements of a list or string into several shorter lists or strings
• pager -- display with paging
• PairsRemaining -- whether to display the number of S-pairs remaining
• Parenthesize
• parsing precedence, in detail
• part -- select terms of a polynomial by degree(s) or weight(s)
• Partition -- a type of list representing a partition of a natural number
• partition -- partition a set or list by values of a function
• Partition _ ZZ -- get element
• partitions -- list the partitions of an integer
• parts -- display terms of a polynomial degree by degree
• pdim -- calculate the projective dimension
• pdim(CoherentSheaf) -- calculate the projective dimension
• pdim(Module) -- calculate the projective dimension of a module
• peek -- examine contents of an object
• peek' -- examine contents of an object
• permanents -- ideal generated by square permanents of a matrix
• permutations -- produce all permutations of a list
• pfaffians -- ideal generated by Pfaffians
• pi
• pivots(Matrix) -- list of pivot locations of a matrix
• plus -- addition
• poincare(ChainComplex) -- assemble degrees of a chain complex into a polynomial
• polarize -- given a monomial ideal, computes the squarefree monomial ideal obtained via polarization
• PolynomialRing -- the class of all ordered monoid rings
• position -- the first element of a list satisfying a condition
• positions -- which elements of a list satisfy a condition
• Power -- the class of all power expressions
• power -- power
• powermod -- powers of integers mod N
• powers
• Precision -- name of an optional argument.
• precision
• prefixPath -- absolute locations of Macaulay2 files
• preimage -- preimage of a map
• prepend -- add an element to the beginning of a list
• presentation -- presentation of a module or ring
• presentation(Module) -- presentation of a module
• presentation(PolynomialRing,QuotientRing) -- presentation of a quotient ring
• pretty -- a pretty printer
• PrimitiveElement -- specify a primitive element
• Print -- top level method for printing results
• print -- print something
• printerr -- print something to stderr
• printingAccuracy -- current accuracy for printing numbers
• printingLeadLimit -- maximum number of leading zeroes to use when printing real numbers
• printingPrecision -- current precision for printing numbers
• printingSeparator -- string used to separate mantissa from exponent when printing real numbers
• printingTimeLimit
• printingTrailLimit -- maximum number of additional trailing digits to use when printing real numbers
• printString -- lowlevel function to print a string, net, or symbol
• printWidth
• Product -- the class of all product expressions
• product
• product(List) -- product of elements
• product(Set) -- product of elements
• product(VirtualTally) -- product of elements
• product(VisibleList,Function) -- product of values of a function
• product(VisibleList,VisibleList,Function) -- product of results of applying a function pairwise
• product(ZZ,Function) -- product of consecutive values of a function
• ProductOrder -- product ordering
• profile -- profile a function
• profileSummary -- display profiling data
• Program -- external program object
• Program << Thing -- run program with input redirection
• programPaths -- user-defined external program paths
• ProgramRun -- result of running an external program
• Proj(Ring) -- make a projective variety
• Projective -- whether to produce a projective Hilbert polynomial
• ProjectiveHilbertPolynomial -- the class of all Hilbert polynomials
• projectiveHilbertPolynomial -- Hilbert polynomial of projective space
• ProjectiveHilbertPolynomial ZZ -- value of polynomial
• ProjectiveVariety -- the class of all projective varieties
• promote -- promote to another ring
• protect -- protect a symbol
• prune -- prune, e.g., compute a minimal presentation
• pruningMap
• Pseudocode -- the class of pseudocodes
• pseudocode -- produce the pseudocode for a function
• pseudoRemainder -- compute the pseudo-remainder
• pushForward(RingMap,Module)
• QQ -- the class of all rational numbers
• QRDecomposition -- compute a QR decomposition of a real matrix
• quotient -- quotient or division
• quotient' -- matrix quotient (opposite)
• quotient(Matrix,GroebnerBasis) -- matrix quotient
• quotientRemainder -- matrix quotient and remainder
• quotientRemainder' -- matrix quotient and remainder (opposite)
• quotientRemainder(RingElement,RingElement) -- quotient and remainder
• QuotientRing -- the class of all quotient rings
• RaiseError -- whether to raise an error
• random -- get a random object

  • random(ZZ,ZZ) -- get a random integer or real number
  • random(Type) -- get a random object of a type
  • random(List) -- shuffle a list randomly
  • random(ZZ,Ideal) -- get a random homogeneous element from a graded ideal
  • random(ZZ,Ring) -- get a random homogeneous element from a graded ring
  • random(Module,Module) -- make a random module map
• randomKRationalPoint -- Pick a random K rational point on the scheme X defined by I
• randomMutableMatrix(ZZ,ZZ,RR,ZZ) -- a random mutable matrix of integers
• Range -- can be assigned a integral interval
• rank -- compute the rank
• read -- read from a file
• read(File) -- read from a file
• read(File,ZZ) -- read from a file
• read(Sequence) -- read from a file
• read(String) -- read from a file
• RealField -- the class of all real fields
• realPart -- real part
• recursionDepth -- the current recursion depth
• recursionLimit -- set the limit on recursion
• reducedRowEchelonForm -- compute the reduced row echelon form of a matrix or mutable matrix over a field
• regularity -- compute the Castelnuovo-Mumford regularity
• regularizedBeta -- regularized beta function
• regularizedGamma -- upper regularized gamma function
• relations -- the defining relations
• relativizeFilename -- relativize a file name
• remainder -- matrix remainder
• remainder' -- matrix quotient and remainder (opposite)
• reorganize -- reorganize a database file
• replace -- replacement in strings and lists
• replace(ZZ,Thing,VisibleList) -- copy a list, replacing an element
• reshape(Module,Module,Matrix) -- reshape a matrix
• Resolution -- the class of all resolution computations
• resolution -- projective resolution
• resolution(...,DegreeLimit=>...) -- compute only up to this degree
• resolution(...,HardDegreeLimit=>...)
• resolution(...,LengthLimit=>...) -- stop when the resolution reaches this length
• resolution(...,PairLimit=>...) -- stop when this number of pairs has been handled
• resolution(...,SortStrategy=>...)
• resolution(...,StopBeforeComputation=>...) -- whether to stop the computation immediately
• resolution(...,Strategy=>...)
• resolution(...,SyzygyLimit=>...) -- stop when this number of syzygies is reached
• resolution(Ideal) -- compute a projective resolution of (the quotient ring corresponding to) an ideal
• resolution(Matrix) -- given a module map represented by a matrix, produce a comparison map between resolutions of its source and target
• resolution(Module) -- compute a free resolution of a module
• return -- return from a function
• returnCode
• reverse -- reverse a list, sequence, or string
• right -- right endpoint of an interval
• right shift
• Ring -- the class of all rings
• ring -- get the associated ring of an object
• Ring / Ideal -- make a quotient ring
• Ring ^ List -- make a free module
• Ring ^ ZZ -- make a free module
• Ring _ List -- make a monomial from a list of exponents
• Ring _ ZZ -- get a ring variable by index
• Ring Array -- the standard way to make a polynomial ring
• Ring List -- make a local polynomial ring
• Ring Monoid -- make a polynomial ring
• RingElement -- the class of all ring elements handled by the engine
• RingElement .. RingElement -- a sequence of consecutive generators of a polynomial ring
• RingElement ..< RingElement -- a sequence of consecutive generators of a polynomial ring
• RingElement / RingElement -- fraction
• RingElement ^ ZZ -- power
• RingElement Array -- substitution of variables
• RingFamily
• RingFamily _* -- a unary postfix operator, used for indicating pushforward maps
• RingMap -- the class of all ring maps
• RingMap RingElement -- apply a ring map
• rootPath
• roots(RingElement) -- compute the roots of a polynomial
• rootURI
• rotate(ZZ,VisibleList) -- rotate a list
• round -- round a number
• row and column operations
• rowAdd -- add a multiple of one row to another
• RowExpression -- the class of all matrix expressions
• rowMult -- multiply a row by a ring element
• rowPermute -- permute some rows
• rowRankProfile -- find the row rank profile of a mutable matrix
• rowSwap -- interchange rows
• RR -- the class of all real numbers
• RR' -- the parent class of all rings of real numbers
• RRi -- the class of all real intervals
• rsort -- sort a list or matrix in reverse order
• rsort(...,DegreeOrder=>...) -- specify Ascending, Descending, or null
• rsort(...,MonomialOrder=>...) -- specify Ascending or Descending monomial order
• RunDirectory -- the directory from which to run a program
• runLengthEncode(VisibleList) -- run length encoding
• runProgram -- run an external program
• same -- whether everything in a list is the same
• scan -- apply a function to each element in a list or sequence
• scanKeys -- apply a function to each key in a hash table or database
• scanLines -- apply a function to each line of a file
• scanPairs -- apply a function to the pairs in a hash table
• scanValues -- apply a function to each value in a hash table or database
• schreyerOrder -- create or obtain free modules with Schreyer monomial orders
• schreyerOrder(Matrix) -- create a matrix with the same entries whose source free module has a Schreyer monomial order
• schreyerOrder(Module) -- obtain Schreyer order information
• Schubert(ZZ,ZZ,VisibleList) -- find the Plücker ideal of a Schubert variety
• scriptCommandLine -- the command line arguments to be used when running a script
• ScriptedFunctor -- the class of all scripted functors
• searchPath(List,String) -- search a path for a file
• sec -- secant
• sech -- hyperbolic secant
• seeParsing -- display parsing precedence table for Macaulay2 operators
• select -- select from a list, hash table, or string
• select(BasicList,Function) -- select elements from a list
• select(BasicList,Type) -- select elements of a given type in a list
• select(HashTable,Function) -- select part of a hash table
• select(Thing,Function) -- select elements from an object with an iterator
• select(ZZ,BasicList,Function) -- select a limited number of elements from a list
• select(ZZ,Function) -- select integers
• select(ZZ,HashTable,Function) -- select a limited number of pairs from a hash table
• selectInSubring -- select columns in a subring
• selectVariables(List,PolynomialRing) -- make a subring of a polynomial ring generated by selected variables
• SelfInitializingType -- the class of all self initializing types
• sequence -- make a sequence
• serialNumber -- serial number of a dictionary, task, symbol, mutable hash table, or mutable list,
• Set -- the class of all sets
• set -- make a set
• Set #? Thing -- test set membership
• Set * Set -- intersection of sets
• Set ** Set -- Cartesian product
• Set + Set -- set union
• Set - Set -- set difference
• setGroupID -- set the process group identifier
• setRandomSeed -- set starting point for random number generator
• setup
• setupEmacs
• sheaf -- make a coherent sheaf
• sheaf(Module) -- make a coherent sheaf
• sheaf(Ring) -- make a coherent sheaf of rings
• sheaf(Variety) -- make a coherent sheaf
• sheaf(Variety,Module) -- make a coherent sheaf
• sheaf(Variety,Ring) -- make a coherent sheaf of rings
• SheafExpression -- the class of sheaf expressions
• sheafExt^ZZ(CoherentSheaf,CoherentSheaf) -- sheaf Ext of coherent sheaves
• sheafHom(CoherentSheaf,CoherentSheaf) -- sheaf Hom
• SheafOfRings -- the class of sheaves of rings
• SheafOfRings ^ List -- make a graded free coherent sheaf
• shield -- shield evaluation from interrupts
• show -- display various objects in an external viewer
• showClassStructure -- display class structure
• showStructure -- display parent structure
• showUserStructure -- show parent structure for those types defined by user
• sin -- compute the sine
• singularLocus -- singular locus
• sinh -- compute the hyperbolic sine
• size -- the size of an object
• size2 -- number of binary digits to the left of the point
• smithNormalForm(Matrix) -- smith normal form for a matrix over ZZ or a PID
• solve -- solve linear equation(s)
• someTerms -- select some terms of a polynomial
• Sort -- a Strategy option value
• sort -- sort a list or columns of a matrix
• sort(...,DegreeOrder=>...) -- specify Ascending, Descending, or null
• sort(...,MonomialOrder=>...) -- specify Ascending or Descending monomial order
• sort(List) -- sort a list
• sort(Matrix) -- sort the columns of a matrix
• sortColumns -- permutation giving sort order
• sortColumns(...,DegreeOrder=>...) -- specify Ascending, Descending, or null
• sortColumns(...,MonomialOrder=>...) -- specify Ascending or Descending monomial order
• SortStrategy -- specify a strategy for sorting S-pairs
• source -- source of a map
• source(ChainComplexMap) -- find the source of a map of chain complexes
• source(GradedModuleMap) -- find the source of a map of graded modules
• source(Matrix) -- find the source module of matrix
• source(RingMap) -- find the source ring for a map of rings
• span -- construct smallest interval
• span(List) -- construct smallest interval
• span(Sequence) -- construct smallest interval
• SparseMonomialVectorExpression -- the class of all sparse monomial vector expressions
• SparseVectorExpression -- the class of all sparse vector expressions
• Spec(Ring) -- make an affine variety
• splice -- remove subsequences from a sequence or list
• splitWWW -- separate an http response into header and body
• sqrt -- square root function
• Standard -- the standard top level printing method
• standardForm -- convert to standard form
• standardPairs -- find the standard pairs of a monomial ideal
• stashValue -- stash values of functions in their arguments
• status -- status of a resolution computation
• status(ProgramRun) -- get the return status of a program run
• stderr -- the standard error output file
• stdio -- the standard input output file
• step -- step by single lines in the debugger
• stopIfError -- whether to stop the program when an error occurs
• StopIteration -- stop iteration
• String .. String -- a sequence of consecutive strings
• String ..< String -- a sequence of consecutive strings, not including the endpoint
• String ^ Sequence -- vertically stacked copies of a string
• String _ Ring -- get a ring variable by name
• sublists -- process interspersed subsequences of a visible list
• submatrix

  • submatrix' -- exclude rows and/or columns of a matrix
  • submatrixByDegrees -- submatrix consisting of rows and columns in an interval or box of degrees
  • submatrix(Matrix,VisibleList) -- select columns
  • submatrix(Matrix,VisibleList,VisibleList) -- select part of a matrix
  • Matrix _ List -- select columns
  • Matrix ^ List -- select rows
  • Matrix _ Array -- component of map corresponding to summand of source
  • Matrix ^ Array -- component of map corresponding to summand of target
• subquotient -- make a subquotient module
• SubringLimit -- stop after finding enough elements of a subring
• Subscript -- the class of all subscript expressions
• subscript -- specify the function in a scripted functor for a subscript
• subsets -- produce the subsets of a set or list
• substituting values for variables
• subtable -- extract a subtable from a table
• Sum -- the class of all sum expressions
• sum -- compute the sum
• sum(ChainComplex) -- direct sum of the components of a chain complex
• sum(ChainComplexMap) -- direct sum of the components of a chain map
• sum(List) -- sum the elements of a list
• sum(Set) -- sum of elements
• sum(VirtualTally) -- sum of elements
• sum(VisibleList,Function) -- sum results of applying a function
• sum(VisibleList,VisibleList,Function) -- sum results of applying a function pairwise
• sum(ZZ,Function) -- sum consecutive values of a function
• SumOfTwists -- the class of all sums of twists
• super -- get the ambient module
• Superscript -- the class of all superscript expressions
• superscript -- specify the function in a scripted functor for a superscript
• support -- list of variables occurring in a polynomial or matrix
• support(Ideal) -- list of variables occurring in the generators of an ideal
• SVD -- singular value decomposition of a matrix
• SVD(...,DivideConquer=>...) -- Use the lapack divide and conquer SVD algorithm
• switch -- copy a list, switching two elements
• Symbol -- the class of all symbols
• symbol -- get a symbol
• Symbol .. Symbol -- sequences of consecutive symbols
• Symbol ..< Symbol -- sequences of consecutive symbols
• Symbol _ Ring -- get a ring variable by name
• Symbol _ Thing -- index variable
• Symbol _ Thing = Thing -- assignment to an indexed variable
• symbolBody -- symbol bodies
• symbols used as the name or value of an optional argument

  • Verify -- an optional argument
  • Generic -- an optional argument
  • Verbosity -- an optional argument
  • Strategy -- an optional argument
  • CodimensionLimit -- an optional argument
  • SyzygyLimit -- an optional argument
  • StopBeforeComputation -- an optional argument
  • DegreeLimit -- an optional argument
  • PairLimit -- an optional argument
  • DegreeGroup -- an optional argument
  • Variables -- an optional argument
  • DegreeLift -- an optional argument
  • VariableBaseName -- an optional argument
  • DegreeRank -- an optional argument
  • Inverses -- an optional argument
  • Degrees -- an optional argument
  • Join -- an optional argument
  • MonomialSize -- an optional argument
  • Local -- an optional argument
  • Heft -- an optional argument
  • Constants -- an optional argument
  • SkewCommutative -- an optional argument
  • DegreeMap -- an optional argument
  • CoefficientRing -- an optional argument
  • FollowLinks -- an optional argument
  • Exclude -- an optional argument
  • Configuration -- an optional argument
  • Version -- an optional argument
  • Reload -- an optional argument
  • PackageExports -- an optional argument
  • Certification -- an optional argument
  • UseCachedExampleOutput -- an optional argument
  • OptionalComponentsPresent -- an optional argument
  • InfoDirSection -- an optional argument
  • PackageImports -- an optional argument
  • Authors -- an optional argument
  • CacheExampleOutput -- an optional argument
  • Keywords -- an optional argument
  • Date -- an optional argument
  • AuxiliaryFiles -- an optional argument
  • DebuggingMode -- an optional argument
  • Options -- an optional argument
  • InstallPrefix -- an optional argument
  • Height -- an optional argument
  • MaximalRank -- an optional argument
  • Density -- an optional argument
  • UpperTriangular -- an optional argument
  • SyzygyMatrix -- an optional argument
  • ChangeMatrix -- an optional argument
  • MinimalMatrix -- an optional argument
  • SeparateExec -- an optional argument
  • RunExamples -- an optional argument
  • MakeHTML -- an optional argument
  • RemakeAllDocumentation -- an optional argument
  • MakeInfo -- an optional argument
  • RerunExamples -- an optional argument
  • IgnoreExampleErrors -- an optional argument
  • MakePDF -- an optional argument
  • FileName -- an optional argument
  • MakeLinks -- an optional argument
  • MakeDocumentation -- an optional argument
  • UserMode -- an optional argument
  • CheckDocumentation -- an optional argument
  • DivideConquer -- an optional argument
  • Minimize -- an optional argument
  • Body -- an optional argument
  • Hermitian -- an optional argument
  • Limit -- an optional argument
  • SourceRing -- an optional argument
  • Truncate -- an optional argument
  • KeepZeroes -- an optional argument
  • Syzygies -- an optional argument
  • StopWithMinimalGenerators -- an optional argument
  • BasisElementLimit -- an optional argument
  • MaxReductionCount -- an optional argument
  • GBDegrees -- an optional argument
  • Algorithm -- an optional argument
  • SyzygyRows -- an optional argument
  • Hilbert -- an optional argument
  • Undo -- an optional argument
  • Dense -- an optional argument
  • LoadDocumentation -- an optional argument
  • Invertible -- an optional argument
  • ClosestFit -- an optional argument
  • Alignment -- an optional argument
  • VerticalSpace -- an optional argument
  • Boxes -- an optional argument
  • BaseRow -- an optional argument
  • HorizontalSpace -- an optional argument
  • Reverse -- an optional argument
  • SizeLimit -- an optional argument
  • UseHilbertFunction -- an optional argument
  • Reduce -- an optional argument
  • First -- an optional argument
  • Result -- an optional argument
  • Prune -- value of an optional argument
  • Binomial -- value of an optional argument
  • Test -- value of an optional argument
  • Center -- value of an optional argument
  • Right -- value of an optional argument
  • Left -- value of an optional argument
  • Quotient -- value of an optional argument
  • Intersection -- value of an optional argument
  • Monomial -- value of an optional argument
  • Default -- value of an optional argument
  • Torsion -- value of an optional argument
  • Iterate -- value of an optional argument
  • Unmixed -- value of an optional argument
  • Decompose -- value of an optional argument
  • CompleteIntersection -- value of an optional argument
  • Bayer -- value of an optional argument
  • Linear -- value of an optional argument
• symmetricAlgebra -- the symmetric algebra of a module
• symmetricPower -- symmetric power
• synonym -- synonym for members of a class
• syz -- the syzygy matrix
• syz(GroebnerBasis) -- retrieve the syzygy matrix
• syz(Matrix) -- compute the syzygy matrix
• syzygyScheme -- construct a syzygy scheme
• Table -- the class of all table expressions
• table -- make a table from a binary function
• take -- Take some elements from a list or sequence.
• Tally -- the class of all tally results
• tally -- tally the elements of a list, sequence, array, or string
• Tally + Tally -- union of tallies
• Tally - Tally -- difference of tallies
• tan -- compute the tangent
• tangentSheaf -- tangent sheaf of a projective variety
• tangentSheaf(ProjectiveVariety) -- tangent sheaf of a projective variety
• tanh -- compute the hyperbolic tangent
• target -- target of a map
• target(ChainComplexMap) -- find the target of a map of chain complexes
• target(GradedModuleMap) -- find the target of a map of graded modules
• target(GroebnerBasis) -- find target of a Gröbner basis
• target(Matrix) -- find the target module of matrix
• target(RingMap) -- find the target ring for a map of rings
• tensor -- tensor product
• tensor(Monoid,Monoid) -- tensor product of monoids
• tensor(RingMap,Matrix) -- tensor product via a ring map
• tensorAssociativity -- associativity isomorphisms for tensor products
• terms -- provide a list of terms of a polynomial
• tests -- locate a package's tests
• tex -- convert to $\TeX$ format
• TeXmacs -- the TeXmacs top level printing method
• texMath -- convert to TeX math format
• the debugger
• then -- condition testing
• Thing -- the class of all things
• Thing // Function -- apply a function
• throw -- throw an exception
• times -- multiplication
• to -- loop control
• toAbsolutePath -- the absolute path version of a file name
• toCC -- convert to high-precision complex number
• toField(Ring) -- declare that a ring is a field
• toList -- create a list
• toLower -- convert to lower case
• topCoefficients -- first variable and its coefficient of a polynomial or matrix
• topLevelMode -- the current top level mode
• Tor -- Tor module
• Tor_ZZ(Module,Module) -- compute a Tor module
• toRR -- convert to high-precision real number
• toRRi -- construct an interval
• toSequence -- convert to sequence
• TotalPairs -- whether to display the total number of S-pairs
• toUpper -- convert to upper case
• trace(Matrix) -- trace of a matrix
• transpose -- transpose a table or a matrix
• transpose(ChainComplexMap) -- transpose a map of chain complexes
• transpose(List) -- transposes a table
• transpose(Matrix) -- transpose a matrix
• trim -- minimize generators and relations
• true
• truncate
• truncate(Number) -- round a number toward zero
• truncateOutput
• tutorial -- convert documentation from tutorial format
• typicalValues -- types of values returned by functions
• ultimate -- ultimate value for an iteration
• unbag(Bag)
• uncurry -- uncurry a function
• undocumented -- declare that something need not be documented
• uniform -- whether all elements of a list are the same class
• Unique -- do not return repeated polynomial roots
• unique -- eliminate duplicates from a list
• uniquePermutations -- produce all unique permutations of a list
• unsequence -- extract the single element from a sequence of length 1
• UpdateOnly -- only copies of newer files should replace files
• URL -- a type representing a URL
• urlEncode -- URL encoding
• use -- install or activate object
• use(Ring) -- install ring variables and ring operations
• userSymbols -- a list of the user's symbols
• utf8substring -- extract part of a utf8 string
• value -- evaluate
• value(IndexedVariable) -- retrieve the value of an indexed variable
• value(Pseudocode) -- execute pseudocode
• value(String) -- evaluate a string
• value(Symbol) -- retrieve the value of a symbol
• Variable -- specify a name for a variable
• variety -- get the variety
• variety(CoherentSheaf) -- get the variety
• variety(Ideal) -- the closed projective subvariety defined by an ideal
• variety(Ring) -- the variety previously associated to a given ring
• variety(SheafOfRings) -- get the variety
• vars -- variables
• vars(List) -- a sequence of variables
• vars(Monoid) -- list of variables in a monoid
• vars(Ring) -- row matrix of the variables
• Vector -- the class of all elements of free modules that are handled by the engine
• vector -- make a vector
• Vector ** Vector -- tensor product
• Vector || Vector -- join Vectors
• VectorExpression -- the class of all vector expressions
• Verbose -- request verbose feedback
• VerticalList -- a type of visible self-initializing list that prints vertically
• VirtualTally
• VirtualTally ** VirtualTally -- Cartesian product of tallies
• VirtualTally + VirtualTally -- union of tallies
• VirtualTally ^** ZZ -- Cartesian power of sets and tallies
• VirtualTally _ Thing -- get a count from a tally
• VisibleList -- the class of all visible lists

  • List -- the class of all lists -- {...}
  • Sequence -- the class of all sequences -- (...)
  • Array -- the class of all arrays -- [...]
• VisibleList / Function -- apply a function to elements of a list
• VisibleList _ List -- get some entries of a list
• VisibleList _ ZZ -- get element from list
• w3
• WebApp -- the web app top level printing method
• wedgeProduct(ZZ,ZZ,Module) -- the exterior multiplication map
• weightRange -- the pair of lowest and highest weights of the monomials
• when -- a keyword
• width -- width of a file or net
• width(File) -- get window width
• Wrap -- a key for methods for wrapping printed output
• wrap
• WrapperType -- a class of lists with abbreviated constructors
• xor -- exclusive disjunction
• youngest -- the youngest member of a sequence
• zero -- whether something is zero
• ZeroExpression -- the class of all zero expressions
• zeta -- Riemann zeta function
• ZZ -- the class of all integers
• ZZ & ZZ -- logical and
• ZZ .. ZZ -- sequences of consecutive integers
• ZZ ..< ZZ -- sequences of consecutive integers
• ZZ : Thing -- repeat an item
• ZZ _ Module -- integers or zero element
• ZZ | ZZ -- logical or
• ZZ ~ -- logical not