Top: Singular Manual

Contents: Table of Contents

Index: Index

About: About this document

Table of Contents

2. Introduction

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

3.5.5 Type conversion and casting

3.5.6 Flow control

3.6 Input and output

3.7.1 Procedure definition

3.7.2 Names in procedures

3.7.3 Parameter list

3.7.4 Procedure commands

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

3.10 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

4.1.1 bigint declarations

4.1.2 bigint expressions

4.1.3 bigint operations

4.1.4 bigint related functions

4.2.1 def declarations

4.3.1 ideal declarations

4.3.2 ideal expressions

4.3.3 ideal operations

4.3.4 ideal related functions

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

4.5.1 intmat declarations

4.5.2 intmat expressions

4.5.3 intmat type cast

4.5.4 intmat operations

4.5.5 intmat related functions

4.6.1 intvec declarations

4.6.2 intvec expressions

4.6.3 intvec operations

4.6.4 intvec related functions

4.7.1 link declarations

4.7.2 link expressions

4.7.3 link related functions

4.7.4 ASCII links

4.7.5.1 MPfile links

4.7.5.2 MPtcp links

4.7.6 DBM links

4.8.1 list declarations

4.8.2 list expressions

4.8.3 list operations

4.8.4 list related functions

4.9.1 map declarations

4.9.2 map expressions

4.9.3 map operations

4.9.4 map related functions

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

4.11.1 module declarations

4.11.2 module expressions

4.11.3 module operations

4.11.4 module related functions

4.12.1 number declarations

4.12.2 number expressions

4.12.3 number operations

4.12.4 number related functions

4.13.1 package declarations

4.13.2 package related functions

4.14.1 poly declarations

4.14.2 poly expressions

4.14.3 poly operations

4.14.4 poly related functions

4.15.1 proc declaration

4.16.1 qring declaration

4.17 resolution

4.17.1 resolution declarations

4.17.2 resolution expressions

4.17.3 resolution related functions

4.18.1 ring declarations

4.18.2 ring related functions

4.18.3 ring operations

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

4.20.1 vector declarations

4.20.2 vector expressions

4.20.3 vector operations

4.20.4 vector related functions

5. Functions and system variables

5.1.5 char_series

5.1.7 cleardenom

5.1.11 contract

5.1.20 division

5.1.22 eliminate

5.1.30 factorize

5.1.33 fglmquot

5.1.34 files, input from

5.1.38 freemodule

5.1.43 groebner

5.1.45 highcorner

5.1.51 indepSet

5.1.53 interred

5.1.54 intersect

5.1.60 killattrib

5.1.62 laguerre

5.1.64 leadcoef

5.1.66 leadmonom

5.1.72 maxideal

5.1.79 mpresmat

5.1.97 preimage

5.1.102 qhweight

5.1.104 quotient

5.1.108 regularity

5.1.111 reservedName

5.1.112 resultant

5.1.113 ringlist

5.1.115 setring

5.1.116 simplex

5.1.117 simplify

5.1.120 sortvec

5.1.121 sprintf

5.1.125 stdfglm

5.1.126 stdhilb

5.1.131 transpose

5.1.134 uressolve

5.1.135 vandermonde

5.1.141 weightKB

5.2 Control structures

5.2.2 breakpoint

5.2.9 importfrom

5.2.10 keepring

5.2.15 ~ (break point)

5.3 System variables

5.3.4 multBound

5.3.6 printlevel

6. Tricks and pitfalls

6.1 Limitations

6.2 Major differences to the C programming language

6.2.1 No rvalue of increments and assignments

6.2.2 Evaluation of logical expressions

6.2.3 No case or switch statement

6.2.4 Usage of commas

6.2.5 Usage of brackets

6.2.6 Behavior of continue

6.2.7 Return type of procedures

6.3 Miscellaneous oddities

6.4 Identifier resolution

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)

7.3.3 eliminate (plural)

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 ncalgebra

7.3.15 nres (plural)

7.3.17 opposite

7.3.18 preimage (plural)

7.3.19 quotient (plural)

7.3.20 reduce (plural)

7.3.21 ringlist (plural)

7.3.22 slimgb (plural)

7.3.23 std (plural)

7.3.24 subst (plural)

7.3.25 syz (plural)

7.3.27 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 center_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

7.5.2.0. annfsOT

7.5.2.0. annfsBM

7.5.2.0. minIntRoot

7.5.2.0. reiffen

7.5.2.0. arrange

7.5.2.0. isHolonomic

7.5.2.0. convloc

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

7.5.5 ncall_lib

7.5.6 ncalg_lib

7.5.6.0. makeUsl2

7.5.6.0. makeUsl

7.5.6.0. makeUgl

7.5.6.0. makeUso5

7.5.6.0. makeUso6

7.5.6.0. makeUso7

7.5.6.0. makeUso8

7.5.6.0. makeUso9

7.5.6.0. makeUso10

7.5.6.0. makeUso11

7.5.6.0. makeUso12

7.5.6.0. makeUsp1

7.5.6.0. makeUsp2

7.5.6.0. makeUsp3

7.5.6.0. makeUsp4

7.5.6.0. makeUsp5

7.5.6.0. makeUg2

7.5.6.0. makeUf4

7.5.6.0. makeUe6

7.5.6.0. makeUe7

7.5.6.0. makeUe8

7.5.6.0. makeQso3

7.5.6.0. makeQsl2

7.5.6.0. makeQsl3

7.5.6.0. Qso3Casimir

7.5.6.0. GKZsystem

7.5.7 ncdecomp_lib

7.5.7.0. CentralQuot

7.5.7.0. CentralSaturation

7.5.7.0. CenCharDec

7.5.7.0. IntersectWithSub

7.5.8 nctools_lib

7.5.8.0. Gweights

7.5.8.0. weightedRing

7.5.8.0. ndcond

7.5.8.0. makeWeyl

7.5.8.0. makeHeisenberg

7.5.8.0. Exterior

7.5.8.0. findimAlgebra

7.5.8.0. ncRelations

7.5.8.0. isCentral

7.5.8.0. UpOneMatrix

7.5.9 perron_lib

7.5.9.0. perron

7.5.10 qmatrix_lib

7.5.10.0. quantMat

7.5.10.0. qminor

7.5.10.0. SymGroup

7.5.10.0. LengthSymElement

7.5.10.0. LengthSym

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.3.8 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

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

D. SINGULAR libraries

D.1 standard_lib

D.2 General purpose

D.2.2 general_lib

D.2.2.3 absValue

D.2.2.4 binomial

D.2.2.5 deleteSublist

D.2.2.6 factorial

D.2.2.7 fibonacci

D.2.2.8 kmemory

D.2.2.9 killall

D.2.2.10 number_e

D.2.2.11 number_pi

D.2.2.12 primes

D.2.2.13 product

D.2.2.16 watchdog

D.2.2.18 primecoeffs

D.2.2.19 primefactors

D.2.2.20 timeStd

D.2.2.21 timeFactorize

D.2.2.22 factorH

D.2.3 inout_lib

D.2.3.1 allprint

D.2.3.4 rMacaulay

D.2.3.6 showrecursive

D.2.3.9 writelist

D.2.4.2 katsura

D.2.4.3 freerank

D.2.4.4 is_zero

D.2.4.6 maxcoef

D.2.4.8 maxdeg1

D.2.4.10 mindeg1

D.2.4.11 normalize

D.2.4.12 rad_con

D.2.4.13 content

D.2.4.14 numerator

D.2.4.15 denominator

D.2.4.16 mod2id

D.2.4.17 id2mod

D.2.4.18 substitute

D.2.4.19 subrInterred

D.2.4.20 newtonDiag

D.2.4.21 hilbPoly

D.2.5 random_lib

D.2.5.1 genericid

D.2.5.2 randomid

D.2.5.3 randommat

D.2.5.4 sparseid

D.2.5.5 sparsematrix

D.2.5.6 sparsemat

D.2.5.7 sparsepoly

D.2.5.8 sparsetriag

D.2.5.9 triagmatrix

D.2.5.10 randomLast

D.2.5.11 randomBinomial

D.2.6.1 changechar

D.2.6.2 changeord

D.2.6.3 changevar

D.2.6.4 defring

D.2.6.5 defrings

D.2.6.6 defringp

D.2.6.7 extendring

D.2.6.8 fetchall

D.2.6.9 imapall

D.2.6.10 mapall

D.2.6.11 ord_test

D.2.6.12 ringtensor

D.2.6.13 ringweights

D.2.6.14 preimageLoc

D.2.6.15 rootofUnity

D.3 Linear algebra

D.3.1 matrix_lib

D.3.1.1 compress

D.3.1.5 flatten

D.3.1.6 genericmat

D.3.1.7 is_complex

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 rm_unitrow

D.3.1.26 rm_unitcol

D.3.1.27 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.3.2.8 charpoly

D.3.2.9 adjoint

D.3.2.11 gaussred

D.3.2.12 gaussred_pivot

D.3.2.13 gauss_nf

D.3.2.14 mat_rk

D.3.2.16 pos_def

D.3.2.17 hessenberg

D.3.2.18 eigenvals

D.3.2.19 minipoly

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.4.3.1 blowup0

D.4.3.4 nselect

D.4.3.7 select1

D.4.4 grwalk_lib

D.4.5 homolog_lib

D.4.5.2 cupproduct

D.4.5.6 fitting

D.4.5.7 flatteningStrat

D.4.5.9 homology

D.4.5.11 isFlat

D.4.5.12 isLocallyFree

D.4.5.14 kernel

D.4.5.16 kontrahom

D.4.5.17 KoszulHomology

D.4.5.18 tensorMod

D.4.6 intprog_lib

D.4.6.1 solve_IP

D.4.7 mprimdec_lib

D.4.7.1 separator

D.4.7.2 PrimdecA

D.4.7.3 PrimdecB

D.4.7.5 zeroMod

D.4.7.7 dec1var

D.4.7.9 splitting

D.4.7.10 primTest

D.4.7.11 preComp

D.4.7.12 indSet

D.4.7.13 GTZopt

D.4.7.14 zeroOpt

D.4.7.15 clrSBmod

D.4.7.16 minSatMod

D.4.7.17 specialModulesEqual

D.4.7.18 stdModulesEqual

D.4.7.19 modulesEqual

D.4.7.20 getData

D.4.8 mregular_lib

D.4.8.1 regIdeal

D.4.8.2 depthIdeal

D.4.8.3 satiety

D.4.8.4 regMonCurve

D.4.8.5 NoetherPosition

D.4.8.7 is_nested

D.4.9 normal_lib

D.4.9.4 primeClosure

D.4.9.5 closureFrac

D.4.9.6 deltaLoc

D.4.10 primdec_lib

D.4.10.2 primdecGTZ

D.4.10.3 primdecSY

D.4.10.4 minAssGTZ

D.4.10.5 minAssChar

D.4.10.6 testPrimary

D.4.10.7 radical

D.4.10.8 radicalEHV

D.4.10.9 equiRadical

D.4.10.10 prepareAss

D.4.10.11 equidim

D.4.10.12 equidimMax

D.4.10.13 equidimMaxEHV

D.4.10.14 zerodec

D.4.10.15 absPrimdecGTZ

D.4.11 primitiv_lib

D.4.11.1 primitive

D.4.11.2 primitive_extra

D.4.11.3 splitring

D.4.12 reesclos_lib

D.4.12.1 ReesAlgebra

D.4.12.2 normalI

D.4.13 resolve_lib

D.4.13.1 blowUp

D.4.13.2 Center

D.4.13.3 resolve

D.4.13.4 blowUpBO

D.4.13.5 createBO

D.4.13.6 CenterBO

D.4.13.8 DeltaList

D.4.13.9 showBO

D.4.14 reszeta_lib

D.4.14.1 intersectionDiv

D.4.14.2 spectralNeg

D.4.14.3 discrepancy

D.4.14.4 zetaDL

D.4.14.5 collectDiv

D.4.14.6 prepEmbDiv

D.4.14.7 abstractR

D.4.15 sheafcoh_lib

D.4.15.1 truncate

D.4.15.2 CM_regularity

D.4.15.3 sheafCohBGG

D.4.15.4 sheafCoh

D.4.15.6 displayCohom

D.4.16 toric_lib

D.4.16.1 toric_ideal

D.4.16.2 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 multseq2charexp

D.5.1.6 charexp2multseq

D.5.1.7 charexp2generators

D.5.1.8 charexp2inter

D.5.1.9 charexp2conductor

D.5.1.10 charexp2poly

D.5.1.11 tau_es2

D.5.2 arcAtPoint_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.5.3.5 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.5.3.12 normalform

D.5.3.13 debug_log

D.5.4 deform_lib

D.5.4.2 mod_versal

D.5.4.3 lift_kbase

D.5.4.4 lift_rel_kb

D.5.5 equising_lib

D.5.5.2 esIdeal

D.5.5.3 esStratum

D.5.5.4 isEquising

D.5.5.5 control_Matrix

D.5.6 gmssing_lib

D.5.6.1 gmsring

D.5.6.3 gmscoeffs

D.5.6.4 bernstein

D.5.6.5 monodromy

D.5.6.6 spectrum

D.5.6.7 sppairs

D.5.6.10 tmatrix

D.5.6.11 endvfilt

D.5.6.13 sppprint

D.5.6.17 spissemicont

D.5.6.18 spsemicont

D.5.6.19 spmilnor

D.5.6.20 spgeomgenus

D.5.6.21 spgamma

D.5.7 gmspoly_lib

D.5.7.2 goodBasis

D.5.8 hnoether_lib

D.5.8.1 hnexpansion

D.5.8.2 develop

D.5.8.3 extdevelop

D.5.8.5 displayHNE

D.5.8.6 invariants

D.5.8.7 displayInvariants

D.5.8.8 multsequence

D.5.8.9 displayMultsequence

D.5.8.10 intersection

D.5.8.11 is_irred

D.5.8.13 newtonpoly

D.5.8.14 is_NND

D.5.8.15 stripHNE

D.5.8.16 puiseux2generators

D.5.8.17 separateHNE

D.5.8.18 squarefree

D.5.8.19 allsquarefree

D.5.8.20 further_hn_proc

D.5.9 mondromy_lib

D.5.9.2 invunit

D.5.9.3 jacoblift

D.5.9.4 monodromyB

D.5.9.5 H2basis

D.5.10 qhmoduli_lib

D.5.10.1 ArnoldAction

D.5.10.2 ModEqn

D.5.10.3 QuotientEquations

D.5.10.4 StabEqn

D.5.10.5 StabEqnId

D.5.10.6 StabOrder

D.5.10.7 UpperMonomials

D.5.11 sing_lib

D.5.11.2 deform

D.5.11.3 dim_slocus

D.5.11.4 is_active

D.5.11.7 is_reg

D.5.11.8 is_regs

D.5.11.9 locstd

D.5.11.10 milnor

D.5.11.11 nf_icis

D.5.11.12 slocus

D.5.11.13 qhspectrum

D.5.11.14 Tjurina

D.5.11.15 tjurina

D.5.11.19 tangentcone

D.5.12 spcurve_lib

D.5.12.1 isCMcod2

D.5.12.2 CMtype

D.5.12.3 matrixT1

D.5.12.4 semiCMcod2

D.5.12.6 qhmatrix

D.5.12.7 relweight

D.5.12.8 posweight

D.5.12.9 KSpencerKernel

D.5.13 spectrum_lib

D.5.13.1 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 cyclotomic

D.6.1.6 group_reynolds

D.6.1.8 reynolds_molien

D.6.1.9 partial_molien

D.6.1.10 evaluate_reynolds

D.6.1.11 invariant_basis

D.6.1.12 invariant_basis_reynolds

D.6.1.13 primary_char0

D.6.1.14 primary_charp

D.6.1.15 primary_char0_no_molien

D.6.1.16 primary_charp_no_molien

D.6.1.17 primary_charp_without

D.6.1.18 primary_char0_random

D.6.1.19 primary_charp_random

D.6.1.20 primary_char0_no_molien_random

D.6.1.21 primary_charp_no_molien_random

D.6.1.22 primary_charp_without_random

D.6.1.23 power_products

D.6.1.24 secondary_char0

D.6.1.25 irred_secondary_char0

D.6.1.26 secondary_charp

D.6.1.27 secondary_no_molien

D.6.1.28 secondary_and_irreducibles_no_molien

D.6.1.29 secondary_not_cohen_macaulay

D.6.1.30 orbit_variety

D.6.1.31 rel_orbit_variety

D.6.1.32 relative_orbit_variety

D.6.1.33 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 faststd

D.7.1.7 findvars

D.7.1.8 hilbvec

D.7.1.9 linearpart

D.7.1.10 tolessvars

D.7.1.11 solvelinearpart

D.7.1.12 sortandmap

D.7.1.13 sortvars

D.7.1.14 valvars

D.7.1.15 idealSplit

D.7.2 solve_lib

D.7.2.1 laguerre_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.7.5.3 Quotient

D.7.5.4 Remainder

D.7.5.6 SQFRNorm

D.7.5.7 ZeroSet

D.7.5.8 EGCDMain

D.7.5.9 FactorMain

D.7.5.10 InvertNumberMain

D.7.5.11 QuotientMain

D.7.5.12 RemainderMain

D.7.5.13 RootsMain

D.7.5.14 SQFRNormMain

D.7.5.15 ContainedQ

D.7.6 rootsmr_lib

D.7.6.1 nrRootsProbab

D.7.6.2 nrRootsDeterm

D.7.6.3 symsignature

D.7.6.4 sturmquery

D.7.6.6 matmult

D.7.6.7 tracemult

D.7.6.9 randcharpoly

D.7.6.10 verify

D.7.6.11 randlinpoly

D.7.6.12 powersums

D.7.6.13 symmfunc

D.7.6.14 univarpoly

D.7.7 signcond_lib

D.7.7.1 signcnd

D.7.7.2 psigncnd

D.7.7.3 firstoct

D.7.8 rootsur_lib

D.7.8.2 whichvariable

D.7.8.3 varsigns

D.7.8.4 boundBuFou

D.7.8.5 boundposDes

D.7.8.6 boundDes

D.7.8.7 allrealst

D.7.8.9 allreal

D.7.8.11 sturmseq

D.7.8.12 sturmha

D.7.8.13 sturmhaseq

D.7.8.14 reverse

D.7.8.15 nrroots

D.7.8.16 isparam

D.8 Visualization

D.8.1 graphics_lib

D.8.1.1 staircase

D.8.1.2 mathinit

D.8.2 latex_lib

D.8.2.1 closetex

D.8.2.2 opentex

D.8.2.4 texdemo

D.8.2.5 texfactorize

D.8.2.7 texname

D.8.2.9 texpoly

D.8.2.10 texproc

D.8.2.11 texring

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.9.1.9 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.10.2.10 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.11.1 krypto_lib

D.11.1.1 decimal

D.11.1.2 exgcdN

D.11.1.3 eexgcdN

D.11.1.6 powerN

D.11.1.7 chineseRem

D.11.1.8 Jacobi

D.11.1.9 primList

D.11.1.10 primL

D.11.1.11 intPart

D.11.1.12 intRoot

D.11.1.13 squareRoot

D.11.1.14 solutionsMod2

D.11.1.15 powerX

D.11.1.16 babyGiant

D.11.1.18 MillerRabin

D.11.1.19 SolowayStrassen

D.11.1.20 PocklingtonLehmer

D.11.1.21 PollardRho

D.11.1.22 pFactor

D.11.1.23 quadraticSieve

D.11.1.24 isOnCurve

D.11.1.25 ellipticAdd

D.11.1.26 ellipticMult

D.11.1.27 ellipticRandomCurve

D.11.1.28 ellipticRandomPoint

D.11.1.29 countPoints

D.11.1.30 ellipticAllPoints

D.11.1.31 ShanksMestre

D.11.1.32 Schoof

D.11.1.33 generateG

D.11.1.34 factorLenstraECM

D.11.2 teachstd_lib

D.11.2.3 sameComponent

D.11.2.4 leadmonomial

D.11.2.5 monomialLcm

D.11.2.7 minEcart

D.11.2.8 NFMora

D.11.2.9 prodcrit

D.11.2.10 chaincrit

D.11.2.11 pairset

D.11.2.12 updatePairs

D.11.2.13 standard

D.11.2.14 localstd

D.11.3 weierstr_lib

D.11.3.1 weierstrDiv

D.11.3.2 weierstrPrep

D.11.3.3 lastvarGeneral

D.11.3.4 generalOrder

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

Top: Singular Manual

Contents: Table of Contents

Index: Index

About: About this document

User manual for Singular version 3-0-2, July 2006, generated by texi2html.