Table of Contents
1. Preface
2. Introduction
2.1 Background
2.2 How to use this manual
2.3 Getting started
2.3.1 First steps
2.3.2 Rings and standard bases
2.3.3 Procedures and libraries
2.3.4 Change of rings
2.3.5 Modules and their annihilator
2.3.6 Resolution
3. General concepts
3.1 Interactive use
3.1.1 How to enter and exit
3.1.2 The SINGULAR prompt
3.1.3 The online help system
3.1.4 Interrupting SINGULAR
3.1.5 Editing input
3.1.6 Command line options
3.1.7 Startup sequence
3.2 Emacs user interface
3.2.1 A quick guide to Emacs
3.2.2 Running SINGULAR under Emacs
3.2.3 Demo mode
3.2.4 Customization of the Emacs interface
3.2.5 Editing SINGULAR input files with Emacs
3.2.6 Top 20 Emacs commands
3.3 Rings and orderings
3.3.1 Examples of ring declarations
3.3.2 General syntax of a ring declaration
3.3.3 Term orderings
3.4 Implemented algorithms
3.5 The SINGULAR language
3.5.1 General command syntax
3.5.2 Special characters
3.5.3 Names
3.5.4 Objects
3.5.5 Type conversion and casting
3.5.6 Flow control
3.6 Input and output
3.7 Procedures
3.7.1 Procedure definition
3.7.2 Names in procedures
3.7.3 Parameter list
3.7.4 Procedure commands
3.8 Libraries
3.8.1 Loading of a library
3.8.2 Format of a library
3.9 Guidelines for writing a library
3.9.1 Procedures in a library
3.9.2 Documentation of a library
3.9.2.1 lib2doc
3.9.3 Typesetting of help strings
3.9.4 The help string of a library
3.9.5 The help string of procedures
3.9.6 template_lib
3.9.6.1 mdouble
3.9.6.2 mtripple
3.9.6.3 msum
3.10 Debugging tools
3.10.1 Tracing of procedures
3.10.2 Source code debugger
3.10.3 Break points
3.10.4 Printing of data
3.10.5 libparse
3.11 Dynamic loading
4. Data types
4.1 bigint
4.1.1 bigint declarations
4.1.2 bigint expressions
4.1.3 bigint operations
4.1.4 bigint related functions
4.2 def
4.2.1 def declarations
4.3 ideal
4.3.1 ideal declarations
4.3.2 ideal expressions
4.3.3 ideal operations
4.3.4 ideal related functions
4.4 int
4.4.1 int declarations
4.4.2 int expressions
4.4.3 int operations
4.4.4 int related functions
4.4.5 boolean expressions
4.4.6 boolean operations
4.5 intmat
4.5.1 intmat declarations
4.5.2 intmat expressions
4.5.3 intmat type cast
4.5.4 intmat operations
4.5.5 intmat related functions
4.6 intvec
4.6.1 intvec declarations
4.6.2 intvec expressions
4.6.3 intvec operations
4.6.4 intvec related functions
4.7 link
4.7.1 link declarations
4.7.2 link expressions
4.7.3 link related functions
4.7.4 ASCII links
4.7.5 MP links
4.7.5.1 MPfile links
4.7.5.2 MPtcp links
4.7.6 DBM links
4.8 list
4.8.1 list declarations
4.8.2 list expressions
4.8.3 list operations
4.8.4 list related functions
4.9 map
4.9.1 map declarations
4.9.2 map expressions
4.9.3 map operations
4.9.4 map related functions
4.10 matrix
4.10.1 matrix declarations
4.10.2 matrix expressions
4.10.3 matrix type cast
4.10.4 matrix operations
4.10.5 matrix related functions
4.11 module
4.11.1 module declarations
4.11.2 module expressions
4.11.3 module operations
4.11.4 module related functions
4.12 number
4.12.1 number declarations
4.12.2 number expressions
4.12.3 number operations
4.12.4 number related functions
4.13 package
4.13.1 package declarations
4.13.2 package related functions
4.14 poly
4.14.1 poly declarations
4.14.2 poly expressions
4.14.3 poly operations
4.14.4 poly related functions
4.15 proc
4.15.1 proc declaration
4.16 qring
4.16.1 qring declaration
4.17 resolution
4.17.1 resolution declarations
4.17.2 resolution expressions
4.17.3 resolution related functions
4.18 ring
4.18.1 ring declarations
4.18.2 ring related functions
4.18.3 ring operations
4.19 string
4.19.1 string declarations
4.19.2 string expressions
4.19.3 string type cast
4.19.4 string operations
4.19.5 string related functions
4.20 vector
4.20.1 vector declarations
4.20.2 vector expressions
4.20.3 vector operations
4.20.4 vector related functions
5. Functions and system variables
5.1 Functions
5.1.1 attrib
5.1.2 bareiss
5.1.3 betti
5.1.4 char
5.1.5 char_series
5.1.6 charstr
5.1.7 chinrem
5.1.8 cleardenom
5.1.9 close
5.1.10 coef
5.1.11 coeffs
5.1.12 contract
5.1.13 dbprint
5.1.14 defined
5.1.15 deg
5.1.16 degree
5.1.17 delete
5.1.18 det
5.1.19 diff
5.1.20 dim
5.1.21 division
5.1.22 dump
5.1.23 eliminate
5.1.24 eval
5.1.25 ERROR
5.1.26 example
5.1.27 execute
5.1.28 exit
5.1.29 extgcd
5.1.30 facstd
5.1.31 factorize
5.1.32 fetch
5.1.33 fglm
5.1.34 fglmquot
5.1.35 files, input from
5.1.36 find
5.1.37 finduni
5.1.38 fprintf
5.1.39 freemodule
5.1.40 frwalk
5.1.41 gcd
5.1.42 gen
5.1.43 getdump
5.1.44 groebner
5.1.45 help
5.1.46 highcorner
5.1.47 hilb
5.1.48 homog
5.1.49 hres
5.1.50 imap
5.1.51 impart
5.1.52 indepSet
5.1.53 insert
5.1.54 interpolation
5.1.55 interred
5.1.56 intersect
5.1.57 jacob
5.1.58 janet
5.1.59 jet
5.1.60 kbase
5.1.61 kill
5.1.62 killattrib
5.1.63 koszul
5.1.64 laguerre
5.1.65 lead
5.1.66 leadcoef
5.1.67 leadexp
5.1.68 leadmonom
5.1.69 LIB
5.1.70 lift
5.1.71 liftstd
5.1.72 listvar
5.1.73 lres
5.1.74 maxideal
5.1.75 memory
5.1.76 minbase
5.1.77 minor
5.1.78 minres
5.1.79 modulo
5.1.80 monitor
5.1.81 mpresmat
5.1.82 mres
5.1.83 mstd
5.1.84 mult
5.1.85 nameof
5.1.86 names
5.1.87 ncols
5.1.88 npars
5.1.89 nres
5.1.90 nrows
5.1.91 nvars
5.1.92 open
5.1.93 option
5.1.94 ord
5.1.95 ordstr
5.1.96 par
5.1.97 pardeg
5.1.98 parstr
5.1.99 preimage
5.1.100 prime
5.1.101 print
5.1.102 printf
5.1.103 prune
5.1.104 qhweight
5.1.105 quote
5.1.106 quotient
5.1.107 random
5.1.108 read
5.1.109 reduce
5.1.110 regularity
5.1.111 repart
5.1.112 res
5.1.113 reservedName
5.1.114 resultant
5.1.115 ringlist
5.1.116 rvar
5.1.117 setring
5.1.118 simplex
5.1.119 simplify
5.1.120 size
5.1.121 slimgb
5.1.122 sortvec
5.1.123 sqrfree
5.1.124 sprintf
5.1.125 sres
5.1.126 status
5.1.127 std
5.1.128 stdfglm
5.1.129 stdhilb
5.1.130 subst
5.1.131 system
5.1.132 syz
5.1.133 trace
5.1.134 transpose
5.1.135 type
5.1.136 typeof
5.1.137 univariate
5.1.138 uressolve
5.1.139 vandermonde
5.1.140 var
5.1.141 variables
5.1.142 varstr
5.1.143 vdim
5.1.144 wedge
5.1.145 weight
5.1.146 weightKB
5.1.147 write
5.2 Control structures
5.2.1 break
5.2.2 breakpoint
5.2.3 continue
5.2.4 else
5.2.5 export
5.2.6 exportto
5.2.7 for
5.2.8 if
5.2.9 importfrom
5.2.10 keepring
5.2.11 load
5.2.12 quit
5.2.13 return
5.2.14 while
5.2.15 ~ (break point)
5.3 System variables
5.3.1 degBound
5.3.2 echo
5.3.3 minpoly
5.3.4 multBound
5.3.5 noether
5.3.6 printlevel
5.3.7 short
5.3.8 timer
5.3.9 TRACE
5.3.10 rtimer
5.3.11 voice
6. Tricks and pitfalls
6.1 Limitations
6.2 System dependent limitations
6.3 Major differences to the C programming language
6.3.1 No rvalue of increments and assignments
6.3.2 Evaluation of logical expressions
6.3.3 No case or switch statement
6.3.4 Usage of commas
6.3.5 Usage of brackets
6.3.6 Behavior of continue
6.3.7 Return type of procedures
6.3.8 First index is 1
6.4 Miscellaneous oddities
6.5 Identifier resolution
7. Non-commutative subsystem PLURAL
7.1 Getting started with PLURAL
7.2 Data types (plural)
7.2.1 ideal (plural)
7.2.1.1 ideal declarations (plural)
7.2.1.2 ideal expressions (plural)
7.2.1.3 ideal operations (plural)
7.2.1.4 ideal related functions (plural)
7.2.2 map (plural)
7.2.2.1 map declarations (plural)
7.2.2.2 map expressions (plural)
7.2.2.3 map (plural) operations
7.2.2.4 map related functions (plural)
7.2.3 module (plural)
7.2.3.1 module declarations (plural)
7.2.3.2 module expressions (plural)
7.2.3.3 module operations (plural)
7.2.3.4 module related functions (plural)
7.2.4 poly (plural)
7.2.4.1 poly declarations (plural)
7.2.4.2 poly expressions (plural)
7.2.4.3 poly operations (plural)
7.2.4.4 poly related functions (plural)
7.2.5 qring (plural)
7.2.5.1 qring declaration (plural)
7.2.5.2 qring related functions (plural)
7.2.6 resolution (plural)
7.2.6.1 resolution declarations (plural)
7.2.6.2 resolution expressions (plural)
7.2.6.3 resolution related functions (plural)
7.2.7 ring (plural)
7.2.7.1 ring declarations (plural)
7.2.7.2 ring operations (plural)
7.2.7.3 ring related functions (plural)
7.3 Functions (plural)
7.3.1 betti (plural)
7.3.2 bracket
7.3.3 eliminate (plural)
7.3.4 envelope
7.3.5 fetch (plural)
7.3.6 imap (plural)
7.3.7 intersect (plural)
7.3.8 kbase (plural)
7.3.9 lift (plural)
7.3.10 liftstd (plural)
7.3.11 minres (plural)
7.3.12 modulo (plural)
7.3.13 mres (plural)
7.3.14 nc_algebra
7.3.15 ncalgebra
7.3.16 nres (plural)
7.3.17 oppose
7.3.18 opposite
7.3.19 preimage (plural)
7.3.20 quotient (plural)
7.3.21 reduce (plural)
7.3.22 ringlist (plural)
7.3.23 slimgb (plural)
7.3.24 std (plural)
7.3.25 subst (plural)
7.3.26 syz (plural)
7.3.27 twostd
7.3.28 vdim (plural)
7.4 Mathematical background (plural)
7.4.1 G-algebras
7.4.2 Groebner bases in G-algebras
7.4.3 Syzygies and resolutions (plural)
7.4.4 References (plural)
7.5 PLURAL libraries
7.5.1 central_lib
7.5.1.0. centralizeSet
7.5.1.0. centralizerVS
7.5.1.0. centralizerRed
7.5.1.0. centerVS
7.5.1.0. centerRed
7.5.1.0. center
7.5.1.0. centralizer
7.5.1.0. sa_reduce
7.5.1.0. sa_poly_reduce
7.5.1.0. inCenter
7.5.1.0. inCentralizer
7.5.1.0. isCartan
7.5.1.0. applyAdF
7.5.1.0. linearMapKernel
7.5.1.0. linearCombinations
7.5.1.0. variablesStandard
7.5.1.0. variablesSorted
7.5.1.0. PBW_eqDeg
7.5.1.0. PBW_maxDeg
7.5.1.0. PBW_maxMonom
7.5.2 dmod_lib
7.5.2.0. annfs
7.5.2.0. annfspecial
7.5.2.0. Sannfs
7.5.2.0. Sannfslog
7.5.2.0. bernsteinBM
7.5.2.0. operatorBM
7.5.2.0. annfsParamBM
7.5.2.0. annfsBMI
7.5.2.0. checkRoot
7.5.2.0. annfsBM
7.5.2.0. annfsLOT
7.5.2.0. annfsOT
7.5.2.0. SannfsBM
7.5.2.0. SannfsLOT
7.5.2.0. SannfsOT
7.5.2.0. annfs0
7.5.2.0. checkRoot1
7.5.2.0. checkRoot2
7.5.2.0. checkFactor
7.5.2.0. arrange
7.5.2.0. reiffen
7.5.2.0. isHolonomic
7.5.2.0. convloc
7.5.2.0. minIntRoot
7.5.2.0. varnum
7.5.3 involut_lib
7.5.3.0. findInvo
7.5.3.0. findInvoDiag
7.5.3.0. findAuto
7.5.3.0. ncdetection
7.5.3.0. involution
7.5.4 gkdim_lib
7.5.4.0. GKdim
7.5.5 ncalg_lib
7.5.5.0. makeUsl2
7.5.5.0. makeUsl
7.5.5.0. makeUgl
7.5.5.0. makeUso5
7.5.5.0. makeUso6
7.5.5.0. makeUso7
7.5.5.0. makeUso8
7.5.5.0. makeUso9
7.5.5.0. makeUso10
7.5.5.0. makeUso11
7.5.5.0. makeUso12
7.5.5.0. makeUsp1
7.5.5.0. makeUsp2
7.5.5.0. makeUsp3
7.5.5.0. makeUsp4
7.5.5.0. makeUsp5
7.5.5.0. makeUg2
7.5.5.0. makeUf4
7.5.5.0. makeUe6
7.5.5.0. makeUe7
7.5.5.0. makeUe8
7.5.5.0. makeQso3
7.5.5.0. makeQsl2
7.5.5.0. makeQsl3
7.5.5.0. Qso3Casimir
7.5.5.0. GKZsystem
7.5.6 ncdecomp_lib
7.5.6.0. CentralQuot
7.5.6.0. CentralSaturation
7.5.6.0. CenCharDec
7.5.6.0. IntersectWithSub
7.5.7 nctools_lib
7.5.7.0. Gweights
7.5.7.0. weightedRing
7.5.7.0. ndcond
7.5.7.0. Weyl
7.5.7.0. makeWeyl
7.5.7.0. makeHeisenberg
7.5.7.0. Exterior
7.5.7.0. findimAlgebra
7.5.7.0. SuperCommutative
7.5.7.0. rightStd
7.5.7.0. ncRelations
7.5.7.0. isCentral
7.5.7.0. isNC
7.5.7.0. UpOneMatrix
7.5.7.0. AltVarStart
7.5.7.0. AltVarEnd
7.5.7.0. IsSCA
7.5.8 perron_lib
7.5.8.0. perron
7.5.9 qmatrix_lib
7.5.9.0. quantMat
7.5.9.0. qminor
7.5.9.0. SymGroup
7.5.9.0. LengthSymElement
7.5.9.0. LengthSym
A. Examples
A.1 Programming
A.1.1 Basic programming
A.1.2 Writing procedures and libraries
A.1.3 Rings associated to monomial orderings
A.1.4 Long coefficients
A.1.5 Parameters
A.1.6 Formatting output
A.1.7 Cyclic roots
A.1.8 Parallelization with MPtcp links
A.1.9 Dynamic modules
A.2 Computing Groebner and Standard Bases
A.2.1 groebner and std
A.2.2 Groebner basis conversion
A.2.3 slim Groebner bases
A.3 Commutative Algebra
A.3.1 Saturation
A.3.2 Finite fields
A.3.3 Elimination
A.3.4 Free resolution
A.3.5 Handling graded modules
A.3.6 Computation of Ext
A.3.7 Depth
A.3.8 Factorization
A.3.9 Primary decomposition
A.3.10 Normalization
A.3.11 Kernel of module homomorphisms
A.3.12 Algebraic dependence
A.4 Singularity Theory
A.4.1 Milnor and Tjurina number
A.4.2 Critical points
A.4.3 Polar curves
A.4.4 T1 and T2
A.4.5 Deformations
A.4.6 Invariants of plane curve singularities
A.4.7 Branches of space curve singularities
A.4.8 Classification of hypersurface singularities
A.4.9 Resolution of singularities
A.5 Invariant Theory
A.5.1 G_a -Invariants
A.5.2 Invariants of a finite group
A.6 Noncommutative Algebra
A.6.1 Left and two-sided Groebner bases
A.6.2 Right Groebner bases and syzygies
A.7 Applications
A.7.1 Solving systems of polynomial equations
A.7.2 AG codes
B. Polynomial data
B.1 Representation of mathematical objects
B.2 Monomial orderings
B.2.1 Introduction to orderings
B.2.2 General definitions for orderings
B.2.3 Global orderings
B.2.4 Local orderings
B.2.5 Module orderings
B.2.6 Matrix orderings
B.2.7 Product orderings
B.2.8 Extra weight vector
C. Mathematical background
C.1 Standard bases
C.2 Hilbert function
C.3 Syzygies and resolutions
C.4 Characteristic sets
C.5 Gauss-Manin connection
C.6 Toric ideals and integer programming
C.6.1 Toric ideals
C.6.2 Algorithms
C.6.2.1 The algorithm of Conti and Traverso
C.6.2.2 The algorithm of Pottier
C.6.2.3 The algorithm of Hosten and Sturmfels
C.6.2.4 The algorithm of Di Biase and Urbanke
C.6.2.5 The algorithm of Bigatti, La Scala and Robbiano
C.6.3 The Buchberger algorithm for toric ideals
C.6.4 Integer programming
C.6.5 Relevant References
C.7 Non-commutative algebra
C.8 References
D. SINGULAR libraries
D.1 standard_lib
D.1.1 qslimgb
D.1.2 par2varRing
D.2 General purpose
D.2.1 all_lib
D.2.2 compregb_lib
D.2.2.1 cgs
D.2.2.2 base2str
D.2.3 general_lib
D.2.3.1 A_Z
D.2.3.2 ASCII
D.2.3.3 absValue
D.2.3.4 binomial
D.2.3.5 deleteSublist
D.2.3.6 factorial
D.2.3.7 fibonacci
D.2.3.8 kmemory
D.2.3.9 killall
D.2.3.10 number_e
D.2.3.11 number_pi
D.2.3.12 primes
D.2.3.13 product
D.2.3.14 sort
D.2.3.15 sum
D.2.3.16 watchdog
D.2.3.17 which
D.2.3.18 primecoeffs
D.2.3.19 primefactors
D.2.3.20 timeStd
D.2.3.21 timeFactorize
D.2.3.22 factorH
D.2.4 inout_lib
D.2.4.1 allprint
D.2.4.2 lprint
D.2.4.3 pmat
D.2.4.4 rMacaulay
D.2.4.5 show
D.2.4.6 showrecursive
D.2.4.7 split
D.2.4.8 tab
D.2.4.9 writelist
D.2.4.10 pause
D.2.5 poly_lib
D.2.5.1 cyclic
D.2.5.2 katsura
D.2.5.3 freerank
D.2.5.4 is_zero
D.2.5.5 lcm
D.2.5.6 maxcoef
D.2.5.7 maxdeg
D.2.5.8 maxdeg1
D.2.5.9 mindeg
D.2.5.10 mindeg1
D.2.5.11 normalize
D.2.5.12 rad_con
D.2.5.13 content
D.2.5.14 numerator
D.2.5.15 denominator
D.2.5.16 mod2id
D.2.5.17 id2mod
D.2.5.18 substitute
D.2.5.19 subrInterred
D.2.5.20 newtonDiag
D.2.5.21 hilbPoly
D.2.6 random_lib
D.2.6.1 genericid
D.2.6.2 randomid
D.2.6.3 randommat
D.2.6.4 sparseid
D.2.6.5 sparsematrix
D.2.6.6 sparsemat
D.2.6.7 sparsepoly
D.2.6.8 sparsetriag
D.2.6.9 triagmatrix
D.2.6.10 randomLast
D.2.6.11 randomBinomial
D.2.7 ring_lib
D.2.7.1 changechar
D.2.7.2 changeord
D.2.7.3 changevar
D.2.7.4 defring
D.2.7.5 defrings
D.2.7.6 defringp
D.2.7.7 extendring
D.2.7.8 fetchall
D.2.7.9 imapall
D.2.7.10 mapall
D.2.7.11 ord_test
D.2.7.12 ringtensor
D.2.7.13 ringweights
D.2.7.14 preimageLoc
D.2.7.15 rootofUnity
D.3 Linear algebra
D.3.1 matrix_lib
D.3.1.1 compress
D.3.1.2 concat
D.3.1.3 diag
D.3.1.4 dsum
D.3.1.5 flatten
D.3.1.6 genericmat
D.3.1.7 is_complex
D.3.1.8 outer
D.3.1.9 power
D.3.1.10 skewmat
D.3.1.11 submat
D.3.1.12 symmat
D.3.1.13 tensor
D.3.1.14 unitmat
D.3.1.15 gauss_col
D.3.1.16 gauss_row
D.3.1.17 addcol
D.3.1.18 addrow
D.3.1.19 multcol
D.3.1.20 multrow
D.3.1.21 permcol
D.3.1.22 permrow
D.3.1.23 rowred
D.3.1.24 colred
D.3.1.25 linear_relations
D.3.1.26 rm_unitrow
D.3.1.27 rm_unitcol
D.3.1.28 headStand
D.3.2 linalg_lib
D.3.2.1 inverse
D.3.2.2 inverse_B
D.3.2.3 inverse_L
D.3.2.4 sym_gauss
D.3.2.5 orthogonalize
D.3.2.6 diag_test
D.3.2.7 busadj
D.3.2.8 charpoly
D.3.2.9 adjoint
D.3.2.10 det_B
D.3.2.11 gaussred
D.3.2.12 gaussred_pivot
D.3.2.13 gauss_nf
D.3.2.14 mat_rk
D.3.2.15 U_D_O
D.3.2.16 pos_def
D.3.2.17 hessenberg
D.3.2.18 eigenvals
D.3.2.19 minipoly
D.3.2.20 spnf
D.3.2.21 spprint
D.3.2.22 jordan
D.3.2.23 jordanbasis
D.3.2.24 jordanmatrix
D.3.2.25 jordannf
D.4 Commutative algebra
D.4.1 absfact_lib
D.4.1.1 absFactorize
D.4.2 algebra_lib
D.4.2.1 algebra_containment
D.4.2.2 module_containment
D.4.2.3 inSubring
D.4.2.4 algDependent
D.4.2.5 alg_kernel
D.4.2.6 is_injective
D.4.2.7 is_surjective
D.4.2.8 is_bijective
D.4.2.9 noetherNormal
D.4.2.10 mapIsFinite
D.4.2.11 finitenessTest
D.4.3 cimonom_lib
D.4.3.1 BelongSemig
D.4.3.2 MinMult
D.4.3.3 CompInt
D.4.4 elim_lib
D.4.4.1 blowup0
D.4.4.2 elim
D.4.4.3 elim1
D.4.4.4 nselect
D.4.4.5 sat
D.4.4.6 select
D.4.4.7 select1
D.4.5 grwalk_lib
D.4.5.1 fwalk
D.4.5.2 twalk
D.4.5.3 awalk1
D.4.5.4 awalk2
D.4.5.5 pwalk
D.4.5.6 gwalk
D.4.6 homolog_lib
D.4.6.1 cup
D.4.6.2 cupproduct
D.4.6.3 depth
D.4.6.4 Ext_R
D.4.6.5 Ext
D.4.6.6 fitting
D.4.6.7 flatteningStrat
D.4.6.8 Hom
D.4.6.9 homology
D.4.6.10 isCM
D.4.6.11 isFlat
D.4.6.12 isLocallyFree
D.4.6.13 isReg
D.4.6.14 kernel
D.4.6.15 kohom
D.4.6.16 kontrahom
D.4.6.17 KoszulHomology
D.4.6.18 tensorMod
D.4.6.19 Tor
D.4.7 intprog_lib
D.4.7.1 solve_IP
D.4.8 lll_lib
D.4.8.1 LLL
D.4.9 modstd_lib
D.4.9.1 modStd
D.4.9.2 modS
D.4.9.3 primeList
D.4.10 mprimdec_lib
D.4.10.1 separator
D.4.10.2 PrimdecA
D.4.10.3 PrimdecB
D.4.10.4 modDec
D.4.10.5 zeroMod
D.4.10.6 GTZmod
D.4.10.7 dec1var
D.4.10.8 annil
D.4.10.9 splitting
D.4.10.10 primTest
D.4.10.11 preComp
D.4.10.12 indSet
D.4.10.13 GTZopt
D.4.10.14 zeroOpt
D.4.10.15 clrSBmod
D.4.10.16 minSatMod
D.4.10.17 specialModulesEqual
D.4.10.18 stdModulesEqual
D.4.10.19 modulesEqual
D.4.10.20 getData
D.4.11 mregular_lib
D.4.11.1 regIdeal
D.4.11.2 depthIdeal
D.4.11.3 satiety
D.4.11.4 regMonCurve
D.4.11.5 NoetherPosition
D.4.11.6 is_NP
D.4.11.7 is_nested
D.4.12 noether_lib
D.4.12.1 NPos_test
D.4.12.2 modNpos_test
D.4.12.3 NPos
D.4.12.4 modNPos
D.4.12.5 nsatiety
D.4.12.6 modsatiety
D.4.12.7 regCM
D.4.12.8 modregCM
D.4.13 normal_lib
D.4.13.1 normal
D.4.13.2 HomJJ
D.4.13.3 genus
D.4.13.4 primeClosure
D.4.13.5 closureFrac
D.4.13.6 deltaLoc
D.4.14 primdec_lib
D.4.14.1 Ann
D.4.14.2 primdecGTZ
D.4.14.3 primdecSY
D.4.14.4 minAssGTZ
D.4.14.5 minAssChar
D.4.14.6 testPrimary
D.4.14.7 radical
D.4.14.8 radicalEHV
D.4.14.9 equiRadical
D.4.14.10 prepareAss
D.4.14.11 equidim
D.4.14.12 equidimMax
D.4.14.13 equidimMaxEHV
D.4.14.14 zerodec
D.4.14.15 absPrimdecGTZ
D.4.15 primitiv_lib
D.4.15.1 primitive
D.4.15.2 primitive_extra
D.4.15.3 splitring
D.4.16 realrad_lib
D.4.16.1 realpoly
D.4.16.2 realzero
D.4.16.3 realrad
D.4.17 reesclos_lib
D.4.17.1 ReesAlgebra
D.4.17.2 normalI
D.4.18 resolve_lib
D.4.18.1 blowUp
D.4.18.2 blowUp2
D.4.18.3 Center
D.4.18.4 resolve
D.4.18.5 showBO
D.4.18.6 presentTree
D.4.18.7 showDataTypes
D.4.18.8 blowUpBO
D.4.18.9 createBO
D.4.18.10 CenterBO
D.4.18.11 Delta
D.4.18.12 DeltaList
D.4.19 reszeta_lib
D.4.19.1 intersectionDiv
D.4.19.2 spectralNeg
D.4.19.3 discrepancy
D.4.19.4 zetaDL
D.4.19.5 collectDiv
D.4.19.6 prepEmbDiv
D.4.19.7 abstractR
D.4.20 sagbi_lib
D.4.20.1 sagbiRreduction
D.4.20.2 sagbiSPoly
D.4.20.3 sagbiNF
D.4.20.4 sagbi
D.4.20.5 sagbiPart
D.4.21 sheafcoh_lib
D.4.21.1 truncate
D.4.21.2 CM_regularity
D.4.21.3 sheafCohBGG
D.4.21.4 sheafCohBGG2
D.4.21.5 sheafCoh
D.4.21.6 dimH
D.4.21.7 displayCohom
D.4.22 toric_lib
D.4.22.1 toric_ideal
D.4.22.2 toric_std
D.5 Singularities
D.5.1 alexpoly_lib
D.5.1.1 resolutiongraph
D.5.1.2 totalmultiplicities
D.5.1.3 alexanderpolynomial
D.5.1.4 semigroup
D.5.1.5 proximitymatrix
D.5.1.6 multseq2charexp
D.5.1.7 charexp2multseq
D.5.1.8 charexp2generators
D.5.1.9 charexp2inter
D.5.1.10 charexp2conductor
D.5.1.11 charexp2poly
D.5.1.12 tau_es2
D.5.2 arcpoint_lib
D.5.2.1 nashmult
D.5.2.2 removepower
D.5.2.3 idealsimplify
D.5.2.4 equalJinI
D.5.3 classify_lib
D.5.3.1 basicinvariants
D.5.3.2 classify
D.5.3.3 corank
D.5.3.4 Hcode
D.5.3.5 init_debug
D.5.3.6 internalfunctions
D.5.3.7 milnorcode
D.5.3.8 morsesplit
D.5.3.9 quickclass
D.5.3.10 singularity
D.5.3.11 A_L
D.5.3.12 normalform
D.5.3.13 debug_log
D.5.3.14 swap
D.5.4 curvepar_lib
D.5.4.1 BlowingUp
D.5.4.2 CurveRes
D.5.4.3 CurveParam
D.5.4.4 WSemigroup
D.5.5 deform_lib
D.5.5.1 versal
D.5.5.2 mod_versal
D.5.5.3 lift_kbase
D.5.5.4 lift_rel_kb
D.5.6 equising_lib
D.5.6.1 tau_es
D.5.6.2 esIdeal
D.5.6.3 esStratum
D.5.6.4 isEquising
D.5.6.5 control_Matrix
D.5.7 gmssing_lib
D.5.7.1 gmsring
D.5.7.2 gmsnf
D.5.7.3 gmscoeffs
D.5.7.4 bernstein
D.5.7.5 monodromy
D.5.7.6 spectrum
D.5.7.7 sppairs
D.5.7.8 vfilt
D.5.7.9 vwfilt
D.5.7.10 tmatrix
D.5.7.11 endvfilt
D.5.7.12 sppnf
D.5.7.13 sppprint
D.5.7.14 spadd
D.5.7.15 spsub
D.5.7.16 spmul
D.5.7.17 spissemicont
D.5.7.18 spsemicont
D.5.7.19 spmilnor
D.5.7.20 spgeomgenus
D.5.7.21 spgamma
D.5.8 gmspoly_lib
D.5.8.1 isTame
D.5.8.2 goodBasis
D.5.9 hnoether_lib
D.5.9.1 hnexpansion
D.5.9.2 develop
D.5.9.3 extdevelop
D.5.9.4 param
D.5.9.5 displayHNE
D.5.9.6 invariants
D.5.9.7 displayInvariants
D.5.9.8 multsequence
D.5.9.9 displayMultsequence
D.5.9.10 intersection
D.5.9.11 is_irred
D.5.9.12 delta
D.5.9.13 newtonpoly
D.5.9.14 is_NND
D.5.9.15 stripHNE
D.5.9.16 puiseux2generators
D.5.9.17 separateHNE
D.5.9.18 squarefree
D.5.9.19 allsquarefree
D.5.9.20 further_hn_proc
D.5.10 kskernel_lib
D.5.10.1 KSker
D.5.10.2 KSconvert
D.5.10.3 KSlinear
D.5.10.4 KScoef
D.5.10.5 StringF
D.5.11 mondromy_lib
D.5.11.1 detadj
D.5.11.2 invunit
D.5.11.3 jacoblift
D.5.11.4 monodromyB
D.5.11.5 H2basis
D.5.12 qhmoduli_lib
D.5.12.1 ArnoldAction
D.5.12.2 ModEqn
D.5.12.3 QuotientEquations
D.5.12.4 StabEqn
D.5.12.5 StabEqnId
D.5.12.6 StabOrder
D.5.12.7 UpperMonomials
D.5.12.8 Max
D.5.12.9 Min
D.5.13 sing_lib
D.5.13.1 codim
D.5.13.2 deform
D.5.13.3 dim_slocus
D.5.13.4 is_active
D.5.13.5 is_ci
D.5.13.6 is_is
D.5.13.7 is_reg
D.5.13.8 is_regs
D.5.13.9 locstd
D.5.13.10 milnor
D.5.13.11 nf_icis
D.5.13.12 slocus
D.5.13.13 qhspectrum
D.5.13.14 Tjurina
D.5.13.15 tjurina
D.5.13.16 T_1
D.5.13.17 T_2
D.5.13.18 T_12
D.5.13.19 tangentcone
D.5.14 spcurve_lib
D.5.14.1 isCMcod2
D.5.14.2 CMtype
D.5.14.3 matrixT1
D.5.14.4 semiCMcod2
D.5.14.5 discr
D.5.14.6 qhmatrix
D.5.14.7 relweight
D.5.14.8 posweight
D.5.14.9 KSpencerKernel
D.5.15 spectrum_lib
D.5.15.1 spectrumnd
D.6 Invariant theory
D.6.1 finvar_lib
D.6.1.1 invariant_ring
D.6.1.2 invariant_ring_random
D.6.1.3 primary_invariants
D.6.1.4 primary_invariants_random
D.6.1.5 invariant_algebra_reynolds
D.6.1.6 invariant_algebra_perm
D.6.1.7 cyclotomic
D.6.1.8 group_reynolds
D.6.1.9 molien
D.6.1.10 reynolds_molien
D.6.1.11 partial_molien
D.6.1.12 evaluate_reynolds
D.6.1.13 invariant_basis
D.6.1.14 invariant_basis_reynolds
D.6.1.15 primary_char0
D.6.1.16 primary_charp
D.6.1.17 primary_char0_no_molien
D.6.1.18 primary_charp_no_molien
D.6.1.19 primary_charp_without
D.6.1.20 primary_char0_random
D.6.1.21 primary_charp_random
D.6.1.22 primary_char0_no_molien_random
D.6.1.23 primary_charp_no_molien_random
D.6.1.24 primary_charp_without_random
D.6.1.25 power_products
D.6.1.26 secondary_char0
D.6.1.27 irred_secondary_char0
D.6.1.28 secondary_charp
D.6.1.29 secondary_no_molien
D.6.1.30 irred_secondary_no_molien
D.6.1.31 secondary_and_irreducibles_no_molien
D.6.1.32 secondary_not_cohen_macaulay
D.6.1.33 orbit_variety
D.6.1.34 rel_orbit_variety
D.6.1.35 relative_orbit_variety
D.6.1.36 image_of_variety
D.6.2 ainvar_lib
D.6.2.1 invariantRing
D.6.2.2 derivate
D.6.2.3 actionIsProper
D.6.2.4 reduction
D.6.2.5 completeReduction
D.6.2.6 localInvar
D.6.2.7 furtherInvar
D.6.2.8 sortier
D.6.3 rinvar_lib
D.6.3.1 HilbertSeries
D.6.3.2 HilbertWeights
D.6.3.3 ImageVariety
D.6.3.4 ImageGroup
D.6.3.5 InvariantRing
D.6.3.6 InvariantQ
D.6.3.7 LinearizeAction
D.6.3.8 LinearActionQ
D.6.3.9 LinearCombinationQ
D.6.3.10 MinimalDecomposition
D.6.3.11 NullCone
D.6.3.12 ReynoldsImage
D.6.3.13 ReynoldsOperator
D.6.3.14 SimplifyIdeal
D.6.4 stratify_lib
D.6.4.1 prepMat
D.6.4.2 stratify
D.7 Symbolic-numerical solving
D.7.1 presolve_lib
D.7.1.1 degreepart
D.7.1.2 elimlinearpart
D.7.1.3 elimpart
D.7.1.4 elimpartanyr
D.7.1.5 fastelim
D.7.1.6 findvars
D.7.1.7 hilbvec
D.7.1.8 linearpart
D.7.1.9 tolessvars
D.7.1.10 solvelinearpart
D.7.1.11 sortandmap
D.7.1.12 sortvars
D.7.1.13 valvars
D.7.1.14 idealSplit
D.7.2 solve_lib
D.7.2.1 laguerre_solve
D.7.2.2 solve
D.7.2.3 ures_solve
D.7.2.4 mp_res_mat
D.7.2.5 interpolate
D.7.2.6 fglm_solve
D.7.2.7 lex_solve
D.7.2.8 simplexOut
D.7.2.9 triangLf_solve
D.7.2.10 triangM_solve
D.7.2.11 triangL_solve
D.7.2.12 triang_solve
D.7.3 triang_lib
D.7.3.1 triangL
D.7.3.2 triangLfak
D.7.3.3 triangM
D.7.3.4 triangMH
D.7.4 ntsolve_lib
D.7.4.1 nt_solve
D.7.4.2 triMNewton
D.7.5 zeroset_lib
D.7.5.1 Quotient
D.7.5.2 Remainder
D.7.5.3 Roots
D.7.5.4 SQFRNorm
D.7.5.5 ZeroSet
D.7.5.6 EGCDMain
D.7.5.7 FactorMain
D.7.5.8 InvertNumberMain
D.7.5.9 QuotientMain
D.7.5.10 RemainderMain
D.7.5.11 RootsMain
D.7.5.12 SQFRNormMain
D.7.5.13 ContainedQ
D.7.5.14 SameQ
D.7.6 signcond_lib
D.7.6.1 signcnd
D.7.6.2 psigncnd
D.7.6.3 firstoct
D.8 Visualization
D.8.1 graphics_lib
D.8.1.1 staircase
D.8.1.2 mathinit
D.8.1.3 mplot
D.8.2 latex_lib
D.8.2.1 closetex
D.8.2.2 opentex
D.8.2.3 tex
D.8.2.4 texdemo
D.8.2.5 texfactorize
D.8.2.6 texmap
D.8.2.7 texname
D.8.2.8 texobj
D.8.2.9 texpoly
D.8.2.10 texproc
D.8.2.11 texring
D.8.2.12 rmx
D.8.2.13 xdvi
D.8.3 resgraph_lib
D.8.3.1 InterDiv
D.8.3.2 ResTree
D.8.3.3 finalCharts
D.8.4 surf_lib
D.8.4.1 plot
D.8.5 surfex_lib
D.8.5.1 plotRotated
D.8.5.2 plotRot
D.8.5.3 plotRotatedList
D.8.5.4 plotRotatedDirect
D.8.5.5 plotRotatedListFromSpecifyList
D.9 Coding theory
D.9.1 brnoeth_lib
D.9.1.1 Adj_div
D.9.1.2 NSplaces
D.9.1.3 BrillNoether
D.9.1.4 Weierstrass
D.9.1.5 extcurve
D.9.1.6 AGcode_L
D.9.1.7 AGcode_Omega
D.9.1.8 prepSV
D.9.1.9 decodeSV
D.9.1.10 closed_points
D.9.1.11 dual_code
D.9.1.12 sys_code
D.9.1.13 permute_L
D.10 System and Control theory
D.10.1 Control theory background
D.10.2 control_lib
D.10.2.1 control
D.10.2.2 controlDim
D.10.2.3 autonom
D.10.2.4 autonomDim
D.10.2.5 leftKernel
D.10.2.6 rightKernel
D.10.2.7 leftInverse
D.10.2.8 rightInverse
D.10.2.9 smith
D.10.2.10 colrank
D.10.2.11 genericity
D.10.2.12 canonize
D.10.2.13 iostruct
D.10.2.14 findTorsion
D.10.2.15 controlExample
D.10.2.16 view
D.11 Teaching
D.11.1 aksaka_lib
D.11.1.1 schnellexpt
D.11.1.2 log2
D.11.1.3 PerfectPowerTest
D.11.1.4 wurzel
D.11.1.5 euler
D.11.1.6 coeffmod
D.11.1.7 powerpolyX
D.11.1.8 ask
D.11.2 atkins_lib
D.11.2.1 newTest
D.11.2.2 bubblesort
D.11.2.3 disc
D.11.2.4 Cornacchia
D.11.2.5 CornacchiaModified
D.11.2.6 maximum
D.11.2.7 expo
D.11.2.8 jOft
D.11.2.9 round
D.11.2.10 HilbertClassPoly
D.11.2.11 rootsModp
D.11.2.12 wUnit
D.11.2.13 Atkin
D.11.3 crypto_lib
D.11.3.1 decimal
D.11.3.2 exgcdN
D.11.3.3 eexgcdN
D.11.3.4 gcdN
D.11.3.5 lcmN
D.11.3.6 powerN
D.11.3.7 chineseRem
D.11.3.8 Jacobi
D.11.3.9 primList
D.11.3.10 primL
D.11.3.11 intPart
D.11.3.12 intRoot
D.11.3.13 squareRoot
D.11.3.14 solutionsMod2
D.11.3.15 powerX
D.11.3.16 babyGiant
D.11.3.17 rho
D.11.3.18 MillerRabin
D.11.3.19 SolowayStrassen
D.11.3.20 PocklingtonLehmer
D.11.3.21 PollardRho
D.11.3.22 pFactor
D.11.3.23 quadraticSieve
D.11.3.24 isOnCurve
D.11.3.25 ellipticAdd
D.11.3.26 ellipticMult
D.11.3.27 ellipticRandomCurve
D.11.3.28 ellipticRandomPoint
D.11.3.29 countPoints
D.11.3.30 ellipticAllPoints
D.11.3.31 ShanksMestre
D.11.3.32 Schoof
D.11.3.33 generateG
D.11.3.34 factorLenstraECM
D.11.3.35 ECPP
D.11.4 hyperel_lib
D.11.4.1 ishyper
D.11.4.2 isoncurve
D.11.4.3 chinrestp
D.11.4.4 norm
D.11.4.5 multi
D.11.4.6 divisor
D.11.4.7 gcddivisor
D.11.4.8 semidiv
D.11.4.9 cantoradd
D.11.4.10 cantorred
D.11.4.11 double
D.11.4.12 cantormult
D.11.5 teachstd_lib
D.11.5.1 ecart
D.11.5.2 tail
D.11.5.3 sameComponent
D.11.5.4 leadmonomial
D.11.5.5 monomialLcm
D.11.5.6 spoly
D.11.5.7 minEcart
D.11.5.8 NFMora
D.11.5.9 prodcrit
D.11.5.10 chaincrit
D.11.5.11 pairset
D.11.5.12 updatePairs
D.11.5.13 standard
D.11.5.14 localstd
D.11.6 weierstr_lib
D.11.6.1 weierstrDiv
D.11.6.2 weierstrPrep
D.11.6.3 lastvarGeneral
D.11.6.4 generalOrder
D.11.7 rootsmr_lib
D.11.7.1 nrRootsProbab
D.11.7.2 nrRootsDeterm
D.11.7.3 symsignature
D.11.7.4 sturmquery
D.11.7.5 matbil
D.11.7.6 matmult
D.11.7.7 tracemult
D.11.7.8 coords
D.11.7.9 randcharpoly
D.11.7.10 verify
D.11.7.11 randlinpoly
D.11.7.12 powersums
D.11.7.13 symmfunc
D.11.7.14 univarpoly
D.11.7.15 qbase
D.11.8 rootsur_lib
D.11.8.1 isuni
D.11.8.2 whichvariable
D.11.8.3 varsigns
D.11.8.4 boundBuFou
D.11.8.5 boundposDes
D.11.8.6 boundDes
D.11.8.7 allrealst
D.11.8.8 maxabs
D.11.8.9 allreal
D.11.8.10 sturm
D.11.8.11 sturmseq
D.11.8.12 sturmha
D.11.8.13 sturmhaseq
D.11.8.14 reverse
D.11.8.15 nrroots
D.11.8.16 isparam
E. Release Notes
E.1 News and changes
E.2 Downloading instructions
E.3 Unix installation instructions
E.4 Windows installation instructions
E.5 Macintosh installation instructions
F. Index
User manual for
Singular
version 3-0-4, Nov 2007, generated by
texi2html
.