Singular Manual: 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

3.3.4 Coefficient rings

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 Parameter list

3.7.3 Help string

3.7.4 Names in procedures

3.7.5 Procedure-specific commands

3.8 Libraries

3.8.1 Libraries in the SINGULAR Documentation

3.8.2 Version string

3.8.3 Category string

3.8.4 Info string

3.8.5 LIB commands

3.8.6 Procedures in a library

3.8.9 Documentation Tool

3.8.10 Typesetting of help and info strings

3.8.11 Loading a library

3.9 Debugging tools

3.9.1 ASSUME

3.9.2 Tracing of procedures

3.9.3 Source code debugger

3.9.4 Break points

3.9.5 Printing of data

4.1.1 bigint declarations

4.1.2 bigint expressions

4.1.3 bigint operations

4.1.4 bigint related functions

4.2 bigintmat

4.2.1 bigintmat declarations

4.2.2 bigintmat expressions

4.2.3 bigintmat type cast

4.2.4 bigintmat operations

4.3 def

4.3.1 def declarations

4.4 ideal

4.4.1 ideal declarations

4.4.2 ideal expressions

4.4.3 ideal operations

4.4.4 ideal related functions

4.5 int

4.5.1 int declarations

4.5.2 int expressions

4.5.3 int operations

4.5.4 int related functions

4.5.5 boolean expressions

4.5.6 boolean operations

4.6 intmat

4.6.1 intmat declarations

4.6.2 intmat expressions

4.6.3 intmat type cast

4.6.4 intmat operations

4.6.5 intmat related functions

4.7 intvec

4.7.1 intvec declarations

4.7.2 intvec expressions

4.7.3 intvec operations

4.7.4 intvec related functions

4.8 link

4.8.1 link declarations

4.8.2 link expressions

4.8.3 link related functions

4.8.4 ASCII links

4.8.5 Ssi links

4.8.5.1 Ssi file links

4.8.5.2 Ssi tcp links

4.8.6 Pipe links

4.8.7 DBM links

4.9 list

4.9.1 list declarations

4.9.2 list expressions

4.9.3 list operations

4.9.4 list related functions

4.10 map

4.10.1 map declarations

4.10.2 map expressions

4.10.3 map operations

4.10.4 map related functions

4.11 matrix

4.11.1 matrix declarations

4.11.2 matrix expressions

4.11.3 matrix type cast

4.11.4 matrix operations

4.11.5 matrix related functions

4.12 module

4.12.1 module declarations

4.12.2 module expressions

4.12.3 module operations

4.12.4 module related functions

4.13 number

4.13.1 number declarations

4.13.2 number expressions

4.13.3 number operations

4.13.4 number related functions

4.14 package

4.14.1 package declarations

4.14.2 package related functions

4.15 poly

4.15.1 poly declarations

4.15.2 poly expressions

4.15.3 poly operations

4.15.4 poly related functions

4.16 proc

4.16.1 proc declaration

4.16.2 proc expression

4.16.3 procs with different argument types

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 qring

4.18.2 ring declarations

4.18.3 ring related functions

4.18.4 ring operations

4.18.5 qring declaration

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

4.21 User defined types

4.21.1 Definition of a user defined type

4.21.2 Declaration of objects of a user defined type

4.21.3 Access to elements of a user defined type

4.21.4 Commands for user defined types

4.21.5 Assignments for user defined types

4.22 cone

4.22.1 coneViaPoints

4.22.2 coneViaInequalities

4.22.3 cone related functions

4.22.3.1 ambientDimension

4.22.3.2 canonicalizeCone

4.22.3.3 codimension

4.22.3.4 coneLink

4.22.3.5 containsAsFace

4.22.3.6 containsInSupport

4.22.3.7 containsPositiveVector

4.22.3.8 containsRelatively

4.22.3.9 convexHull

4.22.3.10 convexIntersection

4.22.3.15 generatorsOfLinealitySpace

4.22.3.16 generatorsOfSpan

4.22.3.17 getLinearForms

4.22.3.18 getMultiplicity

4.22.3.19 inequalities

4.22.3.20 isFullSpace

4.22.3.21 isOrigin

4.22.3.22 isSimplicial

4.22.3.23 linealityDimension

4.22.3.24 linealitySpace

4.22.3.25 negatedCone

4.22.3.26 quotientLatticeBasis

4.22.3.27 randomPoint

4.22.3.28 rays

4.22.3.29 relativeInteriorPoint

4.22.3.30 semigroupGenerator

4.22.3.31 setLinearForms

4.22.3.32 setMultiplicity

4.22.3.33 span

4.22.3.34 uniquePoint

4.23 fan

4.23.1 fan related functions

4.23.1.1 containsInCollection

4.23.1.8 isCompatible

4.23.1.9 isPure

4.23.1.10 nmaxcones

4.23.1.11 ncones

4.23.1.12 numberOfConesOfDimension

4.23.1.13 removeCone

4.24 polytope

4.24.1 polytopeViaPoints

4.24.2 polytopeViaInequalities

4.24.3 polytope related functions

4.24.3.1 dualPolytope

4.24.3.2 newtonPolytope

4.24.3.3 vertices

4.25 pyobject

4.25.1 pyobject declarations

4.25.2 pyobject expressions

4.25.3 pyobject operations

4.25.4 pyobject related functions

4.25.5 python_eval

4.25.6 python_import

4.25.7 python_run

4.26 reference and shared (experimental)

4.26.1 reference declarations

4.26.2 reference expressions

4.26.3 shared declarations

4.26.4 shared expressions

4.26.5 reference and shared operations

4.26.6 reference and shared related functions

5. Functions and system variables

5.1.39 files, input from

5.2 Control structures

5.2.16 ~ (break point)

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

7.1 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 dim (plural)

7.3.4 division (plural)

7.3.5 eliminate (plural)

7.3.6 envelope

7.3.7 fetch (plural)

7.3.8 imap (plural)

7.3.9 intersect (plural)

7.3.10 kbase (plural)

7.3.11 lift (plural)

7.3.12 liftstd (plural)

7.3.13 minres (plural)

7.3.14 modulo (plural)

7.3.21 preimage (plural)

7.3.22 quotient (plural)

7.3.23 reduce (plural)

7.3.24 ringlist (plural)

7.3.25 slimgb (plural)

7.3.26 std (plural)

7.3.27 subst (plural)

7.3.28 syz (plural)

7.3.29 twostd

7.3.30 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 Graded commutative algebras (SCA)

7.6 LETTERPLACE

7.6.1 Free associative algebras

7.6.2 Groebner bases for two-sided ideals in free associative algebras

7.6.3 Letterplace correspondence

7.6.4 Example of use of LETTERPLACE

7.6.5 Release notes of LETTERPLACE

7.7 Non-commutative libraries

7.7.1 bimodules_lib

7.7.1.0. bistd

7.7.1.0. bitrinity

7.7.1.0. liftenvelope

7.7.1.0. CompDecomp

7.7.1.0. isPureTensor

7.7.1.0. isTwoSidedGB

7.7.2.0. pIntersectSyz

7.7.2.0. linReduce

7.7.2.0. linReduceIdeal

7.7.3.0. centralizeSet

7.7.3.0. centralizerVS

7.7.3.0. centralizerRed

7.7.3.0. sa_poly_reduce

7.7.3.0. inCenter

7.7.3.0. inCentralizer

7.7.3.0. isCartan

7.7.3.0. applyAdF

7.7.3.0. linearMapKernel

7.7.3.0. linearCombinations

7.7.3.0. variablesStandard

7.7.3.0. variablesSorted

7.7.3.0. PBW_eqDeg

7.7.3.0. PBW_maxDeg

7.7.3.0. PBW_maxMonom

7.7.4.0. bernsteinLift

7.7.4.0. operatorBM

7.7.4.0. operatorModulo

7.7.4.0. annfsParamBM

7.7.5.0. initialMalgrange

7.7.5.0. initialIdealW

7.7.5.0. inForm

7.7.5.0. restrictionIdeal

7.7.5.0. restrictionModule

7.7.5.0. integralIdeal

7.7.5.0. integralModule

7.7.5.0. deRhamCohom

7.7.5.0. deRhamCohomIdeal

7.7.5.0. inverseFourier

7.7.5.0. insertGenerator

7.7.5.0. deleteGenerator

7.7.6.0. makeMalgrange

7.7.7 fpadim_lib

7.7.7.0. ivDHilbert

7.7.7.0. ivDHilbertSickle

7.7.7.0. lpDHilbertSickle

7.7.8.0. makeLetterplaceRing

7.7.8.0. letplaceGBasis

7.7.8.0. lpNF

7.7.8.0. freeGBasis

7.7.8.0. setLetterplaceAttributes

7.7.8.0. serreRelations

7.7.8.0. fullSerreRelations

7.7.8.0. isVar

7.7.8.0. ademRelations

7.7.9 involut_lib

7.7.9.0. findInvo

7.7.9.0. findInvoDiag

7.7.9.0. findAuto

7.7.9.0. ncdetection

7.7.9.0. involution

7.7.9.0. isInvolution

7.7.11.0. Qso3Casimir

7.7.11.0. GKZsystem

7.7.12 ncdecomp_lib

7.7.12.0. CentralQuot

7.7.12.0. CentralSaturation

7.7.12.0. CenCharDec

7.7.12.0. IntersectWithSub

7.7.13 ncfactor_lib

7.7.13.0. facWeyl

7.7.13.0. facFirstWeyl

7.7.13.0. testNCfac

7.7.13.0. facSubWeyl

7.7.13.0. facShift

7.7.13.0. facFirstShift

7.7.13.0. homogfacNthQWeyl

7.7.13.0. homogfacFirstQWeyl

7.7.13.0. homogfacNthQWeyl_all

7.7.13.0. homogfacFirstQWeyl_all

7.7.13.0. tst_ncfactor

7.7.14 ncpreim_lib

7.7.14.0. eliminateNC

7.7.14.0. preimageNC

7.7.14.0. admissibleSub

7.7.14.0. isUpperTriangular

7.7.14.0. appendWeight2Ord

7.7.14.0. elimWeight

7.7.14.0. extendedTensor

7.7.15 nctools_lib

7.7.15.0. Gweights

7.7.15.0. weightedRing

7.7.15.0. ndcond

7.7.15.0. Weyl

7.7.15.0. makeWeyl

7.7.15.0. makeHeisenberg

7.7.15.0. Exterior

7.7.15.0. findimAlgebra

7.7.15.0. superCommutative

7.7.15.0. rightStd

7.7.15.0. rightNF

7.7.15.0. rightModulo

7.7.15.0. moduloSlim

7.7.15.0. ncRelations

7.7.15.0. isCentral

7.7.15.0. isNC

7.7.15.0. isCommutative

7.7.15.0. isWeyl

7.7.15.0. UpOneMatrix

7.7.15.0. AltVarStart

7.7.15.0. AltVarEnd

7.7.15.0. IsSCA

7.7.15.0. makeModElimRing

7.7.17.0. LengthSymElement

7.7.17.0. LengthSym

7.7.18 purityfiltration_lib

7.7.18.0. projectiveDimension

7.7.18.0. purityFiltration

7.7.18.0. purityTriang

7.7.18.0. gradeNumber

7.7.18.0. showgrades

7.7.18.0. allExtOfLeft

7.7.18.0. allExtOfRight

7.7.18.0. doubleExt

7.7.18.0. allDoubleExt

7.7.19.0. dmodoublext

7.7.19.0. is_cenBimodule

7.7.19.0. is_cenSubbimodule

7.7.20 ratgb_lib

7.7.20.0. ratstd

7.7.21 dmodloc_lib

7.7.21.0. Dlocalization

7.7.21.0. WeylClosure

7.7.21.0. WeylClosure1

7.7.21.0. holonomicRank

7.7.21.0. DsingularLocus

7.7.21.0. polSol

7.7.21.0. polSolFiniteRank

7.7.21.0. ratSol

7.7.21.0. bfctBound

7.7.21.0. annRatSyz

7.7.21.0. dmodGeneralAssumptionCheck

7.7.21.0. extendWeyl

7.7.21.0. polyVars

7.7.21.0. monomialInIdeal

7.7.21.0. vars2pars

7.7.21.0. minIntRoot2

7.7.21.0. maxIntRoot

7.7.21.0. dmodAction

7.7.21.0. dmodActionRat

7.7.21.0. simplifyRat

7.7.21.0. rightNFWeyl

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 ssi 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 Non-commutative 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 Decoding codes with Groebner bases

C.8.1 Codes and the decoding problem

C.8.2 Cooper philosophy

C.8.3 Generalized Newton identities

C.8.4 Fitzgerald-Lax method

C.8.5 Decoding method based on quadratic equations

C.8.6 References for decoding with Groebner bases

C.9 References

D. SINGULAR libraries

D.2.3.5 deleteSublist

D.2.3.19 timeFactorize

D.2.5.6 showrecursive

D.2.7.1 parallelWaitN

D.2.7.2 parallelWaitFirst

D.2.7.3 parallelWaitAll

D.2.7.4 parallelTestAND

D.2.7.5 parallelTestOR

D.2.8.18 subrInterred

D.2.9.9 sparseHomogIdeal

D.2.9.10 triagmatrix

D.2.9.11 randomLast

D.2.9.12 randomBinomial

D.2.11.13 ringweights

D.2.11.14 preimageLoc

D.2.11.15 rootofUnity

D.2.11.16 optionIsSet

D.2.11.17 hasNumericCoeffs

D.2.11.18 hasCommutativeVars

D.2.11.19 hasGlobalOrdering

D.2.11.20 hasMixedOrdering

D.2.11.21 hasFieldCoefficient

D.2.11.22 hasAlgExtensionCoefficient

D.2.11.23 isQuotientRing

D.2.11.24 isSubModule

D.2.11.25 changeordTo

D.2.13.4 compareTasks

D.2.13.9 waitAllTasks

D.2.13.10 pollTask

D.2.13.11 getCommand

D.2.13.12 getArguments

D.3.1.25 linear_relations

D.3.1.26 rm_unitrow

D.3.1.27 rm_unitcol

D.3.1.28 headStand

D.3.1.29 symmetricBasis

D.3.1.30 exteriorBasis

D.3.1.31 symmetricPower

D.3.1.32 exteriorPower

D.3.2.5 orthogonalize

D.3.2.12 gaussred_pivot

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.4.1.2 absFactorizeBCG

D.4.2 algebra_lib

D.4.2.1 algebra_containment

D.4.2.2 module_containment

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.4.2.12 nonZeroEntry

D.4.3 assprimeszerodim_lib

D.4.3.1 zeroRadical

D.4.3.2 assPrimes

D.4.4 cisimplicial_lib

D.4.4.1 minMult

D.4.4.2 belongSemigroup

D.4.4.3 oneDimBelongSemigroup

D.4.6 ellipticcovers_lib

D.4.6.1 makeGraph

D.4.6.2 printGraph

D.4.6.3 propagator

D.4.6.4 computeConstant

D.4.6.5 evalutateIntegral

D.4.6.6 gromovWitten

D.4.6.7 computeGromovWitten

D.4.8.8 flatteningStrat

D.4.8.13 isLocallyFree

D.4.8.18 KoszulHomology

D.4.8.19 tensorMod

D.4.8.20 Tor

D.4.9 integralbasis_lib

D.4.9.1 integralBasis

D.4.14 monomialideal_lib

D.4.14.5 membershipMon

D.4.14.6 intersectMon

D.4.14.7 quotientMon

D.4.14.8 radicalMon

D.4.14.9 isprimeMon

D.4.14.10 isprimaryMon

D.4.14.11 isirreducibleMon

D.4.14.12 isartinianMon

D.4.14.13 isgenericMon

D.4.14.14 dimMon

D.4.14.15 irreddecMon

D.4.16.5 NoetherPosition

D.4.17.2 modNpos_test

D.4.18.6 primeClosure

D.4.18.12 getSmallest

D.4.18.13 getOneVar

D.4.18.14 changeDenominator

D.4.19 normaliz_lib

D.4.19.1 intclToricRing

D.4.19.2 normalToricRing

D.4.19.3 normalToricRingFromBinomials

D.4.19.4 ehrhartRing

D.4.19.5 intclMonIdeal

D.4.19.6 torusInvariants

D.4.19.7 finiteDiagInvariants

D.4.19.8 diagInvariants

D.4.19.9 intersectionValRings

D.4.19.10 intersectionValRingIdeals

D.4.19.11 showNuminvs

D.4.19.12 exportNuminvs

D.4.19.13 setNmzOption

D.4.19.14 showNmzOptions

D.4.19.15 normaliz

D.4.19.16 setNmzExecPath

D.4.19.17 writeNmzData

D.4.19.18 readNmzData

D.4.19.19 setNmzFilename

D.4.19.20 setNmzDataPath

D.4.19.21 writeNmzPaths

D.4.19.22 startNmz

D.4.19.23 rmNmzFiles

D.4.19.24 mons2intmat

D.4.19.25 intmat2mons

D.4.19.26 binomials2intmat

D.4.20 pointid_lib

D.4.20.1 nonMonomials

D.4.20.2 cornerMonomials

D.4.21.10 testPrimary

D.4.21.11 testPrimaryE

D.4.21.12 radical

D.4.21.13 radicalEHV

D.4.21.14 equiRadical

D.4.21.15 prepareAss

D.4.21.16 equidim

D.4.21.17 equidimMax

D.4.21.18 equidimMaxEHV

D.4.21.19 zerodec

D.4.21.20 absPrimdecGTZ

D.4.21.21 absPrimdecGTZE

D.4.21.22 sep

D.4.22 primdecint_lib

D.4.23.2 primitive_extra

D.4.26.5 algebraicDependence

D.5 Algebraic geometry

D.5.2.3 Singular2bertini

D.5.2.4 bertini2Singular

D.5.2.5 ReJunkUseHomo

D.5.2.12 NumPrimDecom

D.5.3.4 parametrizeOrbit

D.5.3.5 maxZeros

D.5.4 paraplanecurves_lib

D.5.4.1 adjointIdeal

D.5.4.2 invertBirMap

D.5.4.3 paraPlaneCurve

D.5.4.4 rncAntiCanonicalMap

D.5.4.5 rationalPointConic

D.5.4.6 mapToRatNormCurve

D.5.4.7 rncItProjOdd

D.5.4.8 rncItProjEven

D.5.4.9 paraConic

D.5.4.10 testParametrization

D.5.4.11 testPointConic

D.5.5 resbinomial_lib

D.5.5.1 BINresol

D.5.5.2 Eresol

D.5.5.3 determinecenter

D.5.5.22 constructblwup

D.5.5.23 constructlastblwup

D.5.5.29 multiplylist

D.5.8.7 showDataTypes

D.5.9.1 intersectionDiv

D.5.10.2 truncateFast

D.5.10.3 CM_regularity

D.5.10.4 sheafCohBGG

D.5.10.5 sheafCohBGG2

D.5.10.6 sheafCoh

D.5.10.7 dimH

D.5.10.8 dimGradedPart

D.5.10.9 displayCohom

D.5.11 JMBTest_lib

D.5.11.1 Minimus

D.5.11.2 Maximus

D.5.11.3 StartOrderingV

D.5.11.4 TestJMark

D.5.12 JMSConst_lib

D.5.12.1 BorelCheck

D.5.12.2 JMarkedScheme

D.6 Singularities

D.6.1 alexpoly_lib

D.6.1.1 resolutiongraph

D.6.1.2 totalmultiplicities

D.6.1.3 alexanderpolynomial

D.6.1.4 semigroup

D.6.1.5 proximitymatrix

D.6.1.6 multseq2charexp

D.6.1.7 charexp2multseq

D.6.1.8 charexp2generators

D.6.1.9 charexp2inter

D.6.1.10 charexp2conductor

D.6.1.11 charexp2poly

D.6.2.3 idealsimplify

D.6.2.4 equalJinI

D.6.3 classify_lib

D.6.3.1 basicinvariants

D.6.3.6 internalfunctions

D.6.3.16 complexSingType

D.6.3.17 prepRealclassify

D.6.4 classifyceq_lib

D.6.6.6 MultiplicitySequence

D.6.6.7 CharacteristicExponents

D.6.6.8 IntersectionMatrix

D.6.6.9 ContactMatrix

D.6.6.10 plainInvariants

D.6.8.5 control_Matrix

D.6.9.17 spissemicont

D.6.11.7 displayInvariants

D.6.11.8 multsequence

D.6.11.9 displayMultsequence

D.6.11.10 intersection

D.6.11.16 puiseux2generators

D.6.11.17 separateHNE

D.6.11.18 squarefree

D.6.11.19 allsquarefree

D.6.11.20 further_hn_proc

D.6.14.1 ArnoldAction

D.6.14.2 ModEqn

D.6.14.3 QuotientEquations

D.6.14.4 StabEqn

D.6.14.5 StabEqnId

D.6.14.6 StabOrder

D.6.14.7 UpperMonomials

D.6.15.19 tangentcone

D.6.16.9 KSpencerKernel

D.6.17 spectrum_lib

D.6.17.1 spectrumnd

D.6.18 surfacesignature_lib

D.6.18.1 signatureBrieskorn

D.6.18.2 signaturePuiseux

D.6.18.3 signatureNemethi

D.7 Invariant theory

D.7.1 finvar_lib

D.7.1.1 invariant_ring

D.7.1.2 invariant_ring_random

D.7.1.3 primary_invariants

D.7.1.4 primary_invariants_random

D.7.1.5 invariant_algebra_reynolds

D.7.1.6 invariant_algebra_perm

D.7.1.7 cyclotomic

D.7.1.8 group_reynolds

D.7.1.9 molien

D.7.1.10 reynolds_molien

D.7.1.11 partial_molien

D.7.1.12 evaluate_reynolds

D.7.1.13 invariant_basis

D.7.1.14 invariant_basis_reynolds

D.7.1.15 primary_char0

D.7.1.16 primary_charp

D.7.1.17 primary_char0_no_molien

D.7.1.18 primary_charp_no_molien

D.7.1.19 primary_charp_without

D.7.1.20 primary_char0_random

D.7.1.21 primary_charp_random

D.7.1.22 primary_char0_no_molien_random

D.7.1.23 primary_charp_no_molien_random

D.7.1.24 primary_charp_without_random

D.7.1.25 power_products

D.7.1.26 secondary_char0

D.7.1.27 irred_secondary_char0

D.7.1.28 secondary_charp

D.7.1.29 secondary_no_molien

D.7.1.30 irred_secondary_no_molien

D.7.1.31 secondary_and_irreducibles_no_molien

D.7.1.32 secondary_not_cohen_macaulay

D.7.1.33 orbit_variety

D.7.1.34 rel_orbit_variety

D.7.1.35 relative_orbit_variety

D.7.1.36 image_of_variety

D.7.2 ainvar_lib

D.7.2.1 invariantRing

D.7.2.2 derivate

D.7.2.3 actionIsProper

D.7.2.4 reduction

D.7.2.5 completeReduction

D.7.3.1 HilbertSeries

D.7.3.2 HilbertWeights

D.7.3.3 ImageVariety

D.7.3.4 ImageGroup

D.7.3.5 InvariantRing

D.7.3.6 InvariantQ

D.7.3.7 LinearizeAction

D.7.3.8 LinearActionQ

D.7.3.9 LinearCombinationQ

D.7.3.10 MinimalDecomposition

D.7.3.11 NullCone

D.7.3.12 ReynoldsImage

D.7.3.13 ReynoldsOperator

D.7.3.14 SimplifyIdeal

D.7.4 stratify_lib

D.7.4.1 prepMat

D.7.4.2 stratify

D.8 Symbolic-numerical solving

D.8.2.2 elimlinearpart

D.8.2.10 solvelinearpart

D.8.3.1 laguerre_solve

D.8.3.9 triangLf_solve

D.8.3.10 triangM_solve

D.8.3.11 triangL_solve

D.8.3.12 triang_solve

D.8.6.8 invertNumberMain

D.8.6.9 quotientMain

D.8.6.10 remainderMain

D.8.6.11 rootsMain

D.8.6.12 sqfrNormMain

D.9.4.3 plotRotatedList

D.9.4.4 plotRotatedDirect

D.9.4.5 plotRotatedListFromSpecifyList

D.10.1.3 BrillNoether

D.10.1.4 Weierstrass

D.10.1.5 extcurve

D.10.1.6 AGcode_L

D.10.1.7 AGcode_Omega

D.10.1.8 prepSV

D.10.1.9 decodeSV

D.10.1.10 closed_points

D.10.2.2 sysCRHTMindist

D.10.2.10 randomCheck

D.10.2.11 genMDSMat

D.10.2.12 mindist

D.10.2.13 decode

D.10.2.14 decodeRandom

D.10.2.15 decodeCode

D.10.2.16 vanishId

D.10.2.17 sysFL

D.10.2.18 decodeRandomFL

D.11 System and Control theory

D.11.1 Control theory background

D.11.2.8 rightInverse

D.11.2.13 findTorsion

D.11.2.14 controlExample

D.11.4.1 findifs_example

D.12.1.3 PerfectPowerTest

D.12.2.5 CornacchiaModified

D.12.2.11 HilbertClassPoly

D.12.3.12 solutionsMod2

D.12.3.13 powerX

D.12.3.14 babyGiant

D.12.3.15 rho

D.12.3.16 MillerRabin

D.12.3.17 SolowayStrassen

D.12.3.18 PocklingtonLehmer

D.12.3.19 PollardRho

D.12.3.20 pFactor

D.12.3.21 quadraticSieve

D.12.3.22 isOnCurve

D.12.3.23 ellipticAdd

D.12.3.24 ellipticMult

D.12.3.25 ellipticRandomCurve

D.12.3.26 ellipticRandomPoint

D.12.3.27 countPoints

D.12.3.28 ellipticAllPoints

D.12.3.29 ShanksMestre

D.12.3.30 Schoof

D.12.3.31 generateG

D.12.3.32 factorLenstraECM

D.12.3.33 ECPP

D.12.3.34 calculate_ordering

D.12.3.35 is_primitive_root

D.12.3.36 find_first_primitive_root

D.12.3.37 binary_add

D.12.3.38 inverse_modulus

D.12.3.39 is_prime

D.12.3.40 find_index

D.12.3.41 subset_sum01

D.12.3.42 subset_sum02

D.12.3.43 unbounded_knapsack

D.12.3.44 multidimensional_knapsack

D.12.3.45 naccache_stern_generation

D.12.3.46 naccache_stern_encryption

D.12.3.47 naccache_stern_decryption

D.12.3.48 m_merkle_hellman_transformation

D.12.3.49 m_merkle_hellman_encryption

D.12.3.50 m_merkle_hellman_decryption

D.12.3.51 merkle_hellman_encryption

D.12.3.52 merkle_hellman_decryption

D.12.3.53 super_increasing_knapsack

D.12.3.54 h_increasing_knapsack

D.12.3.55 injective_knapsack

D.12.3.56 calculate_max_sum

D.12.3.57 set_is_injective

D.12.3.58 is_h_injective

D.12.3.59 is_fix_injective

D.12.3.60 three_elements

D.12.5.3 sameComponent

D.12.5.4 leadmonomial

D.12.5.12 updatePairs

D.12.6.2 weierstrPrep

D.12.6.3 lastvarGeneral

D.12.6.4 generalOrder

D.12.7 rootsmr_lib

D.12.7.1 nrRootsProbab

D.12.7.2 nrRootsDeterm

D.12.7.3 symsignature

D.12.7.9 randcharpoly

D.12.7.10 verify

D.12.7.11 randlinpoly

D.12.8.2 whichvariable

D.13 Tropical Geometry

D.13.1 polymake_lib

D.13.1.1 polymakePolytope

D.13.1.2 newtonPolytopeP

D.13.1.3 newtonPolytopeLP

D.13.1.4 normalFanL

D.13.1.5 groebnerFanP

D.13.1.6 triangulations

D.13.1.7 secondaryPolytope

D.13.1.8 secondaryFan

D.13.1.9 cycleLength

D.13.1.10 splitPolygon

D.13.1.11 eta

D.13.1.12 findOrientedBoundary

D.13.1.13 cyclePoints

D.13.1.14 latticeArea

D.13.1.15 picksFormula

D.13.1.16 ellipticNF

D.13.1.17 ellipticNFDB

D.13.2 realizationMatroids_lib

D.13.2.1 realizationDim

D.13.2.2 irrRealizationDim

D.13.2.3 realizationDimPoly

D.13.3 tropical_lib

D.13.3.1 tropicalLifting

D.13.3.2 displayTropicalLifting

D.13.3.3 puiseuxExpansion

D.13.3.4 displayPuiseuxExpansion

D.13.3.5 tropicalCurve

D.13.3.6 drawTropicalCurve

D.13.3.7 drawNewtonSubdivision

D.13.3.8 tropicalJInvariant

D.13.3.9 weierstrassForm

D.13.3.10 jInvariant

D.13.3.11 conicWithTangents

D.13.3.12 tropicalise

D.13.3.13 tropicaliseSet

D.13.3.14 tInitialForm

D.13.3.15 tInitialIdeal

D.13.3.16 initialForm

D.13.3.17 initialIdeal

D.13.3.18 texNumber

D.13.3.19 texPolynomial

D.13.3.20 texMatrix

D.13.3.21 texDrawBasic

D.13.3.22 texDrawTropical

D.13.3.23 texDrawNewtonSubdivision

D.13.3.24 texDrawTriangulation

D.13.3.25 radicalMemberShip

D.13.3.26 tInitialFormPar

D.13.3.27 tInitialFormParMax

D.13.3.28 solveTInitialFormPar

D.13.3.29 detropicalise

D.13.3.30 tDetropicalise

D.13.3.31 dualConic

D.13.3.32 parameterSubstitute

D.13.3.33 tropicalSubst

D.13.3.34 randomPolyInT

D.13.3.35 cleanTmp

D.14 Miscellaneous libraries

D.14.1 phindex_lib

D.14.1.1 signatureL

D.14.1.2 signatureLqf

D.14.2.2 boolean_poly_ring

D.14.2.3 boolean_constant

D.14.2.4 boolean_poly

D.14.2.5 direct_boolean_poly

D.14.2.6 recursive_boolean_poly

D.14.2.7 boolean_ideal

D.14.2.8 boolean_set

D.14.2.9 from_boolean_constant

D.14.2.10 from_boolean_poly

D.14.2.11 direct_from_boolean_poly

D.14.2.12 recursive_from_boolean_poly

D.14.2.13 from_boolean_ideal

D.14.2.14 from_boolean_set

D.15 Experimental libraries

D.15.1.5 arrPrintMatrix

D.15.1.12 arrCentered

D.15.1.13 arrCentralize

D.15.1.14 arrCoordChange

D.15.1.15 arrCoordNormalize

D.15.1.16 arrCone

D.15.1.17 arrDecone

D.15.1.18 arrLocalize

D.15.1.19 arrRestrict

D.15.1.20 arrIsEssential

D.15.1.21 arrEssentialize

D.15.1.27 arrRandomCentral

D.15.1.28 arrEdelmanReiner

D.15.1.29 arrOrlikSolomon

D.15.1.30 arrDer

D.15.1.31 arrIsFree

D.15.1.32 arrExponents

D.15.1.33 arr2multarr

D.15.1.34 multarr2arr

D.15.1.35 multarrRestrict

D.15.1.36 multarrMultRestrict

D.15.1.37 arrFlats

D.15.1.38 arrLattice

D.15.1.39 moebius

D.15.1.40 arrCharPoly

D.15.1.41 arrPoincare

D.15.1.42 arrChambers

D.15.1.43 arrBoundedChambers

D.15.1.44 printMoebius

D.15.2 brillnoether_lib

D.15.2.1 RiemannRochBN

D.15.3 chern_lib

D.15.3.1 symm

D.15.3.2 symNsym

D.15.3.3 CompleteHomog

D.15.3.8 sum_of_powers

D.15.3.13 ChernRootsSum

D.15.3.14 chSum

D.15.3.15 ChernRootsDual

D.15.3.16 chDual

D.15.3.17 ChernRootsProd

D.15.3.22 ChernRootsHom

D.15.3.23 chHom

D.15.3.24 ChernRootsSymm

D.15.3.25 ChernRootsWedge

D.15.3.41 eulerChProj

D.15.3.42 chNumbersProj

D.15.3.43 classpoly

D.15.3.44 chernPoly

D.15.3.45 chernCharPoly

D.15.5.4 negativedivisor

D.15.5.5 normalForm

D.15.5.6 isEqualDivisor

D.15.5.7 globalSections

D.15.5.8 degreeDivisor

D.15.5.9 linearlyEquivalent

D.15.5.10 effective

D.15.5.11 makeFormalDivisor

D.15.5.12 evaluateFormalDivisor

D.15.5.13 formaldivisorplus

D.15.5.14 negativeformaldivisor

D.15.5.15 multformaldivisor

D.15.5.16 degreeFormalDivisor

D.15.5.17 makePDivisor

D.15.5.18 evaluatePDivisor

D.15.5.19 pdivisorplus

D.15.6 finitediff_lib

D.15.6.13 setinitials

D.15.6.14 errormap

D.15.6.15 matrixsystem

D.15.6.16 timestep

D.15.6.17 fouriersystem

D.15.6.18 PartitionVar

D.15.6.19 ComplexValue

D.15.6.20 VarToPar

D.15.6.21 ParToVar

D.15.6.22 qepcad

D.15.6.23 qepcadsystem

D.15.7 graal_lib

D.15.7.1 graalMixed

D.15.7.2 dimensionOfLocalization

D.15.7.3 systemOfParametersOfLocalization

D.15.7.4 isLocalizationRegular

D.15.7.5 warkedPreimageStd

D.15.7.6 resolutionInLocalization

D.15.9.1 desingularization

D.15.10 gradedModules_lib

D.15.10.11 grtranspose

D.15.10.12 grgens

D.15.10.13 grpres

D.15.10.14 grorder

D.15.10.15 grtranspose1

D.15.10.16 TestGRRes

D.15.10.17 KeneshlouMatrixPresentation

D.15.10.22 grgroebner

D.15.10.28 grlifting2

D.15.10.29 mappingcone

D.15.10.30 grlifting3

D.15.10.31 mappingcone3

D.15.10.32 grrange

D.15.10.33 grneg

D.15.10.34 matrixpres

D.15.11 hess_lib

D.15.11.1 RiemannRochHess

D.15.13 multigrading_lib

D.15.13.1 setBaseMultigrading

D.15.13.2 getVariableWeights

D.15.13.3 getGradingGroup

D.15.13.4 getLattice

D.15.13.5 createGroup

D.15.13.6 createQuotientGroup

D.15.13.7 createTorsionFreeGroup

D.15.13.8 printGroup

D.15.13.9 isGroup

D.15.13.10 isGroupHomomorphism

D.15.13.11 isGradedRingHomomorphism

D.15.13.12 createGradedRingHomomorphism

D.15.13.13 setModuleGrading

D.15.13.14 getModuleGrading

D.15.13.15 isSublattice

D.15.13.16 imageLattice

D.15.13.17 intRank

D.15.13.18 kernelLattice

D.15.13.19 latticeBasis

D.15.13.20 preimageLattice

D.15.13.21 projectLattice

D.15.13.22 intersectLattices

D.15.13.23 isIntegralSurjective

D.15.13.24 isPrimitiveSublattice

D.15.13.25 intInverse

D.15.13.26 integralSection

D.15.13.27 primitiveSpan

D.15.13.28 factorgroup

D.15.13.29 productgroup

D.15.13.30 multiDeg

D.15.13.31 multiDegBasis

D.15.13.32 multiDegPartition

D.15.13.33 isTorsionFree

D.15.13.34 isPositive

D.15.13.35 isZeroElement

D.15.13.36 areZeroElements

D.15.13.37 isHomogeneous

D.15.13.38 equalMultiDeg

D.15.13.39 multiDegGroebner

D.15.13.40 multiDegSyzygy

D.15.13.41 multiDegModulo

D.15.13.42 multiDegResolution

D.15.13.43 multiDegTensor

D.15.13.44 multiDegTor

D.15.13.45 defineHomogeneous

D.15.13.46 pushForward

D.15.13.47 gradiator

D.15.13.48 hermiteNormalForm

D.15.13.49 smithNormalForm

D.15.13.50 hilbertSeries

D.15.13.51 lll

D.15.14 nfmodstd_lib

D.15.14.1 chinrempoly

D.15.17 realclassify_lib

D.15.17.1 realclassify

D.15.17.2 realmorsesplit

D.15.17.3 milnornumber

D.15.17.4 determinacy

D.15.18 deRham_lib

D.15.18.1 deRhamCohomology

D.16.1.2 homogeneitySpace

D.16.1.3 groebnerCone

D.16.1.4 maximalGroebnerCone

D.16.1.5 groebnerFan

D.16.1.6 lowerHomogeneitySpace

D.16.1.7 groebnerComplex

D.16.1.8 tropicalVariety

D.16.2 polymake_so

D.16.2.1 boundaryLatticePoints

D.16.2.2 ehrhartPolynomialCoeff

D.16.2.3 facetVertexLatticeDistances

D.16.2.4 facetWidth

D.16.2.5 facetWidths

D.16.2.6 fVectorP

D.16.2.7 gorensteinIndex

D.16.2.8 gorensteinVector

D.16.2.9 hilbertBasis

D.16.2.10 hStarVector

D.16.2.11 hVector

D.16.2.12 interiorLatticePoints

D.16.2.13 isBounded

D.16.2.14 isCanonical

D.16.2.15 isCompressed

D.16.2.16 isGorenstein

D.16.2.17 isLatticeEmpty

D.16.2.18 isNormal

D.16.2.19 isReflexive

D.16.2.20 isSmooth

D.16.2.21 isTerminal

D.16.2.22 isVeryAmple

D.16.2.23 latticeCodegree

D.16.2.24 latticeDegree

D.16.2.25 latticePoints

D.16.2.26 latticeVolume

D.16.2.27 maximalFace

D.16.2.28 maximalValue

D.16.2.29 minimalFace

D.16.2.30 minimalValue

D.16.2.31 minkowskiSum

D.16.2.32 nBoundaryLatticePoints

D.16.2.33 nHilbertBasis

D.16.2.34 nInteriorLatticePoints

D.16.2.35 nLatticePoints

D.16.2.36 normalFan

D.16.2.37 vertexAdjacencyGraph

D.16.2.38 vertexEdgeGraph

D.16.2.39 visual

E. Release Notes

E.1 News and changes

E.2 Singular 3 and Singular 4

E.2.1 Version schema for Singular

E.2.2 Notes for Singular users

E.2.3 Notes for developers

E.2.4 Building Singular

E.2.5 Side-by-side installation

E.3 Download instructions

E.4 Unix installation instructions

E.5 Windows installation instructions

E.6 Macintosh installation instructions

F. Index

User manual for Singular version 4-0-3, 2016, generated by texi2html.