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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)
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)
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)
diff(ProjectiveHilbertPolynomial,ZZ)
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)
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)
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)
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)
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