Index P

[____] [____] [_____] [____] [__] [Index] [Root]

Index P

P-key

p-key

p-Quotient

p-Quotient (FINITELY PRESENTED GROUPS)
p-Quotients (Process Version) (FP GROUPS - ADVANCED FEATURES)

p_sylow_creation

Construction of p-Sylow Subgroups (GENERIC ABELIAN GROUPS)

package

FUNCTIONS, PROCEDURES AND PACKAGES
Packages (FUNCTIONS, PROCEDURES AND PACKAGES)

PackageUserAttributes

Func_PackageUserAttributes (Example H2E13)

Packing

SpherePackingBound(K, n, d) : FldFin, RngIntElt, RngIntElt -> RngIntElt

Pad

PadCode(C, n) : Code, RngIntElt -> Code
PadCode(C, n) : Code, RngIntElt -> Code

PadCode

PadCode(C, n) : Code, RngIntElt -> Code
PadCode(C, n) : Code, RngIntElt -> Code

pAdic

PrimeField(L) : FldLoc -> FldLoc
pAdicRing(L) : RngLoc -> RngLoc
pAdicField(L) : FldLoc -> FldLoc
PrimeRing(L) : RngLoc -> RngLoc
pAdicRing(L) : RngLoc -> RngLoc
pAdicEllipticLogarithm(P, p: parameters): PtEll, RngIntElt -> FldLocElt
pAdicEmbeddings(f, p) : ModFrmElt, RngIntElt -> List
pAdicRing(p, n) : RngIntElt, RngIntElt -> RngLoc
pAdicRing(p, n) : RngIntElt, RngIntElt -> RngLoc
pAdicRing(p, n) : RngIntElt, RngIntElt -> RngLoc
pAdicRing(p, n) : RngIntElt, RngIntElt -> RngLoc
pAdicRing(p, n) : RngIntElt, RngIntElt -> RngLoc
pAdicRing(p, n) : RngIntElt, RngIntElt -> RngLoc
pAdicRing(p, n) : RngIntElt, RngIntElt -> RngLoc

padic

p-ADIC RINGS AND FIELDS

pAdicEllipticLogarithm

pAdicEllipticLogarithm(P, p: parameters): PtEll, RngIntElt -> FldLocElt

pAdicEmbeddings

pAdicEmbeddings(f, p) : ModFrmElt, RngIntElt -> List

pAdicField

PrimeField(L) : FldLoc -> FldLoc
pAdicRing(L) : RngLoc -> RngLoc
pAdicField(L) : FldLoc -> FldLoc
PrimeRing(L) : RngLoc -> RngLoc
pAdicRing(p, n) : RngIntElt, RngIntElt -> RngLoc
pAdicRing(p, n) : RngIntElt, RngIntElt -> RngLoc

pAdicRing

Pair

PairReduce(L) : Lat -> Lat, AlgMatElt
PairReduce(X) : ModMatRngElt -> ModMatRngElt, AlgMatElt
PairReduceGram(F) : ModMatRngElt -> ModMatRngElt, AlgMatElt, RngIntElt

pair

Pair Reduction (LATTICES)

pair-reduce

Pair Reduction (LATTICES)

Pairing

HeightPairing(P, Q: parameters) : PtEll, PtEll -> FldPrElt
HeightPairing(P, Q: Precision) : JacHypPt, JacHypPt -> FldPrElt
HeightPairingMatrix(S: Precision) : [JacHypPt] -> AlgMat
IntersectionPairing(x, y) : ModSymElt, ModSymElt -> FldRatElt
WeilPairing(P, Q, m) : JacHypPt, JacHypPt, RngIntElt -> RngElt
WeilPairing(P, Q, n) : PtEll, PtEll, RngIntElt -> RngElt

PairReduce

PairReduce(L) : Lat -> Lat, AlgMatElt
PairReduce(X) : ModMatRngElt -> ModMatRngElt, AlgMatElt

PairReduceGram

PairReduceGram(F) : ModMatRngElt -> ModMatRngElt, AlgMatElt, RngIntElt

Paley

PaleyGraph(q) : RngIntElt -> GrphUnd
PaleyTournament(q) : RngIntElt -> GrphDir

PaleyGraph

PaleyGraph(q) : RngIntElt -> GrphUnd

PaleyTournament

PaleyTournament(q) : RngIntElt -> GrphDir

Parabolic

IsStandardParabolicSubgroup( W, H ) : GrpCox -> GrpCox
StandardParabolicSubgroup( W, s ) : GrpCox, {} -> GrpCox

Parabolics

MaximalParabolics(C) : CosetGeom -> SetIndx
MaxParabolics(C) : CosetGeom -> SetIndx

Parallel

AllParallelClasses(D) : Inc -> SeqEnum
HasParallelClass(D) : Inc -> BoolElt, { IncBlk }
IsParallel(P, l, m) : Plane, PlaneLn, PlaneLn -> BoolElt
IsParallelClass(D, B, C) : Inc, IncBlk, IncBlk -> BoolElt, { IncBlk }
ParallelClass(P, l) : Plane, PlaneLn -> { PlaneLn }
ParallelClasses(P) : PlaneAff -> { { PlaneLn } }

ParallelClass

ParallelClass(P, l) : Plane, PlaneLn -> { PlaneLn }

ParallelClasses

ParallelClasses(P) : PlaneAff -> { { PlaneLn } }

Parallelism

HasParallelism(D: parameters) : Inc, RngIntElt -> BoolElt, { SetEnum }
IsParallelism(D, P) : Inc, SetEnum[SetEnum] -> BoolElt, RngIntElt

Parallelisms

AllParallelisms(D) : Inc -> SeqEnum

Parameter

UniformizingParameter(P) : PlcCrvElt -> FldFunRatMElt
UniformizingParameter(p) : Pt -> FldFunRatMElt

parameter

Intrinsics (OVERVIEW)
Options and Controls (FINITELY PRESENTED ALGEBRAS)

Parameters

ExtraSpecialParameters(G) : GrpMat -> [RngIntElt, RngIntElt]
Parameters(D) : Dsgn -> Record
SetProcessParameters(~P: parameters) : GrpFPCosetEnumProc ->
Func_Parameters (Example H2E2)

parameters

Graph Colouring and Parameter Setting: A General Discussion (GRAPHS)

Parametrisation

Parametrisation(X,C,p) : Sch,Crv,Pt -> MapSch

parametrisation

Parametrisation of Rational Curves (RATIONAL CURVES AND CONICS)

Parametrization

Parametrization(X,C) : Sch,CrvCon -> MapSch
ParametrizationMatrix(C) : CrvCon -> ModMatRngElt
ParametrizationToPuiseux(T) : Tup -> SeqEnum
PuiseuxToParametrization(S) : RngSerElt -> Tup

ParametrizationMatrix

ParametrizationMatrix(C) : CrvCon -> ModMatRngElt

ParametrizationToPuiseux

ParametrizationToPuiseux(T) : Tup -> SeqEnum

Parent

GetParent(SQP) : SQProc -> List
Parent(u) : AlgFPElt -> AlgFP
Parent(a) : AlgGenElt -> AlgGen
Parent(a) : AlgMatElt -> AlgMat
Parent(u) : GrpAbElt -> GrpAb
Parent(r) : GrpAbRel -> GrpAb
Parent(w) : GrpAtcElt -> GrpAtc
Parent(g) : GrpElt -> Grp
Parent(w) : GrpFPElt -> GrpFP
Parent(r) : GrpFPRel -> GrpFP
Parent(x) : GrpGPCElt -> GrpGPC
Parent(G) : GrpMatElt -> GrpMat
Parent(x) : GrpPCElt -> GrpPC
Parent(g) : GrpPermElt -> GrpPerm
Parent(w) : GrpRWSElt -> GrpRWS
Parent(u) : GrpSLPElt -> GrpSLP
Parent(f) : MapCrvHyp -> HomCrvHyp
Parent(x) : ModBrdtElt -> ModBrdt
Parent(V) : ModFld -> SetPow
Parent(u) : ModTupElt -> ModRng
Parent(u) : ModTupElt -> ModRng
Parent(w): ModTupRngElt -> ModTupRng
Parent(w): ModTupRngElt -> ModTupRng
Parent(w) : MonRWSElt -> MonRWS
Parent(G) : GrpGPC -> PowerStructure
Parent(P) : PtEll -> SetPtEll
Parent(R) : Rng -> Pow
Parent(r) : RngElt -> Rng
Parent(x) : RngLocElt -> RngLoc
Parent(S) : Seq -> Struct
Parent(R) : Set -> Struct
Parent(u) : SgpFPElt -> SgpFP
Parent(T) : Tup -> SetCart
ParentGraph(s) : GrphVert -> Grph
ParentGraph(S) : GrphVertSet -> Grph
ParentPlane(l) : PlaneLn -> Plane, PlanePtSet, PlaneLnSet
ParentPlane(L) : PlaneLnSet -> Plane, PlanePtSet, PlaneLnSet
ParentPlane(p) : PlanePt -> Plane, PlanePtSet, PlaneLnSet
ParentPlane(V) : PlanePtSet -> Plane, PlanePtSet, PlaneLnSet
ParentRing(N) : NwtnPgon -> Rng

parent

parent-category

parent-type

Type(r) : RngElt -> Cat
Parent and Category (INTRODUCTION [BASIC RINGS])

ParentGraph

ParentGraph(s) : GrphVert -> Grph
ParentGraph(S) : GrphVertSet -> Grph

parentheses

Expression (OVERVIEW)

parenthesis

Expression (OVERVIEW)

ParentPlane

ParentPlane(l) : PlaneLn -> Plane, PlanePtSet, PlaneLnSet
ParentPlane(L) : PlaneLnSet -> Plane, PlanePtSet, PlaneLnSet
ParentPlane(p) : PlanePt -> Plane, PlanePtSet, PlaneLnSet
ParentPlane(V) : PlanePtSet -> Plane, PlanePtSet, PlaneLnSet

ParentRing

ParentRing(N) : NwtnPgon -> Rng

Parity

ParityCheckMatrix(C) : Code -> ModMatFldElt
ParityCheckMatrix(C) : Code -> ModMatRngElt
SubfieldRepresentationParityCode(C, K) : Code, FldFin -> Code

ParityCheckMatrix

ParityCheckMatrix(C) : Code -> ModMatFldElt
ParityCheckMatrix(C) : Code -> ModMatRngElt

parsing

Parsing Strings (INPUT AND OUTPUT)

Part

ALGEBRAIC GEOMETRY (PART)
ALGEBRAS (PART)
BASIC RINGS (PART)
CODING THEORY AND CRYPTOGRAPHY (PART)
COMMUTATIVE ALGEBRA (PART)
CRYPTOGRAPHY (PART)
EXTENSIONS OF RINGS (PART)
FINITE INCIDENCE STRUCTURES (PART)
GROUPS (PART)
HOMOLOGICAL ALGEBRA (PART)
LATTICES AND QUADRATIC FORMS (PART)
LIE THEORY (PART)
LINEAR ALGEBRA AND MODULE THEORY (PART)
OPTIMIZATION (PART)
REPRESENTATION THEORY (PART)
SEMIGROUPS AND MONOIDS (PART)
SETS, SEQUENCES, AND MAPPINGS (PART)
THE MAGMA LANGUAGE (PART)
ContentAndPrimitivePart(f) : RngMPolElt -> RngIntElt, RngMPolElt
ContentAndPrimitivePart(p) : RngUPolElt -> RngIntElt, RngUPolElt
LRatioOddPart(M, j) : ModSym, RngIntElt -> FldRatElt
PrimitivePart(f) : RngMPolElt -> RngMPolElt
PrimitivePart(p) : RngUPolElt -> RngUPolElt
SquarefreePart(f) : RngMPolElt -> RngMPolElt

Partial

IsPartialRoot(f, c) : RngUPolElt, RngSerElt -> BoolElt
IsUniquePartialRoot(f, c) : RngUPolElt, RngSerElt -> BoolElt
PartialFactorization(S) : [ RngIntElt ] -> [ RngIntEltFact ]
PartialFractionDecomposition(f) : FldFunRatUElt -> [ <RngUPolElt, RngIntElt, RngUPolElt> ]
SquarefreePartialFractionDecomposition(f) : FldFunRatUElt -> [ <RngUPolElt, RngIntElt, RngUPolElt> ]

partial

Creation of Partial Maps (MAPPINGS)
Partial Fraction Decomposition (RATIONAL FUNCTION FIELDS)
Partial Mappings (OVERVIEW)

partial-fraction

Partial Fraction Decomposition (RATIONAL FUNCTION FIELDS)

partial-mapping

Creation of Partial Maps (MAPPINGS)

PartialFact

RngInt_PartialFact (Example H40E8)

PartialFactorization

PartialFactorization(S) : [ RngIntElt ] -> [ RngIntEltFact ]

PartialFractionDecomposition

PartialFractionDecomposition(f) : FldFunRatUElt -> [ <RngUPolElt, RngIntElt, RngUPolElt> ]
FldFunRat_PartialFractionDecomposition (Example H46E3)

PartialMap

Partial Mappings (OVERVIEW)

Partition

ConjugatePartition(P) : SeqEnum -> SeqEnum
DistancePartition(u) : GrphVert -> [ { GrphVert } ]
DistancePartition(u) : GrphVert -> [ { GrphVert } ]
EquitablePartition(P, G) : { { GrphVert } }, GrphUnd -> { { GrphVert } }
IndexOfPartition(P) : SeqEnum -> RngIntElt
IsPartition(S) : SeqEnum -> BoolElt
MaximalPartition(G) : GrpPerm -> GSet
MinimalPartition(G: parameters) : GrpPerm -> GSet
OrbitsPartition(G : parameters ) : GrphUnd -> [ { GrphVert } ]
Partition(S, p) : SeqEnum, RngIntElt -> SeqEnum(SeqEnum)
Partition(S, P) : SeqEnum, [RngIntElt] -> SeqEnum(SeqEnum)
PartitionCovers(P1, P2) : SeqEnum, SeqEnum -> BoolElt

partition

Action on a G-invariant Partition (PERMUTATION GROUPS)

partition-action

Action on a G-invariant Partition (PERMUTATION GROUPS)

PartitionCovers

PartitionCovers(P1, P2) : SeqEnum, SeqEnum -> BoolElt

Partitions

AllPartitions(G) : GrpPerm -> SetEnum
MinimalPartitions(G: parameters) : GrpPerm -> [ GSet ]
NumberOfPartitions(n) : RngIntElt -> RngIntElt
NumberOfPartitions(n) : RngIntElt -> RngIntElt
Partitions(n) : RngIntElt -> [ [ RngIntElt ] ]
Partitions(n) : RngIntElt -> [ [ RngIntElt ] ]
Partitions(n, k) : RngIntElt, RngIntElt -> [ [ RngIntElt ] ]
RestrictedPartitions(n, k, M) : RngIntElt, SetEnum -> [ [ RngIntElt ] ]
RestrictedPartitions(n, k, M) : RngIntElt, SetEnum -> [ [ RngIntElt ] ]
RestrictedPartitions(n, M) : RngIntElt, SetEnum -> [ [ RngIntElt ] ]
RestrictedPartitions(n, Q) : RngIntElt, SetEnum -> [ [ RngIntElt ] ]
EnumComb_Partitions (Example H92E2)

partitions

Partitions (ENUMERATIVE COMBINATORICS)

Pascal

PascalTriangle(D) : Dsgn -> SeqEnum

PascalTriangle

PascalTriangle(D) : Dsgn -> SeqEnum

Passants

AllPassants(P, A) : Plane, { PlanePt } -> { PlaneLn }
ExternalLines(P, A) : Plane, { PlanePt } -> { PlaneLn }

Patch

AffinePatch(C,i) : Crv,RngIntElt -> SeqEnum
AffinePatch(X,p) : Sch,Pt -> Sch,Pt
AffinePatch(X,i) : Sch,RngIntElt -> Sch
RestrictionToPatch(f,j) : MapSch,RngIntElt -> MapSch
RestrictionToPatch(f,i,j) : MapSch,RngIntElt,RngIntElt -> MapSch

PATH

MAGMA_PATH

Path

BranchVertexPath(u,v) : GrphVert,GrphVert -> SeqEnum
DiameterPath(G) : Grph -> [GrphVert]
IsPath(G) : Grph -> BoolElt
PathGraph(p) : RngIntElt -> GrphUnd
PathTree(B, i) : AlgBas, RngIntElt -> ModRng
SetPath(s) : MonStgElt ->
VertexPath(u,v) : GrphSplVert,GrphSplVert -> SeqEnum,SeqEnum
VertexPath(u,v) : GrphVert,GrphVert -> SeqEnum

path

Connectedness, Paths and Circuits (GRAPHS)

PathGraph

PathGraph(p) : RngIntElt -> GrphUnd

PathTree

PathTree(B, i) : AlgBas, RngIntElt -> ModRng

pc

Groups (OVERVIEW)
Power-Conjugate Presentations (FINITE SOLUBLE GROUPS)
Transfer from GrpPC (FINITE SOLUBLE GROUPS)
Transfer to GrpPC (FINITE SOLUBLE GROUPS)

pc-presentations

Power-Conjugate Presentations (FINITE SOLUBLE GROUPS)

pc-to-perm

GrpPC_pc-to-perm (Example H25E23)

pc_hom

GrpPC_pc_hom (Example H25E5)

pc_quotient

GrpPC_pc_quotient (Example H25E19)

PCClass

WeightClass(x) : GrpPCElt -> RngIntElt
PCClass(x) : GrpPCElt -> RngIntElt

pCentral

pCentralSeries(G, p) : GrpFin, RngIntElt -> [ GrpFin ]
pCentralSeries(G, p) : GrpMat, RngIntElt -> [ GrpMat ]
pCentralSeries(G, p) : GrpPC, RngIntElt -> [GrpPC]
pCentralSeries(G, p) : GrpPerm, RngIntElt -> [ GrpPerm ]

pCentralSeries

PCExponents

PCExponents(G) : GrpGPC -> [RngIntElt]

PCGenerators

PCGenerators(G) : GrpGPC -> {@ GrpGPCElt @}
Generators(G) : GrpGPC -> {@ GrpGPCElt @}
Generators(H, G) : GrpGPC, GrpGPC -> {@ GrpGPCElt @}
NumberOfGenerators(G) : GrpGPC -> RngIntElt
NumberOfPCGenerators(G) : GrpPC -> RngIntElt
NumberOfPCGenerators(P) : Process(pQuot) -> RngIntElt
PCGenerators(G) : GrpPC -> SetIndx

PCGroup

PCGroup(G) : Grp -> GrpPC, Hom(Grp)
PCGroup(A) : GrpAb -> GrpPC, Hom(Grp)
PCGroup(G) : GrpMat -> GrpPC, Map
PCGroup(G): GrpMat -> GrpPC, Map
PCGroup(G) : GrpPerm -> GrpPC, Map
PCGroup(Q : parameters ) : [RngIntElt] -> GrpPC
PCGroup(G) : GrpGPC -> GrpPC, Map

pcgroup

GrpPC_pcgroup (Example H25E22)

pClass

pClass(G) : GrpPC -> RngIntElt
pClass(P) : Process(pQuot) -> RngIntElt

PCMap

MakePCMap(A, P, S) : Aff, Prj, SeqEnum ->
MakeProjectiveClosureMap(A, P, S) : Aff,Prj,SeqEnum ->
ProjectiveClosureMap(A) : Aff -> MapSch

pCore

pCore(G, p) : GrpAb, RngIntElt -> GrpAb
pCore(G, p) : GrpFin, RngIntElt -> GrpFin
pCore(G, p) : GrpMat, RngIntElt -> GrpMat
pCore(G, S) : GrpPC, { RngIntElt } -> GrpPC
pCore(G, S) : GrpPC, { RngIntElt } -> GrpPC
pCore(G, p) : GrpPerm, RngIntElt -> GrpPerm

pCover

pCover(G, F, p) : GrpFin, GrpFinFP, RngIntElt -> GrpFinFP
pCover(G, F, p) : GrpPerm, GrpFP, RngIntElt -> GrpFP

pCovering

pCoveringGroup(~P) : Process(pQuot) ->

pCoveringGroup

pCoveringGroup(~P) : Process(pQuot) ->

PCPrimes

PCPrimes(G) : GrpPC -> [RngIntElt]

pElementary

pElementaryAbelianNormalSubgroup(G, p) : GrpPerm, RngIntElt -> GrpPerm

pElementaryAbelianNormalSubgroup

pElementaryAbelianNormalSubgroup(G, p) : GrpPerm, RngIntElt -> GrpPerm

Pencil

Pencil(P, p) : Plane, PlanePt -> { PlaneLn }

pencil

Creation from Pencils (RESOLUTION GRAPHS AND SPLICE DIAGRAMS)
GrphRes_pencil (Example H83E2)

Perfect

IsNearlyPerfect(C) : Code -> BoolElt
IsPerfect(C) : Code -> BoolElt
IsPerfect(G) : GrpAb -> BoolElt
IsPerfect(G) : GrpFin -> BoolElt
IsPerfect(G) : GrpGPC -> BoolElt
IsPerfect(G) : GrpMat -> BoolElt
IsPerfect(G) : GrpPC -> BoolElt
IsPerfect(G) : GrpPerm -> BoolElt
PerfectGroupDatabase() : -> DB
PerfectSubgroups(G: parameters) : GrpFin -> [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
PerfectSubgroups(G: parameters) : GrpPerm -> [ rec< GrpPerm, RngIntElt, RngIntElt, GrpFP> ]
RngInt_Perfect (Example H40E7)

PerfectGroupDatabase

PerfectGroupDatabase() : -> DB

PerfectSubgroups

PerfectSubgroups(G: parameters) : GrpFin -> [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
PerfectSubgroups(G: parameters) : GrpPerm -> [ rec< GrpPerm, RngIntElt, RngIntElt, GrpFP> ]

perfgps

GrpData_perfgps (Example H34E4)

pergps

Database of Some Permutation Groups (OVERVIEW)

Period

ClassicalPeriod(M, j, prec) : ModSym, RngIntElt, RngIntElt -> FldPrElt
PeriodMapping(M, prec) : ModSym, RngIntElt -> Map
RealPeriod(E: parameters) : CrvEll -> FldPRElt

period

The Period Map (MODULAR SYMBOLS)

period-map

The Period Map (MODULAR SYMBOLS)

PeriodMapping

PeriodMapping(M, prec) : ModSym, RngIntElt -> Map

Periods

Periods(M, prec) : ModSym, RngIntElt -> SeqEnum
Periods(E: parameters) : CrvEll -> [ FldPRElt ]

Perm

DualMatrixToPerm( W, M ) : GrpCox, AlgMatElt -> GrpPermElt
MatrixToPerm( W, M ) : GrpCox, AlgMatElt -> GrpPermElt
PermToMatrix( W, M ) : GrpCox, GrpPermElt -> AlgMatElt
PermToWord( W, p ) : GrpCox, GrpPermElt -> SeqEnum
WordToPerm( W, w ) : GrpCox, [] -> GrpPermElt

permgp

Permutation Group Databases (DATABASES OF GROUPS)

permgp-data

Permutation Group Databases (DATABASES OF GROUPS)

permreps

Induced Permutation Representations (FP GROUPS - ADVANCED FEATURES)

PermToDualMatrix

PermToDualMatrix( W, M ) : GrpCox, GrpPermElt -> AlgMatElt
PermToMatrix( W, M ) : GrpCox, GrpPermElt -> AlgMatElt

PermToMatrix

PermToDualMatrix( W, M ) : GrpCox, GrpPermElt -> AlgMatElt
PermToMatrix( W, M ) : GrpCox, GrpPermElt -> AlgMatElt

PermToWord

PermToWord( W, p ) : GrpCox, GrpPermElt -> SeqEnum

Permutation

permutation

Database of Some Permutation Groups (OVERVIEW)
Identification as a Permutation Group (PERMUTATION GROUPS)
Permutation Character (CHARACTERS OF FINITE GROUPS)
Permutation Group Actions on Polynomials (INVARIANT RINGS OF FINITE GROUPS)
PERMUTATION GROUPS
Permutation Modules (MODULES OVER A MATRIX ALGEBRA)
Permutation Polynomials (FINITE FIELDS)
Permutation Polynomials (UNIVARIATE POLYNOMIAL RINGS)

permutation-modules

Permutation Modules (MODULES OVER A MATRIX ALGEBRA)

permutation-polynomials

Permutation Polynomials (FINITE FIELDS)
Permutation Polynomials (UNIVARIATE POLYNOMIAL RINGS)

PermutationActionD8

AlgFP_PermutationActionD8 (Example H74E3)

PermutationAutomorphism

PermutationAutomorphism(A,g) : Sch,GrpPermElt -> IsoSch

PermutationCharacter

PermutationCharacter(G, H) : Grp, Grp -> AlgChtrElt
PermutationCharacter(G, H) : GrpFin, GrpFin -> AlgChtrElt
PermutationCharacter(G, H) : GrpMat, GrpMat -> AlgChtrElt
PermutationCharacter(G) : GrpPerm -> AlgChtrElt
PermutationCharacter(G) : GrpPerm -> AlgChtrElt
PermutationCharacter(G) : GrpPerm -> AlgChtrElt
PermutationCharacter(G, H) : GrpPerm, GrpPerm -> AlgChtrElt

PermutationCode

PermutationCode(u, G) : ModTupRngElt, GrpPerm -> Code
PermutationCode(u, G) : ModTupRngElt, GrpPerm -> Code
CodeFld_PermutationCode (Example H97E3)
CodeRng_PermutationCode (Example H98E3)

PermutationGroup

PermutationGroup(C) : Code -> GrpPerm, PowMap, Map
PermutationGroup(A) : GrpAb -> GrpPerm, Hom(Grp)
PermutationGroup(D, i: parameters): DB, RngIntElt -> GrpFP
PermutationGroup< n | L > : RngIntElt, List -> GrpPerm
PermutationGroup< X | L > : Set, List -> GrpPerm
PermutationGroup< X | L > : Set, List -> GrpPerm, Hom

PermutationModule

PermutationModule(G, K) : Grp, Fld -> ModGrp
PermutationModule(G, H, K) : Grp, Grp, Fld -> ModGrp
PermutationModule(G, H, R) : Grp, Grp, Rng -> ModGrp
PermutationModule(G, V) : Grp, ModTup -> ModGrp
PermutationModule(G, u) : Grp, ModTupElt -> ModGrp
PermutationModule(G, H, R) : GrpFin, GrpFin, Rng -> ModGrpFin
PermutationModule(G, H, R) : GrpMat, GrpMat, Rng -> ModGrp
PermutationModule(G, R) : GrpPerm, Rng -> ModGrp
PermutationModule(G, R) : GrpPerm, Rng -> ModGrpFin

PermutationRepresentation

ClassAction(A) : GrpAuto -> Map, GrpPerm, SetIndx
PermutationRepresentation(A) : GrpAuto -> Map, GrpPerm, SetIndx
PermutationRepresentation(D, i: parameters): DB, RngIntElt -> Hom(Grp), GrpFP, GrpPerm

Permutations

NumberOfPermutations(n, k) : RngIntElt, RngIntElt -> RngIntElt
Permutations(S) : SetEnum -> SetEnum;
Permutations(S) : SetEnum -> SetEnum;
Permutations(S, k) : SetEnum, RngIntElt -> SetEnum;
Permutations(S, k) : SetEnum, RngIntElt -> SetEnum;
ReflectionPermutations( RD ) : RootDtm -> []
SimpleReflectionPermutations( RD ) : RootDtm -> []
TensorInducedPermutations(G) : GrpMat -> SeqEnum
GrpPerm_Permutations (Example H20E2)

permutations

Permutations (PERMUTATION GROUPS)

pfgps

Database of Perfect Groups (DATABASES OF GROUPS)

pFundamental

pFundamentalUnits(O, p) : RngOrd, RngIntElt -> GrpAb, Map

pFundamentalUnits

pFundamentalUnits(O, p) : RngOrd, RngIntElt -> GrpAb, Map

PGamma

PGammaL(arguments)
ProjectiveGammaLinearGroup(arguments)
ProjectiveGammaUnitaryGroup(arguments)

PGammaL

PGammaL(arguments)
ProjectiveGammaLinearGroup(arguments)

PGammaU

PGammaU(arguments)
ProjectiveGammaUnitaryGroup(arguments)

PGL

PGL(arguments)
ProjectiveGeneralLinearGroup(arguments)

PGO

ProjectiveGeneralOrthogonalGroup(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}
PGO(arguments)

PGOMinus

ProjectiveGeneralOrthogonalGroupMinus(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}
PGOMinus(arguments)

PGOPlus

ProjectiveGeneralOrthogonalGroupPlus(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}
PGOPlus(arguments)

PGroup

PGroupStrong(G) : GrpMat -> GrpFP, Hom(Grp)
FPGroupStrong(G) : GrpPerm -> GrpFP, Hom(Grp)
NumberOfSubgroupsAbelianPGroup (A) : SeqEnum -> SeqEnum
OrderAutomorphismGroupAbelianPGroup (A) : SeqEnum -> RngIntElt
PGroupSection(SQP, p: parameter) : SQProc, RngIntElt -> BoolElt, SQProc

PGroupSection

NilpotentSection(SQP: parameter) : SQProc -> BoolElt, SQProc
PGroupSection(SQP, p: parameter) : SQProc, RngIntElt -> BoolElt, SQProc

PGroupStrong

PGroupStrong(G) : GrpMat -> GrpFP, Hom(Grp)
FPGroupStrong(G) : GrpPerm -> GrpFP, Hom(Grp)

PGU

PGU(arguments)
ProjectiveGeneralUnitaryGroup(arguments)

Phi

EulerPhi(n) : RngIntElt -> RngIntElt
FactoredEulerPhi(n) : RngIntElt -> RngIntEltFact
IsogenyMapPhi(I) : Map -> RngUPolElt
IsogenyMapPhiMulti(I) : Map -> RngUPolElt

PHom

PHom(M,N) : ModAlg, ModAlg -> ModMatFld

Pi

Pi(R) : FldPr -> FldPrElt

pi

Hall pi-Subgroups and Sylow Systems (FINITE SOLUBLE GROUPS)

Picard

PicardNumber(O) : RngQuad -> RngIntElt
PicardGroup(O) : RngQuad -> GrpAb, Map

PicardGroup

PicardNumber(O) : RngQuad -> RngIntElt
PicardGroup(O) : RngQuad -> GrpAb, Map

PicardNumber

PicardNumber(O) : RngQuad -> RngIntElt
PicardGroup(O) : RngQuad -> GrpAb, Map

PID

IsPrincipalIdealDomain(R) : Rng -> BoolElt
IsPID(R) : Rng -> BoolElt

Pipe

Pipe(C, S) : MonStgElt, MonStgElt -> MonStgElt

PIR

IsPrincipalIdealRing(R) : Rng -> BoolElt
IsPIR(R) : Rng -> BoolElt

Place

CurvePlace(P) : PlcFunElt -> PlcCrvElt
S ! P : PlcCrv, PlcFunElt -> PlcCrvElt
S ! I : PlcFun, RngFunOrdIdl -> PlcFunElt
FunctionFieldPlace(P) : PlcCrvElt -> PlcFunElt
IsWeierstrassPlace(P) : PlcCrvElt -> BoolElt
IsWeierstrassPlace(P) : PlcFunElt -> BoolElt
Place(C,m) : Crv,RngIntElt -> BoolElt,PlcCrvElt
Place(F, m) : FldFun, RngIntElt -> BoolElt, PlcFunElt
Place(F, m) : FldFun, RngIntElt -> PlcFunElt
Place(p) : Pt -> PlcCrvElt
Place(I) : RngFunOrdIdl -> PlcFunElt
Place(Q) : SeqEnum -> PlcCrvElt
RandomPlace(F, m) : FldFun, RngIntElt -> BoolElt, PlcFunElt
RandomPlace(F, m) : FldFun, RngIntElt -> BoolElt, PlcFunElt

place-equations

Crv_place-equations (Example H82E13)

Places

Places (ALGEBRAIC FUNCTION FIELDS)
NumberOfPlaces(F, m) : FldFun, RngIntElt -> RngIntElt
NumberOfPlaces(F, m) : FldFun, RngIntElt -> RngIntElt
NumberOfPlacesOfDegreeOne(F) : FldFun -> RngIntElt
NumberOfPlacesOfDegreeOne(F, m) : FldFun, RngIntElt -> RngIntElt
NumberOfPlacesOfDegreeOne(F, m) : FldFun, RngIntElt -> RngIntElt
NumberOfPlacesOfDegreeOneBound(F) : FldFun -> RngIntElt
NumberOfPlacesOfDegreeOneBound(F, m) : FldFun, RngIntElt -> RngIntElt
Places(C) : Crv -> PlcCrv
Places(C,m) : Crv,RngIntElt -> SeqEnum
Places(F) : FldFun -> PlcFun
Places(F) : FldFun -> PlcFun
Places(F) : FldFun -> PlcFun
Places(F, m) : FldFun, RngIntElt -> SeqEnum[PlcFunElt]
Places(F, m) : FldFun, RngIntElt -> SeqEnum[PlcFunElt]
Places(F, m) : FldFun, RngIntElt -> SeqEnum[PlcFunElt]
Places(p) : Pt -> SeqEnum
WeierstrassPlaces(D) : DivCrvElt -> SeqEnum
WeierstrassPlaces(F) : FldFunG -> [PlcFunElt]
WeierstrassPlaces(F) : FldFunG -> [PlcFunElt]
WeierstrassPlaces(D) : DivFunElt -> [PlcFunElt]

places

Functions related to Places and Divisors (ALGEBRAIC FUNCTION FIELDS)
Functions related to Places and Divisors (ALGEBRAIC FUNCTION FIELDS)
Places (PLANE ALGEBRAIC CURVES)
Places (PLANE ALGEBRAIC CURVES)
Sets of Places (PLANE ALGEBRAIC CURVES)
FldFunG_places (Example H57E11)

Planar

[Future release] IsPlanar(G) : GrphUnd -> BoolElt

Plane

Combinatorial and Geometrical Structures (OVERVIEW)
AffineSpace(k,2) : Rng, RngIntElt -> Aff
FiniteAffinePlane(D) : Inc -> Plane, PlanePtSet, PlaneLnSet
FiniteAffinePlane(W) : ModFld -> PlaneAff
FiniteAffinePlane< v | X : parameters > : RngIntElt, List -> PlaneAff
FiniteAffinePlane(P, l) : PlaneProj, PlaneLn -> PlaneAff, PlanePtSet, PlaneLnSet, Map
FiniteProjectivePlane(D) : Inc -> Plane, PlanePtSet, PlaneLnSet
FiniteProjectivePlane(W) : ModTupFld -> PlaneProj
FiniteProjectivePlane< v | X : parameters > : RngIntElt, List -> PlaneProj
ParentPlane(l) : PlaneLn -> Plane, PlanePtSet, PlaneLnSet
ParentPlane(L) : PlaneLnSet -> Plane, PlanePtSet, PlaneLnSet
ParentPlane(p) : PlanePt -> Plane, PlanePtSet, PlaneLnSet
ParentPlane(V) : PlanePtSet -> Plane, PlanePtSet, PlaneLnSet
ProjectiveSpace(k,2) : Rng,RngIntElt -> Prj
UpperHalfPlaneWithCusps() : -> SpcHyp

plane

FINITE PLANES
Genus and Singularities (PLANE ALGEBRAIC CURVES)
Planes (PLANE ALGEBRAIC CURVES)
The Upper Half Plane (SUBGROUPS OF PSL_2(R))

plane-curvepl

Genus and Singularities (PLANE ALGEBRAIC CURVES)

plane-points

Crv_plane-points (Example H82E1)

PlaneLn

Combinatorial and Geometrical Structures (OVERVIEW)

PlaneLnSet

Combinatorial and Geometrical Structures (OVERVIEW)

PlanePt

Combinatorial and Geometrical Structures (OVERVIEW)

PlanePtSet

Combinatorial and Geometrical Structures (OVERVIEW)

planes

Planes in Magma (FINITE PLANES)
Translation Planes (FINITE PLANES)

planes-in-magma

Planes in Magma (FINITE PLANES)

PlayWithPoints

CrvEll_PlayWithPoints (Example H85E12)

Plotkin

PlotkinAsymptoticBound(K, delta) : FldFin, FldPrElt -> FldPrElt
PlotkinBound(K, n, d) : FldFin, RngIntElt, RngIntElt -> RngIntElt
PlotkinSum(C, D) : Code, Code -> Code
PlotkinSum(C, D) : Code, Code -> Code

PlotkinAsymptoticBound

PlotkinAsymptoticBound(K, delta) : FldFin, FldPrElt -> FldPrElt

PlotkinBound

PlotkinBound(K, n, d) : FldFin, RngIntElt, RngIntElt -> RngIntElt

PlotkinSum

PlotkinSum(C, D) : Code, Code -> Code
PlotkinSum(C, D) : Code, Code -> Code

Plus

GOPlus(arguments)
GeneralOrthogonalGroupPlus(arguments)
OmegaPlus(arguments)
PGOPlus(arguments)
PSOPlus(arguments)
ProjectiveOmegaPlus(arguments)
SpecialOrthogonalGroupPlus(arguments)

plus

Operators (OVERVIEW)

pmap

Partial Mappings (OVERVIEW)
pmap< A -> B | x : -> e(x) > : Struct, Struct -> Map
pmap< A -> B | x : -> e(x), y : -> i(y) > : Struct, Struct -> Map
pmap< A -> B | x : -> e(x), y : -> i(y) > : Struct, Struct -> Map
pmap< A -> B | G > : Struct, Struct -> Map

pMaximal

pMaximalOrder(O, p) : RngOrd, RngIntElt -> RngOrd

pMaximalOrder

pMaximalOrder(O, p) : RngOrd, RngIntElt -> RngOrd

pMinimal

pMinimalWeierstrassModel(C, p) : CrvHyp, RngIntElt -> CrvHyp, MapCrvHyp

pMinimalWeierstrassModel

pMinimalWeierstrassModel(C, p) : CrvHyp, RngIntElt -> CrvHyp, MapCrvHyp

pMinus1

pMinus1(n) : RngIntElt -> RngIntEltFact, [ RngIntElt ]

pMultiplicator

pMultiplicator(G, p) : GrpFin, RngIntElt -> [ RngIntElt ]
pMultiplicator(G, p) : GrpPerm, RngIntElt -> [ RngIntElt ]
pMultiplicatorRank(G) : GrpPC -> RngIntElt

pMultiplicatorRank

pMultiplicatorRank(G) : GrpPC -> RngIntElt

pNormal

pNormalModel(C, p) : CrvHyp, RngIntElt -> CrvHyp, MapCrvHyp

pNormalModel

pNormalModel(C, p) : CrvHyp, RngIntElt -> CrvHyp, MapCrvHyp

Point

PointSet(E, m) : CrvEll, Map -> SetPtEll
E(m) : CrvEll, Map -> SetPtEll
E(L) : CrvEll, Rng -> SetPtEll
BasePoint(G, i) : GrpMat, RngIntElt -> Elt
BasePoint(G, i) : GrpPerm, RngIntElt -> Elt
CollineationGroup(P) : Plane -> GrpPerm, GSet, GSet, PowMap, Map
HasNonsingularPoint(X) : Sch -> BoolElt,Pt
HasPoint(C) : CrvCon -> BoolElt, Pt
HasReducedAffinePoint(C) : CrvCon -> BoolElt, Pt
IsBasePointFree(L) : LinSys -> BoolElt
IsDoublePoint(p) : Crv,Pt -> BoolElt
IsInflectionPoint(p) : Sch,Pt -> BoolElt,RngIntElt
IsPoint(C, S) : CrvHyp, SeqEnum -> BoolElt, PtHyp
IsPoint(N,p) : NwtnPgon,Tup -> BoolElt
IsPoint(H, x) : SetPtEll, RngElt -> BoolElt, PtEll
IsPoint(H, S) : SetPtEll, [ RngElt ] -> BoolElt, PtEll
IsPoint(K, S) : SrfKum, [RngElt] -> BoolElt, SrfKumPt
IsPointRegular(D) : IncNsp -> BoolElt, RngIntElt
IsPointTransitive(D) : Inc -> BoolElt
IsPointTransitive(P) : Plane -> BoolElt
Point(D, i) : Inc, RngIntElt -> IncPt
PointDegree(D, p) : Inc, IncPt -> RngIntElt
PointDegrees(D) : Inc -> [ RngIntElt ]
PointGraph(D) : Inc -> Grph
PointGraph(D) : Inc -> GrphUnd
PointGraph(P) : Plane -> GrphUnd;
PointGroup(D) : Inc -> GrpPerm, GSet
PointSet(D) : Inc -> IncPtSet
PointSet(P) : Plane -> PlanePtSet
ProjectionFromNonsingularPoint(X,p) : Sch,Pt -> Sch,MapSch,Sch
RepresentativePoint(P) : PlcCrv -> Pt
X(L) : Sch,Rng -> SetPt

point

EltSeq(P): PtEll -> [ RngElt ]
Access Operations (ELLIPTIC CURVES)
Arithmetic (ELLIPTIC CURVES)
Associated Structures (ELLIPTIC CURVES)
Combinatorial and Geometrical Structures (OVERVIEW)
Creating Points and Blocks (INCIDENCE STRUCTURES AND DESIGNS)
Creation of Points (ELLIPTIC CURVES)
Creation of Points (MODULAR CURVES)
Creation Predicates (ELLIPTIC CURVES)
Operations on Points (ELLIPTIC CURVES)
Operations on Points and Blocks (INCIDENCE STRUCTURES AND DESIGNS)
Point Order (ELLIPTIC CURVES)
Points (PLANE ALGEBRAIC CURVES)
Predicates on Points (ELLIPTIC CURVES)
The Point--Set and Block--Set of an Incidence Structure (INCIDENCE STRUCTURES AND DESIGNS)
The Point-Set and Line-Set of a Plane (FINITE PLANES)
The Set of Points and Set of Lines (FINITE PLANES)
Using the Point-Set and Line-Set to Create Points and Lines (FINITE PLANES)

point-access

EltSeq(P): PtEll -> [ RngElt ]
Access Operations (ELLIPTIC CURVES)

point-arithmetic

Arithmetic (ELLIPTIC CURVES)

point-block

Creating Points and Blocks (INCIDENCE STRUCTURES AND DESIGNS)

point-block-set

The Point--Set and Block--Set of an Incidence Structure (INCIDENCE STRUCTURES AND DESIGNS)

point-category

Curve(P) : SetPtEll -> CrvEll
Associated Structures (ELLIPTIC CURVES)

point-creation

Creation of Points (ELLIPTIC CURVES)
Creation of Points (MODULAR CURVES)

point-creation_predicates

Creation Predicates (ELLIPTIC CURVES)

point-line

The Set of Points and Set of Lines (FINITE PLANES)
Using the Point-Set and Line-Set to Create Points and Lines (FINITE PLANES)

point-line-set

The Point-Set and Line-Set of a Plane (FINITE PLANES)

point-order

Point Order (ELLIPTIC CURVES)

point-predicates

Predicates on Points (ELLIPTIC CURVES)

point_access_curve

ElementToSequence(P) : PtHyp -> SeqEnum
Access Operations (HYPERELLIPTIC CURVES)

point_access_jacobian

Access Operations (HYPERELLIPTIC CURVES)
Access Operations (HYPERELLIPTIC CURVES)

point_access_kummer

ElementToSequence(P) : PtHyp -> SeqEnum
Access Operations (HYPERELLIPTIC CURVES)

point_arithmetic_curve

Involution(P) : PtHyp -> PtHyp
Arithmetic of Points (HYPERELLIPTIC CURVES)

point_counting

Point Counting (ELLIPTIC CURVES)

point_creation_jacobian

Creation of Points (HYPERELLIPTIC CURVES)

point_enumeration_curve

Enumeration and Counting Points (HYPERELLIPTIC CURVES)

point_order_jacobian

Order of Points on the Jacobian (HYPERELLIPTIC CURVES)

point_predicates

Predicates on Points (HYPERELLIPTIC CURVES)

point_predicates_jacobian

IsIdentity(P) : JacHypPt -> BoolElt
Booleans and Predicates for Points (HYPERELLIPTIC CURVES)

point_predicates_kummer

Predicates on Points (HYPERELLIPTIC CURVES)

point_reduction

Reducing a Point (RATIONAL CURVES AND CONICS)

point_structures_jacobian

Rational Points and Group Structure (HYPERELLIPTIC CURVES)

PointArithmetic

CrvEll_PointArithmetic (Example H85E10)

PointDegree

PointDegree(D, p) : Inc, IncPt -> RngIntElt

PointDegrees

PointDegrees(D) : Inc -> [ RngIntElt ]

PointEnumeration

CrvHyp_PointEnumeration (Example H86E6)

PointGraph

PointGraph(D) : Inc -> Grph
PointGraph(D) : Inc -> GrphUnd
PointGraph(P) : Plane -> GrphUnd;

PointGroup

AutomorphismGroup(P) : Plane -> GrpPerm, GSet, GSet, PowMap, Map
PointGroup(P) : Plane -> GrpPerm, GSet, GSet, PowMap, Map
CollineationGroup(P) : Plane -> GrpPerm, GSet, GSet, PowMap, Map
PointGroup(D) : Inc -> GrpPerm, GSet

Points

BasePoints(L) : LinSys -> SeqEnum
BasePoints(f) : MapSch -> SetEnum
DefiningPoints(N) : NwtnPgon -> SeqEnum
EllipticPoints(G) : GrpPSL2, SpcHyp -> [SpcHypElt]
FixedPoints(g,H) : GrpPSL2Elt, SpcHyp -> SeqEnum
Flexes(C) : Sch -> SeqEnum
GoodBasePoints(G: parameters) : GrpMat -> []
HasPointsOverExtension(X) : Sch -> BoolElt
HasSingularPointsOverExtension(C) : Sch -> BoolElt
IntegralPoints(E) : CrvEll -> [ PtEll ], [ Tup ]
IntegralQuarticPoints(Q) : [ RngIntElt ] -> [ SeqEnum ]
IntegralQuarticPoints(Q, P) : [ RngIntElt ], [ RngIntElt ] -> [ SeqEnum ]
ModuliPoints(X,E) : CrvMod, CrvEll -> SeqEnum
NumberOfPoints(D) : Inc -> RngInt
NumberOfPoints(P) : Plane -> RngIntElt
Points(C) : CrvHyp -> SetIndx
Points(C, x) : CrvHyp, RngElt -> SetIndx
Points(D) : Inc -> { IncPt }
Points(D) : IncGeom -> SetIndx
Points(J) : JacHyp -> SetIndx
Points(J, P) : JacHyp, SrfKumPt -> SetIndx
Points(P) : Plane -> { PlanePt }
Points(X) : Sch -> SetIndx
Points(G) : SchGrpEll -> SetIndx
Points(H) : SetPtEll -> @ PtEll @
Points(H, x) : SetPtEll, RngElt -> [ PtEll ]
Points(K,[x1, x2, x3]) : SrfKum, [RngElt] -> SetIndx
PointsAtInfinity(C) : Crv -> SetEnum
PointsAtInfinity(C) : CrvHyp -> SetIndx
PointsAtInfinity(H) : SetPtEll -> @ PtEll @
PointsKnown(C) : CrvHyp -> BoolElt
RationalPoints(J, a, d) : JacHyp, RngUPolElt, RngIntElt -> SetIndx
RationalPoints(X) : Sch -> SetIndx
RationalPoints(K, Q) : SrfKum, [RngElt] -> SetIndx
SIntegralDesbovesPoints(Q, S) : [ RngIntElt ], [ RngIntElt ] -> [ SeqEnum ]
SIntegralLjunggrenPoints(Q, S) : [ RngIntElt ], [ RngIntElt ] -> [ SeqEnum ]
SIntegralPoints(E, S) : CrvEll, SeqEnum -> [ PtEll ], [ Tup ]
SIntegralQuarticPoints(Q, S) : [ RngIntElt ], [ RngIntElt ] -> [ SeqEnum ]
SingularPoints(C) : Sch -> SeqEnum
WeierstrassPlaces(D) : DivCrvElt -> SeqEnum

points

Arithmetic of Points (HYPERELLIPTIC CURVES)
Creation of Points on Curves (PLANE ALGEBRAIC CURVES)
Cusps and elliptic points of congruence subgroups (SUBGROUPS OF PSL_2(R))
Enumeration of Points (ELLIPTIC CURVES)
Maps and Points (SCHEMES)
Points (SCHEMES)
Points and Point Sets (SCHEMES)
Points of Subgroup Schemes (ELLIPTIC CURVES)
Points on Hyperelliptic Curves (HYPERELLIPTIC CURVES)
Points on the Jacobian (HYPERELLIPTIC CURVES)
Prelude to Points (SCHEMES)
Random Points (HYPERELLIPTIC CURVES)
The Fixed-point Space of a Module (MODULES OVER A MATRIX ALGEBRA)

points-blocks

Design_points-blocks (Example H94E2)

points-jac

Points on the Jacobian (HYPERELLIPTIC CURVES)

points-lines

Plane_points-lines (Example H95E2)

points_creation_kummer

Creation of Points (HYPERELLIPTIC CURVES)

points_kummer

RationalPoints(J, P) : JacHyp, SrfKumPt -> SetIndx
Points on the Kummer Surface (HYPERELLIPTIC CURVES)

PointsAtInfinity

PointsAtInfinity(C) : Crv -> SetEnum
PointsAtInfinity(C) : CrvHyp -> SetIndx
PointsAtInfinity(H) : SetPtEll -> @ PtEll @

PointSet

PointSet(E, m) : CrvEll, Map -> SetPtEll
E(m) : CrvEll, Map -> SetPtEll
E(L) : CrvEll, Rng -> SetPtEll
PointSet(D) : Inc -> IncPtSet
PointSet(P) : Plane -> PlanePtSet
X(L) : Sch,Rng -> SetPt

pointset

Associated Structures (ELLIPTIC CURVES)
Creation of Point Sets (ELLIPTIC CURVES)
Operations on Point Sets (ELLIPTIC CURVES)
Predicates on Point Sets (ELLIPTIC CURVES)

pointset-category

Associated Structures (ELLIPTIC CURVES)

pointset-creation

PointSet(E, m) : CrvEll, Map -> SetPtEll
Creation of Point Sets (ELLIPTIC CURVES)

pointset-predicates

Predicates on Point Sets (ELLIPTIC CURVES)

PointSets

CrvEll_PointSets (Example H85E9)

PointsKnown

PointsKnown(C) : CrvHyp -> BoolElt

pol-is

Newton_pol-is (Example H58E7)

Polar

ComplexToPolar(c) : FldComElt -> FldReElt, FldReElt
PolarToComplex(m, a) : FldReElt, FldReElt -> FldComElt

PolarToComplex

PolarToComplex(m, a) : FldReElt, FldReElt -> FldComElt

Poles

Poles(a) : FldFunElt -> SeqEnum[PlcFunElt]
Poles(a) : FldFunElt -> [ PlcFunElt ]
Poles(a) : RngFunOrdElt -> SeqEnum[PlcFunElt]
Zeros(C,f) : DivCrv, FldFunElt -> SeqEnum

Pollard

PollardRho(n) : RngIntElt -> RngIntEltFact, [ RngIntElt ]

PollardRho

PollardRho(n) : RngIntElt -> RngIntEltFact, [ RngIntElt ]

Poly

PolyMapKernel(f) : Map -> RngMPol

poly

Operations on Polynomials which use Newton Polygons (NEWTON POLYGONS)

Poly-Hensel

RngLoc_Poly-Hensel (Example H59E17)
RngPad_Poly-Hensel (Example H42E15)

poly-ops

Operations on Polynomials which use Newton Polygons (NEWTON POLYGONS)

poly-ops-ex

Newton_poly-ops-ex (Example H58E6)

Polycyclic

PolycyclicGroup< X | R > : List(Identifiers), List(GrpFPRel) -> GrpPC, Hom
AbelianGroup< X | R > : List(Identifiers), List(GrpAbRel) -> GrpAb, Hom(GrpAb)
Group< X | R > : List(Identifiers), List(GrpFPRel) -> GrpFP, Hom(Grp)
PolycyclicGenerators(G) : GrpMat -> [ GrpPCElt ]
PolycyclicGroup< x_1, ..., x_n | R : parameters > : List(Identifiers), List(GrpFPRel) -> GrpPC, Map
PolycyclicGroup< x_1, ..., x_n | R : parameters > : List(Identifiers), List(GrpFPRel) -> GrpGPC, Map

polycyclic

Introduction (POLYCYCLIC GROUPS)
POLYCYCLIC GROUPS
Polycyclic Groups and Polycyclic Presentations (POLYCYCLIC GROUPS)

polycyclic-groups

Polycyclic Groups and Polycyclic Presentations (POLYCYCLIC GROUPS)

polycyclic-groups-introduction

Introduction (POLYCYCLIC GROUPS)

PolycyclicGenerators

PolycyclicGenerators(G) : GrpMat -> [ GrpPCElt ]

PolycyclicGroup

PolycyclicGroup< X | R > : List(Identifiers), List(GrpFPRel) -> GrpPC, Hom
AbelianGroup< X | R > : List(Identifiers), List(GrpAbRel) -> GrpAb, Hom(GrpAb)
Group< X | R > : List(Identifiers), List(GrpFPRel) -> GrpFP, Hom(Grp)
PolycyclicGroup< x_1, ..., x_n | R : parameters > : List(Identifiers), List(GrpFPRel) -> GrpPC, Map
PolycyclicGroup< x_1, ..., x_n | R : parameters > : List(Identifiers), List(GrpFPRel) -> GrpGPC, Map
GrpGPC_PolycyclicGroup (Example H24E2)
GrpPC_PolycyclicGroup (Example H25E2)
Grp_PolycyclicGroup (Example H19E4)

Polygon

IsPolygon(G) : Grph -> BoolElt
NewtonPolygon(C) : Crv -> NwtnPgon
NewtonPolygon(f) : RngMPolElt -> NwtnPgon
NewtonPolygon(f) : RngUPolElt -> NwtnPgon
NewtonPolygon(g) : RngUPolElt -> NwtnPgon
NewtonPolygon(g) : RngUPolElt -> NwtnPgon
NewtonPolygon(V) : SeqEnum -> NwtnPgon
PolygonGraph(p) : RngIntElt -> GrphUnd

polygon

NEWTON POLYGONS

PolygonGraph

PolygonGraph(p) : RngIntElt -> GrphUnd

Polygons

DisplayPolygons(P,filename) : SeqEnum, MonStgElt ->

Polylog

Polylog(m, s) : FldPrElt -> FldPrElt
Polylog(m, f) : RngIntElt, RngSerElt -> RngSerElt
PolylogD(m, s) : FldPrElt -> FldPrElt
PolylogP(m, s) : FldPrElt -> FldPrElt

PolylogD

PolylogDold(m, s) : FldPrElt -> FldPrElt
PolylogP(m, s) : FldPrElt -> FldPrElt
PolylogD(m, s) : FldPrElt -> FldPrElt

PolylogDold

PolylogDold(m, s) : FldPrElt -> FldPrElt
PolylogP(m, s) : FldPrElt -> FldPrElt
PolylogD(m, s) : FldPrElt -> FldPrElt

PolylogP

PolylogDold(m, s) : FldPrElt -> FldPrElt
PolylogP(m, s) : FldPrElt -> FldPrElt
PolylogD(m, s) : FldPrElt -> FldPrElt

PolyMapKernel

PolyMapKernel(f) : Map -> RngMPol

Polynomial

polynomial

Action on a Polynomial Ring (MODULES OVER A MATRIX ALGEBRA)
Database of Galois Group Polynomials (OVERVIEW)
Minimal and Characteristic Polynomial (FINITE FIELDS)
Minimal Polynomial, Norm and Trace (ALGEBRAICALLY CLOSED FIELDS)
MULTIVARIATE POLYNOMIAL RINGS
Polynomials for Finite Fields (FINITE FIELDS)
Rings, Fields, and Algebras (OVERVIEW)
The Bernoulli Polynomial (UNIVARIATE POLYNOMIAL RINGS)
The Generator Polynomial (LINEAR CODES OVER FINITE FIELDS)
UNIVARIATE POLYNOMIAL RINGS

polynomial-ring-action

Action on a Polynomial Ring (MODULES OVER A MATRIX ALGEBRA)

PolynomialAlgebra

PolynomialRing(R) : Rng -> RngUPol
PolynomialAlgebra(R) : Rng -> RngUPol
PolynomialRing(R, n) : Rng, RngIntElt -> RngMPol
PolynomialRing(R, n) : Rng, RngIntElt -> RngMPol
PolynomialRing(R, n, order) : Rng, RngIntElt, MonStgElt, ... -> RngMPol
PolynomialRing(R, n, order) : Rng, RngIntElt, MonStgElt, ... -> RngMPol
PolynomialRing(R, Q) : Rng, [ RngIntElt ] -> RngMPol

PolynomialMap

PolynomialMap(L) : LinSys -> RngMPolElt

PolynomialRing

PolynomialRing(R) : Rng -> RngUPol
PolynomialAlgebra(R) : Rng -> RngUPol
PolynomialRing(E) : CrvEll -> RngMPol, Map
PolynomialRing(R, n) : Rng, RngIntElt -> RngMPol
PolynomialRing(R, n) : Rng, RngIntElt -> RngMPol
PolynomialRing(R, n, order) : Rng, RngIntElt, MonStgElt, ... -> RngMPol
PolynomialRing(R, n, order) : Rng, RngIntElt, MonStgElt, ... -> RngMPol
PolynomialRing(R, Q) : Rng, [ RngIntElt ] -> RngMPol
PolynomialRing(R) : RngInvar -> RngMPol

Polynomials

AllIrreduciblePolynomials(F, m) : FldFin, RngIntElt -> { RngPolElt }
DefiningPolynomials(C) : CrvHyp -> RngUPolElt, RngUPolElt
FactoredMinimalAndCharacteristicPolynomials(A: parameters) : Mtrx -> [<RngUPolElt, RngIntElt>], [<RngUPolElt, RngIntElt>]
MinimalAndCharacteristicPolynomials(A: parameter) : Mtrx -> RngUPolElt, RngUPolElt
NumberOfPrimePolynomials(q, d) : RngIntElt, RngIntElt -> RngIntElt
PrimePolynomials(R, d) : RngUPol, RngIntElt -> SeqEnum[ RngUPolElt ]
RngPol_Polynomials (Example H44E2)

polynomials

Orthogonal Polynomials (UNIVARIATE POLYNOMIAL RINGS)
Permutation Polynomials (FINITE FIELDS)
Permutation Polynomials (UNIVARIATE POLYNOMIAL RINGS)
Polynomials (LOCAL RINGS AND FIELDS)
Polynomials (p-ADIC RINGS AND FIELDS)
Polynomials Associated with Newton Polygons (NEWTON POLYGONS)
Polynomials over series rings (POWER, LAURENT AND PUISEUX SERIES)
Special Families of Polynomials (UNIVARIATE POLYNOMIAL RINGS)

PolynomialSieve

PolynomialSieve( T ) : Tup -> SeqEnum

POmega

POmega(arguments)
ProjectiveOmega(arguments)
ProjectiveOmegaMinus(arguments)
ProjectiveOmegaPlus(arguments)

POmegaMinus

POmegaMinus(arguments)
ProjectiveOmegaMinus(arguments)

POmegaPlus

POmegaPlus(arguments)
ProjectiveOmegaPlus(arguments)

Pop

IndentPop() : ->

POpen

POpen(S, T) : MonStgElt, MonStgElt -> File

Pos

NumPosRoots( W ) : GrpCox -> RngIntElt
NumberOfPositiveRoots( W ) : GrpCox -> RngIntElt
NumberOfPositiveRoots( G ) : GrpLie -> RngIntElt
NumberOfPositiveRoots( RD ) : RootDtm -> RngIntElt

poset

Operations on Poset Elements (GROUPS)
Operations on Subgroup Class Posets (GROUPS)
The Poset of Subgroup Classes (GROUPS)

poset-element

Operations on Poset Elements (GROUPS)

poset-operation

Operations on Subgroup Class Posets (GROUPS)

Position

Position(s, t) : MonStgElt, MonStgElt -> RngIntElt
Index(s, t) : MonStgElt, MonStgElt -> RngIntElt
Index(S, x) : SeqEnum, Elt -> RngIntElt
Index(S, x) : SetIndx, Elt -> RngIntElt
RootPosition( W, v ) : GrpCox, . -> {@@}
RootPosition( G, v ) : GrpLie, . -> {@@}
RootPosition( RD, v ) : RootDtm, . -> {@@}

Positions

IdempotentPositions(B) : AlgBas -> SeqEnum

Positive

IsPositive(D) : DivCrvElt -> BoolElt
IsEffective(D) : DivCrvElt -> BoolElt
IsPositive( RD, r ) : RootDtm, RngIntElt -> BoolElt
IsPositiveDefinite(F) : ModMatRngElt -> BoolElt
IsPositiveSemiDefinite(F) : ModMatRngElt -> BoolElt
NumberOfPositiveRoots( W ) : GrpCox -> RngIntElt
NumberOfPositiveRoots( G ) : GrpLie -> RngIntElt
NumberOfPositiveRoots( RD ) : RootDtm -> RngIntElt
PositiveDefiniteForm(G) : GrpMat -> AlgMatElt
PositiveRoots( W ) : GrpCox -> {@@}
PositiveRoots( G ) : GrpLie -> {@@}
PositiveRoots( RD ) : RootDtm -> {@@}
PositiveSum(m, i) : Map, RngIntElt -> FldPrElt

positive

Positive and simple roots (ROOT DATA FOR LIE THEORY)

positive-simple-roots

Positive and simple roots (ROOT DATA FOR LIE THEORY)

PositiveCoroots

PositiveCoroots( W ) : GrpCox -> {@@}
PositiveRoots( W ) : GrpCox -> {@@}
PositiveRoots( G ) : GrpLie -> {@@}
PositiveRoots( RD ) : RootDtm -> {@@}

PositiveDefiniteForm

PositiveDefiniteForm(G) : GrpMat -> AlgMatElt
Lat_PositiveDefiniteForm (Example H66E21)

PositiveRoots

PositiveCoroots( W ) : GrpCox -> {@@}
PositiveRoots( W ) : GrpCox -> {@@}
PositiveRoots( G ) : GrpLie -> {@@}
PositiveRoots( RD ) : RootDtm -> {@@}

PositiveSum

PositiveSum(m, i) : Map, RngIntElt -> FldPrElt

Power

power

Operators (OVERVIEW)
Parents of Sets and Sequences (INTRODUCTION [SETS, SEQUENCES, AND MAPPINGS])
Power Groups (POLYCYCLIC GROUPS)
Power Sequences (SEQUENCES)
Power Sets (SETS)
POWER, LAURENT AND PUISEUX SERIES
PowerGroup (FINITE SOLUBLE GROUPS)
Rings, Fields, and Algebras (OVERVIEW)

power-group

Power Groups (POLYCYCLIC GROUPS)
PowerGroup (FINITE SOLUBLE GROUPS)

power-sequence

Power Sequences (SEQUENCES)

power-set

Power Sets (SETS)

power-set-sequence

Parents of Sets and Sequences (INTRODUCTION [SETS, SEQUENCES, AND MAPPINGS])

PowerFormalSet

PowerFormalSet(R) : Struct -> PowSetIndx

PowerGroup

PowerGroup(G) : GrpPC -> PowerGroup

PowerGroupTwo

GrpPC_PowerGroupTwo (Example H25E29)

PowerIndexedSet

PowerIndexedSet(R) : Struct -> PowSetIndx

powering

AlgGrp_powering (Example H73E5)

PowerMap

PowerMap(G) : GrpAb -> Map
PowerMap(G) : GrpFin -> Map
PowerMap(G) : GrpMat -> Map
PowerMap(G) : GrpPC -> Map
PowerMap(G) : GrpPerm -> Map

PowerMultiset

PowerMultiset(R) : Struct -> PowSetMulti

PowerRelation

PowerRelation(r, k: parameters) : FldPrElt, RngIntElt -> RngUPolElt

PowerResidueCode

PowerResidueCode(K,n,p) : FldFin, RngIntElt, RngIntElt -> Code

PowerSequence

PowerSequence(R) : Struct -> PowSeqEnum
Seq_PowerSequence (Example H8E2)

PowerSeries

PowerSeries(f) : ModFrmElt -> RngSerPowElt

PowerSeriesRing

PowerSeriesRing(R) : Rng -> RngSerPow

PowerSet

PowerSet(R) : Struct -> PowSetEnum
Set_PowerSet (Example H7E6)

pPrimary

pPrimaryComponent(A, p) : GrpAb, RngIntElt -> GrpAb
pPrimaryInvariants(A, p) : GrpAb, RngIntElt -> [ RngIntElt ]

pPrimaryComponent

pPrimaryComponent(A, p) : GrpAb, RngIntElt -> GrpAb

pPrimaryInvariants

pPrimaryInvariants(A, p) : GrpAb, RngIntElt -> [ RngIntElt ]

pQuotient

[Future release] pQuotient(G, p, c) : GrpMat, RngIntElt, RngIntElt -> GrpPC, Map, SeqEnum
pQuotient(G, p, c) : GrpPerm, RngIntElt, RngIntElt -> GrpPC, Map, SeqEnum
pQuotient( F, p, c : parameters ) : GrpFP, RngIntElt, RngIntElt -> GrpPC, Map
pQuotient( G, p, c : parameters ) : GrpPC, RngIntElt, RngIntElt -> GrpPC, Map
pQuotient(F, p, c: parameters) : GrpFP, RngIntElt, RngIntElt -> GrpPC, Map
pQuotient(F, p, c: parameters) : GrpFP, RngIntElt, RngIntElt -> GrpPC, Map, SeqEnum
pQuotientProcess(F, p, c: parameters) : GrpFP, RngIntElt, RngIntElt -> Process

pQuotient1

GrpFP_1_pQuotient1 (Example H22E20)

pQuotient2

GrpFP_1_pQuotient2 (Example H22E21)

pQuotient3

GrpFP_1_pQuotient3 (Example H22E22)

pQuotient4

GrpFP_1_pQuotient4 (Example H22E23)

pQuotient5

GrpFP_2_pQuotient5 (Example H23E9)

pQuotient6

GrpFP_2_pQuotient6 (Example H23E10)

pQuotient7

GrpFP_2_pQuotient7 (Example H23E11)

pQuotient8

GrpFP_2_pQuotient8 (Example H23E12)

pQuotientProcess

pQuotientProcess(F, p, c: parameters) : GrpFP, RngIntElt, RngIntElt -> Process

pRadical

pRadical(O, p) : RngOrd, RngIntElt -> RngOrdIdl

PRank

ClassGroupPRank(F) : FldFunG -> RngIntElt
ClassGroupPRank(F) : FldFunG -> RngIntElt

pRank

pRank(P) : Plane -> RngIntElt
pRank(P, p) : Plane -> RngIntElt

pRanks

pRanks(G) : GrpPC-> [ RngIntElt ]

prec

Precision and Valuation (LOCAL RINGS AND FIELDS)
Precision and Valuation (p-ADIC RINGS AND FIELDS)

prec-val

Precision and Valuation (LOCAL RINGS AND FIELDS)
Precision and Valuation (p-ADIC RINGS AND FIELDS)

Precision

AbsolutePrecision(x) : RngLocElt -> RngIntElt
AbsolutePrecision(f) : RngSerElt -> RngIntElt
ChangePrecision(L, r) : RngLoc, RngIntElt -> RngLoc
ChangePrecision(P, r) : RngLoc, RngIntElt -> RngLoc
ChangePrecision(x, m) : RngLocElt, RngIntElt -> RngLocElt
ExpandToPrecision(f, c, n) : RngUPolElt, RngSerPuisElt -> RngSerPuisElt
IsSinglePrecision(n) : RngIntElt -> BoolElt
Precision(R) : FldCom -> RngIntElt
Precision(s) : FldPrElt -> RngIntElt
Precision(r) : FldReElt -> RngIntElt
Precision(M) : ModFrm -> RngIntElt
Precision(L) : RngLoc -> RngIntElt
Precision(P) : RngLoc -> RngIntElt
Precision(x) : RngLocElt -> RngIntElt
Precision(R) : RngSer -> Rng
PrecisionBound(M : parameters) : ModFrm -> RngIntElt
RelativePrecision(x) : RngLocElt -> RngIntElt
RelativePrecision(f) : RngSerElt -> RngIntElt
SetKantPrecision(O, n) : RngOrd, RngIntElt ->
SetPrecision(M, prec) : ModFrm, RngIntElt ->
SuggestedPrecision(f) : RngUPolElt -> RngIntElt
SuggestedPrecision(f) : RngUPolElt -> RngIntElt

precision

Fixed Precision Real Numbers (REAL AND COMPLEX FIELDS)
Free and Fixed Precision (POWER, LAURENT AND PUISEUX SERIES)
Precision (POWER, LAURENT AND PUISEUX SERIES)
Precision (POWER, LAURENT AND PUISEUX SERIES)
Precision (REAL AND COMPLEX FIELDS)

PrecisionBound

PrecisionBound(M : parameters) : ModFrm -> RngIntElt

pred

Predicates on Elements (ALGEBRAIC FUNCTION FIELDS)
Structure Predicates (ALGEBRAIC FUNCTION FIELDS)

predicate

Booleans (OVERVIEW)
Ideal Predicates (IDEAL THEORY AND GRÖBNER BASES)
Predicates (RING OF INTEGERS)
Predicates and Boolean Operations (INTRODUCTION [BASIC RINGS])
Predicates on Ring Elements (VALUATION RINGS)
Ring Predicates and Booleans (CHARACTERS OF FINITE GROUPS)
Ring Predicates and Booleans (FINITE FIELDS)
Ring Predicates and Booleans (GALOIS RINGS)
Ring Predicates and Booleans (RATIONAL FUNCTION FIELDS)
Ring Predicates and Booleans (RING OF INTEGERS)
Ring Predicates and Properties (ALGEBRAICALLY CLOSED FIELDS)

Predicates

Predicates (CHAIN COMPLEXES)
AlgLie_Predicates (Example H75E7)
ModForm_Predicates (Example H90E10)

predicates

Basic Attributes (SUBGROUPS OF PSL_2(R))
Basic Attributes (SUBGROUPS OF PSL_2(R))
Basic Functions (SUBGROUPS OF PSL_2(R))
Creation Predicates (ELLIPTIC CURVES)
Creation Predicates (HYPERELLIPTIC CURVES)
Predicates (BASIC ALGEBRAS)
Predicates (CHAIN COMPLEXES)
Predicates (FINITE SOLUBLE GROUPS)
Predicates (LIE ALGEBRAS)
Predicates (MATRICES)
Predicates (MODULAR FORMS)
Predicates (MODULES OVER AFFINE ALGEBRAS)
Predicates (MODULES OVER AFFINE ALGEBRAS)
Predicates and Booleans on Lattices (LATTICES)
Predicates for Elements (FINITE SOLUBLE GROUPS)
Predicates for Subgroups (FINITE SOLUBLE GROUPS)
Predicates on Algebras (QUATERNION ALGEBRAS)
Predicates on Curve Models (ELLIPTIC CURVES)
Predicates on Curve Models (HYPERELLIPTIC CURVES)
Predicates on Elements (CYCLOTOMIC FIELDS)
Predicates on Elliptic Curves (ELLIPTIC CURVES)
Predicates on Point Sets (ELLIPTIC CURVES)
Predicates on Points (ELLIPTIC CURVES)
Predicates on Subgroup Schemes (ELLIPTIC CURVES)
Structure Predicates (ALGEBRAIC FUNCTION FIELDS)
Structure Predicates (ALGEBRAIC FUNCTION FIELDS)
Structure Predicates (ALGEBRAIC FUNCTION FIELDS)
Structure Predicates (ALGEBRAIC FUNCTION FIELDS)

predicates-GrpPsl2

Basic Attributes (SUBGROUPS OF PSL_2(R))

predicates-GrpPSL2Elt

Basic Functions (SUBGROUPS OF PSL_2(R))

preds

Predicates on Elements (ALGEBRAS)

Preface

PREFACE
PREFACE

Prefix

AssignNamePrefix(A, S) : FldAC, MonStgElt ->

Preimage

IsGlobalUnitWithPreimage(a) : FldFunElt -> BoolElt, GrpAbElt
IsGlobalUnitWithPreimage(a) : FldFunElt -> BoolElt, GrpAbElt
IsSUnitWithPreimage(a, S) : FldFunElt, SetEnum[PlcFunElt] -> BoolElt, GrpAbElt
IsUnitWithPreimage(a) : RngFunOrdElt -> BoolElt, GrpAbElt
PreimageIdeal(I) : RngMPolRes -> RngMPol
PreimageRing(I) : RngMPolRes -> RngMPol
PreimageRing(Q) : RngUPolRes -> RngUPol

preimage

Images and Preimages (MAPPINGS)

PreimageIdeal

PreimageIdeal(I) : RngMPolRes -> RngMPol

PreimageRing

PreimageRing(I) : RngMPolRes -> RngMPol
PreimageRing(Q) : RngUPolRes -> RngUPol

prelude

Gathering the Data (THE K3 DATABASE)

Preparata

PreparataCode(m): RngIntElt, RngUPolElt -> Code
PreparataCode(m, h): RngIntElt, RngUPolElt -> Code

PreparataCode

PreparataCode(m): RngIntElt, RngUPolElt -> Code
PreparataCode(m, h): RngIntElt, RngUPolElt -> Code

Preprune

Preprune(C) : ModCpx -> ModCpx
Preprune(C,n) : ModCpx, RngIntElt -> ModCpx

Presentation

CompactPresentation(G) : GrpPC -> [RngIntElt]
IsIdenticalPresentation(G, H) : GrpPC, GrpPC -> BoolElt
NilpotentPresentation(G) : GrpGPC -> GrpGPC, Map
PresentationIsSmall(G) : GrpGPC -> BoolElt
PresentationLength(G) : GrpFP -> RngIntElt
PresentationLength(P) : Process(Tietze) -> RngIntElt
Simplify(~P : parameters) : Process(Tietze) ->
SpecialPresentation(G) : GrpPC -> GrpPC
StandardPresentation(G): GrpPC -> GrpPC, Map

presentation

Actions (COXETER GROUPS)
Braid groups (COXETER GROUPS)
CompactPresentation (FINITE SOLUBLE GROUPS)
Conditioned Presentations (FINITE SOLUBLE GROUPS)
Constructing a Presentation for a Subgroup (FINITELY PRESENTED GROUPS)
Construction of a Quotient: Specification of a Presentation (FINITELY PRESENTED ALGEBRAS)
Conversion (COXETER GROUPS)
Creation (COXETER GROUPS)
Finitely presented Coxeter groups (COXETER GROUPS)
Generators and Relations (MATRIX GROUPS)
Isomorphism testing and Standard Presentations (p-GROUPS)
Operations on FP Coxeter groups (COXETER GROUPS)
Operations on words (COXETER GROUPS)
Presentation of Submodules (FREE MODULES)
Special Presentations (FINITE SOLUBLE GROUPS)
Specification of a Presentation (ABELIAN GROUPS)
Specification of a Presentation (FINITELY PRESENTED SEMIGROUPS)
Structuring Presentations (FINITELY PRESENTED ALGEBRAS)
The Presentation of Submodules (INTRODUCTION [LINEAR ALGEBRA AND MODULE THEORY])

presentation-action

Actions (COXETER GROUPS)

presentation-braid

Braid groups (COXETER GROUPS)

presentation-convert

Conversion (COXETER GROUPS)

presentation-create

Creation (COXETER GROUPS)

presentation-elt-op

Operations on words (COXETER GROUPS)

presentation-op

Operations on FP Coxeter groups (COXETER GROUPS)

PresentationIsSmall

PresentationIsSmall(G) : GrpGPC -> BoolElt

PresentationLength

PresentationLength(G) : GrpFP -> RngIntElt
PresentationLength(P) : Process(Tietze) -> RngIntElt

presentations

Modifying Presentations (FP GROUPS - ADVANCED FEATURES)
More About Presentations (FINITE SOLUBLE GROUPS)
Power-Conjugate Presentations (FINITE SOLUBLE GROUPS)
Presentations (PERMUTATION GROUPS)

presented

FINITELY PRESENTED ALGEBRAS
Finitely Presented Algebras (FINITELY PRESENTED ALGEBRAS)
FINITELY PRESENTED GROUPS
Finitely Presented Modules (FINITELY PRESENTED ALGEBRAS)
FINITELY PRESENTED SEMIGROUPS
FP GROUPS - ADVANCED FEATURES
Rings, Fields, and Algebras (OVERVIEW)

ClearPrevious() : ->
GetPreviousSize() : -> RngIntElt
PreviousPrime(n) : RngIntElt -> RngIntElt
SetPreviousSize(n) : RngIntElt ->
ShowPrevious() : ->
ShowPrevious(i) : RngIntElt ->

PrimeDivisors(n) : RngIntElt -> [RngIntElt]
Other Functions Relating to Primes (RING OF INTEGERS)

PreviousPrime

PreviousPrime(n) : RngIntElt -> RngIntElt

primality

Primality (RING OF INTEGERS)

Primary

IsPrimary(I) : RngMPol -> BoolElt
IsPrimary(I) : RngMPolRes -> BoolElt
Primary(a) : RngQuadElt -> RngQuadElt
PrimaryAlgebra(R) : RngInvar -> RngMPol
PrimaryComponents(X) : Sch -> SeqEnum
PrimaryDecomposition(I) : RngMPol -> [ RngMPol ], [ RngMPol ]
PrimaryDecomposition(I) : RngMPolRes -> [ RngMPolRes ], [ RngMPolRes ]
PrimaryIdeal(R) : RngInvar -> RngMPol
PrimaryInvariantFactors(a) : AlgMatElt -> [ <RngUPolElt, RngIntElt ]
PrimaryInvariantFactors(A) : Mtrx -> [ <RngUPolElt, RngIntElt> ]
R`PrimaryInvariants
PrimaryInvariants(A) : GrpAb -> [ RngIntElt ]
PrimaryInvariants(R) : RngInvar -> [ RngMPolElt ]
PrimaryRationalForm(a) : AlgMatElt -> AlgMatElt, AlgMatElt, [ <RngUPolElt, RngIntElt ]
PrimaryRationalForm(A) : Mtrx -> AlgMatElt, AlgMatElt, [ <RngUPolElt, RngIntElt ]

primary

Primary Decomposition (IDEAL THEORY AND GRÖBNER BASES)
Primary Invariants (INVARIANT RINGS OF FINITE GROUPS)

primary-decomposition

Primary Decomposition (IDEAL THEORY AND GRÖBNER BASES)

PrimaryAlgebra

PrimaryAlgebra(R) : RngInvar -> RngMPol

PrimaryComponents

PrimaryComponents(X) : Sch -> SeqEnum

PrimaryDecomposition

PrimaryDecomposition(I) : RngMPol -> [ RngMPol ], [ RngMPol ]
PrimaryDecomposition(I) : RngMPolRes -> [ RngMPolRes ], [ RngMPolRes ]
GB_PrimaryDecomposition (Example H50E18)

PrimaryIdeal

PrimaryIdeal(R) : RngInvar -> RngMPol

PrimaryInvariantFactors

PrimaryInvariantFactors(a) : AlgMatElt -> [ <RngUPolElt, RngIntElt ]
PrimaryInvariantFactors(A) : Mtrx -> [ <RngUPolElt, RngIntElt> ]

PrimaryInvariants

R`PrimaryInvariants
PrimaryInvariants(A) : GrpAb -> [ RngIntElt ]
PrimaryInvariants(R) : RngInvar -> [ RngMPolElt ]

PrimaryRationalForm

PrimaryRationalForm(a) : AlgMatElt -> AlgMatElt, AlgMatElt, [ <RngUPolElt, RngIntElt ]
PrimaryRationalForm(A) : Mtrx -> AlgMatElt, AlgMatElt, [ <RngUPolElt, RngIntElt ]

Prime

DegreeOnePrimeIdeals(O, B) : RngOrd, RngIntElt -> [ RngOrdIdl ]
IsPrime(x) : RngElt -> BoolElt
IsPrime(I) : RngFunOrdIdl -> BoolElt
IsPrime(n) : RngIntElt -> BoolElt
IsPrime(n) : RngIntElt -> BoolElt
IsPrime(n) : RngIntElt -> BoolElt
IsPrime(I) : RngMPol -> BoolElt
IsPrime(I) : RngMPolRes -> BoolElt
IsPrime(I) : RngOrdIdl -> BoolElt, RngOrdIdl
IsPrimePower(n) : RngIntElt -> BoolElt, RngIntElt, RngIntElt
IsProbablePrime(n: parameter) : RngIntElt -> BoolElt
KeepPrimePower(SQP, p) : SQProc, RngIntElt -> SeqEnum
NextPrime(n) : RngIntElt -> RngIntElt
NumberOfPrimePolynomials(q, d) : RngIntElt, RngIntElt -> RngIntElt
PreviousPrime(n) : RngIntElt -> RngIntElt
Prime(P) : FldLoc -> RngIntElt
Prime(L) : RngLoc -> RngIntElt
Prime(G) : SymGenLoc -> RngIntElt
PrimeBasis(n) : RngIntElt -> [RngIntElt]
PrimeBasis(n) : RngIntElt -> [RngIntElt]
PrimeComponents(X) : Sch -> SeqEnum
PrimeField(F) : Fld -> Fld
PrimeField(F) : FldFin -> FldFin
PrimeForm(Q, p) : QuadBin, RngIntElt -> QuadBinElt
PrimeIdeal(S,p) : AlgQuatOrd, RngIntElt -> AlgQuatOrd
PrimePolynomials(R, d) : RngUPol, RngIntElt -> SeqEnum[ RngUPolElt ]
PrimeRing(F) : FldFun -> Rng
PrimeRing(R) : Rng -> Rng
PrimeRing(F) : RngFunOrd -> Rng
PrimeRing(L) : RngLoc -> RngLoc
pAdicRing(L) : RngLoc -> RngLoc
RandomPrime(n: parameter) : RngIntElt -> RngIntElt
RandomPrime(n: parameter) : RngIntElt -> RngIntElt
RandomPrime(n, a, b, x: parameter) :RngIntElt, RngIntElt, RngIntElt -> BoolElt, RngIntElt
RandomPrime(n, a, b, x: parameter) :RngIntElt, RngIntElt, RngIntElt -> BoolElt, RngIntElt
RandomPrimePolynomial(R, d) : RngUPol, RngIntElt -> RngUPolElt

prime

PrimeDivisors(n) : RngIntElt -> [RngIntElt]
Primes and Primality Testing (RING OF INTEGERS)

PrimeBasis

PrimeDivisors(n) : RngIntElt -> [RngIntElt]
PrimeBasis(n) : RngIntElt -> [RngIntElt]
PrimeBasis(n) : RngIntElt -> [RngIntElt]

PrimeComponents

PrimeComponents(X) : Sch -> SeqEnum

PrimeDivisors

PrimeDivisors(n) : RngIntElt -> [RngIntElt]
PrimeBasis(n) : RngIntElt -> [RngIntElt]
PrimeBasis(n) : RngIntElt -> [RngIntElt]

PrimeField

PrimeField(F) : Fld -> Fld
PrimeField(F) : FldFin -> FldFin
PrimeRing(F) : FldFun -> Rng
PrimeRing(L) : RngLoc -> RngLoc

PrimeForm

PrimeForm(Q, p) : QuadBin, RngIntElt -> QuadBinElt

PrimeIdeal

PrimeIdeal(S,p) : AlgQuatOrd, RngIntElt -> AlgQuatOrd

PrimePolynomials

PrimePolynomials(R, d) : RngUPol, RngIntElt -> SeqEnum[ RngUPolElt ]

PrimeRing

PrimeField(F) : FldFun -> Rng
PrimeRing(F) : FldFun -> Rng
PrimeRing(R) : Rng -> Rng
PrimeRing(F) : RngFunOrd -> Rng
PrimeRing(L) : RngLoc -> RngLoc

Primes

AddPrimes(SQP, p: IsComplete): SQProc, RngIntElt ->
BadPrimes(E) : CrvEll -> [ RngIntElt ]
BadPrimes(C) : CrvHyp -> SeqEnum
BadPrimes(J) : JacHyp -> SeqEnum
ExtensionPrimes(D, Q) : DB, MonStgElt -> SetEnum
GModulePrimes(G, A) : GrpFP, GrpFP -> SetMulti
GModulePrimes(G, A) : GrpGPC, GrpGPC -> SetMulti
GModulePrimes(G, A, B) : GrpFP, GrpFP, GrpFP -> SetMulti
GModulePrimes(G, A, B) : GrpGPC, GrpGPC, GrpGPC -> SetMulti
GetPrimes(SQP) : SQProc -> SetEnum, BoolElt
Primes (SQP): SQProc ->
PrintPrimes(SQP : parameters) : SQProc ->
RamifiedPrimes(A) : AlgQuat -> SeqEnum
ReplacePrimes(SQP, m: IsComplete): SQProc, SetEnum ->

primes

Calculating the Relevant Primes (FP GROUPS - ADVANCED FEATURES)
Relevant Primes (FP GROUPS - ADVANCED FEATURES)

Primitive

primitive

Contpp(f) : RngMPolElt -> RngIntElt, RngMPolElt
Content and Primitive Part (MULTIVARIATE POLYNOMIAL RINGS)
Finding Special Elements (ORDERS AND ALGEBRAIC FIELDS)
Primitive Unitary Reflection Groups (REFLECTION GROUPS)
Special Elements (FINITE FIELDS)

primitive-unitary-reflection-groups

Primitive Unitary Reflection Groups (REFLECTION GROUPS)

PrimitiveElement

PrimitiveElement(F) : FldFin -> FldFinElt
PrimitiveElement(K) : FldNum -> FldNumElt
PrimitiveElement(R) : RngIntRes -> RngIntResElt
PrimitiveElement(I) : RngOrdIdl -> RngOrdElt

PrimitiveGroup

PrimitiveGroup(d) : RngIntElt -> GrpPerm, MonStgElt
TransitiveGroup(d) : RngIntElt -> GrpPerm, MonStgElt
TransitiveGroup(d, f) : RngIntElt, Program -> GrpPerm, MonStgElt
TransitiveGroup(d, n) : RngIntElt, RngIntElt -> GrpPerm, MonStgElt
TransitiveGroup(S, f) : [RngIntElt], Program -> GrpPerm, MonStgElt

PrimitiveGroupDatabaseLimit

PrimitiveGroupDatabaseLimit() : -> RngIntElt
TransitiveGroupDatabaseLimit() : -> RngIntElt

PrimitiveGroupDescription

PrimitiveGroupDescription(d, n) : RngIntElt, RngIntElt -> MonStgElt
TransitiveGroupDescription(d, n) : RngIntElt, RngIntElt -> MonStgElt

PrimitiveGroupProcess

PrimitiveGroupProcess(d) : RngIntElt -> Process
TransitiveGroupProcess(d) : RngIntElt -> Process
TransitiveGroupProcess(d, f) : RngIntElt, Program -> Process
TransitiveGroupProcess(S) : [RngIntElt] -> Process
TransitiveGroupProcess(S, f) : [RngIntElt], Program -> Process

PrimitiveGroups

PrimitiveGroups(d: parameters) : RngIntElt -> [GrpPerm]
TransitiveGroups(d: parameters) : RngIntElt -> [GrpPerm]
TransitiveGroups(S: parameters) : [RngIntElt] -> [GrpPerm]
TransitiveGroups(d, f) : RngIntElt, Program -> [GrpPerm]
TransitiveGroups(S, f) : [RngIntElt], Program -> [GrpPerm]

PrimitivePart

PrimitivePart(f) : RngMPolElt -> RngMPolElt
PrimitivePart(p) : RngUPolElt -> RngUPolElt

PrimitivePolynomial

PrimitivePolynomial(F, m) : FldFin, RngIntElt -> RngPolElt

PrimitiveQuotient

PrimitiveQuotient(G) : GrpPerm -> GrpPerm, Hom, GrpPerm

PrimitiveRoot

PrimitiveRoot(R) : RngIntRes -> RngIntResElt
PrimitiveElement(R) : RngIntRes -> RngIntResElt
PrimitiveRoot(m) : RngIntElt -> RngIntElt

PrimitiveStructure

GrpPerm_PrimitiveStructure (Example H20E26)

PrimitiveWreathProduct

PrimitiveWreathProduct(G, H) : GrpPerm, GrpPerm -> GrpPerm
PrimitiveWreathProduct(Q) : [ GrpPerm ] -> GrpPerm

primitivity

Primitivity Testing (MATRIX GROUPS)

Principal

PrincipalDivisor(a) : FldFunGElt -> DivFunElt
Divisor(a) : FldFunGElt -> DivFunElt
Divisor(a) : RngFunOrdElt -> DivFunElt
Id(R) : AlgChtr -> AlgChtrElt
IsPID(R) : Rng -> BoolElt
IsPIR(R) : Rng -> BoolElt
IsPrincipal(D) : DivCrvElt -> BoolElt,FldFunRatMElt
IsPrincipal(I) : RngFunOrdIdl -> BoolElt, FldFunElt
IsPrincipal(I) : RngMPol -> BoolElt, RngMPolElt
IsPrincipal(I) : RngOrdFracIdl -> BoolElt, FldOrdElt
IsPrincipalIdealRing(F) : FldAlg -> BoolElt
IsPrincipalIdealRing(O) : RngOrd -> BoolElt
PrincipalDivisor(Div,f) : DivCrv, FldFunElt -> DivCrvElt
PrincipalDivisorMap(F) : FldFun -> Map
PrincipalIdealMap(O) : RngFunOrd -> Map

PrincipalCharacter

Identity(R) : AlgChtr -> AlgChtrElt
One(R) : AlgChtr -> AlgChtrElt
PrincipalCharacter(G) : Grp -> AlgChtrElt
Id(R) : AlgChtr -> AlgChtrElt

PrincipalDivisor

PrincipalDivisor(a) : FldFunGElt -> DivFunElt
Divisor(a) : FldFunGElt -> DivFunElt
Divisor(a) : RngFunOrdElt -> DivFunElt
PrincipalDivisor(Div,f) : DivCrv, FldFunElt -> DivCrvElt

PrincipalDivisorMap

PrincipalDivisorMap(F) : FldFun -> Map

PrincipalIdealMap

PrincipalIdealMap(O) : RngFunOrd -> Map

Print

print

Access Functions (FP GROUPS - ADVANCED FEATURES)
Automatic Printing (INPUT AND OUTPUT)
Print Names (MULTIVARIATE POLYNOMIAL RINGS)
Printing (INPUT AND OUTPUT)
The print-Statement (INPUT AND OUTPUT)
The print statement (OVERVIEW)
print expression;

print-access

Access Functions (FP GROUPS - ADVANCED FEATURES)

PrintCollector

PrintCollector (SQP : parameters) : SQProc ->

PrintExtensions

PrintExtensions(SQP : parameters) : SQProc ->

printf

The printf and fprintf Statements (INPUT AND OUTPUT)
printf format, expression, ..., expression;
IO_printf (Example H3E4)
IO_printf (Example H3E6)

printf2

IO_printf2 (Example H3E5)

PrintFile

Write(F, x) : MonStgElt, Var ->
PrintFile(F, x) : MonStgElt, Var ->
PrintFile(F, x, L) : MonStgElt, Var, MonStgElt ->

PrintFileMagma

PrintFileMagma(F, x) : MonStgElt, Var ->

Printing

SetKantPrinting(f) : BoolElt -> BoolElt
SetPowerPrinting(F, l) : FldFin, BoolElt ->

printing

Printing to a File (INPUT AND OUTPUT)

printing-file

Printing to a File (INPUT AND OUTPUT)

PrintModules

PrintModules(SQP : parameters) : SQProc ->

printname

Generator Assignment (OVERVIEW)

PrintPrimes

PrintPrimes(SQP : parameters) : SQProc ->

PrintProcess

PrintProcess(SQP : parameters) : SQProc ->

PrintQuotient

PrintQuotient (SQP : parameters) : SQProc ->

PrintRelat

PrintRelat(SQP : parameters) : SQProc ->

PrintSeries

PrintSeries(SQP : parameters) : SQProc ->

Probable

IsProbablyPrime(n: parameter) : RngIntElt -> BoolElt
IsProbablePrime(n: parameter) : RngIntElt -> BoolElt
ProbableRadicalDecomposition(I) : RngMPol -> [ RngMPol ]

ProbableRadicalDecomposition

ProbableRadicalDecomposition(I) : RngMPol -> [ RngMPol ]

Probably

IsProbablyPrime(n: parameter) : RngIntElt -> BoolElt
IsProbablePrime(n: parameter) : RngIntElt -> BoolElt
IsProbablyMaximal(G, H: parameters) : GrpPerm, GrpPerm -> BoolElt
IsProbablyPermutationPolynomial(p) : RngUPolElt -> BoolElt
IsProbablySupersingular(E) : CrvEll -> BoolElt

proc

Procedure Expressions (OVERVIEW)
p := proc< x_1, ..., x_n: parameters | expression >;

procedure

Functions and Procedures (FUNCTIONS, PROCEDURES AND PACKAGES)
FUNCTIONS, PROCEDURES AND PACKAGES
Functions, Procedures, and Mappings (OVERVIEW)
Procedure Expressions (MAGMA SEMANTICS)
Procedures (FUNCTIONS, PROCEDURES AND PACKAGES)
Procedures (OVERVIEW)
p := procedure(x_1, ..., x_n: parameters) statements : ->

procedure-expression

Procedure Expressions (MAGMA SEMANTICS)

Procedures

Func_Procedures (Example H2E4)

Process

CentralExtensionProcess (G, U) : GrpPC, GrpPC -> Proc
CloseVectorsProcess(L, w, u) : Lat, ModTupRngElt, RngElt -> LatEnumProc
CosetEnumerationProcess(G, H: parameters) : GrpFP, GrpFP -> GrpFPCosetEnumProc
ExtensionProcess(G, M, F) : GrpFin, ModRng, GrpFinFP -> Process
ExtensionProcess(G, M, F) : GrpPerm, ModRng, GrpFP -> Process
IsolProcess() : -> Process
IsolProcessOfDegree(d) : . -> Process
IsolProcessOfDegreeField(d, p) : ., . -> Process
IsolProcessOfField(p) : . -> Process
LowIndexProcess(G, R : parameters) : GrpFP, RngIntElt -> Process(Lix)
PrintProcess(SQP : parameters) : SQProc ->
RandomProcess(G) : GrpAb -> Process
RandomProcess(G) : GrpFin -> Process
RandomProcess(G) : GrpMat -> Process
RandomProcess(G) : GrpPC -> Process
RandomProcess(G) : GrpPerm -> Process
RandomProcess(G) : GrpSLP -> Process
RandomProcess(G) : GrpGPC -> Process
SetProcessParameters(~P: parameters) : GrpFPCosetEnumProc ->
ShortVectorsProcess(L, u) : Lat, RngElt -> LatEnumProc
SmallGroupProcess(o: parameters) : RngIntElt -> Process
SmallGroupProcess(o, f: parameters) : RngIntElt, Program -> Process
SmallGroupProcess(S: parameters) : [RngIntElt] -> Process
SmallGroupProcess(S, f: parameters) : [RngIntElt], Program -> Process
SolubleQuotientProcess(F : parameters): GrpFP -> SQProc
TietzeProcess(G: parameters) : GrpFP -> Process(Tietze)
TransitiveGroupProcess(d) : RngIntElt -> Process
TransitiveGroupProcess(d, f) : RngIntElt, Program -> Process
TransitiveGroupProcess(S) : [RngIntElt] -> Process
TransitiveGroupProcess(S, f) : [RngIntElt], Program -> Process
TransversalProcess(G, H) : GrpPerm, GrpPerm -> GrpPermTransProc
TransversalProcessNext(P) : GrpPermTransProc -> GrpPermElt
TransversalProcessRemaining(P) : GrpPermTransProc -> RngIntElt
pQuotientProcess(F, p, c: parameters) : GrpFP, RngIntElt, RngIntElt -> Process

process

Calculation of Standard Sections (FP GROUPS - ADVANCED FEATURES)
Initialisation (FP GROUPS - ADVANCED FEATURES)
Miscellaneous Functions (FP GROUPS - ADVANCED FEATURES)
Processes (DATABASES OF GROUPS)
Processes (DATABASES OF GROUPS)
Processes (DATABASES OF GROUPS)
Short and Close Vector Processes (LATTICES)
Soluble Quotient Process Tools (FP GROUPS - ADVANCED FEATURES)
Soluble Quotient Processes (FP GROUPS - ADVANCED FEATURES)
The p-Quotient Process (FP GROUPS - ADVANCED FEATURES)

Product

InnerProduct(u, v) : ModTupFldElt, ModTupFldElt -> FldElt
(u, v) : ModTupFldElt, ModTupFldElt -> FldElt
(u, v) : ModTupRngElt, ModTupRngElt -> RngElt
(u, v) : ModTupRngElt, ModTupRngElt : -> RngElt
(u, v) : ModTupRngElt, ModTupRngElt : -> RngElt
BasisProduct(A, i, j) : AlgGen, RngIntElt, RngIntElt -> AlgGenElt
CartesianProduct(G, H) : GrphDir, GrphDir -> GrphDir
CartesianProduct(R, S) : Str, ..., Str -> SetCart
DirectProduct(C, D) : Code, Code -> Code
DirectProduct(C, D) : Code, Code -> Code
DirectProduct(G, H) : GrpGPC, GrpGPC -> GrpGPC, [Map], [Map]
DirectProduct(G, H) : Grp, Grp -> Grp
DirectProduct(G, H) : GrpFP, GrpFP -> GrpFP
DirectProduct(G, H) : GrpMat, GrpMat -> GrpMat
DirectProduct(G, H) : GrpPC, GrpPC -> GrpPC, [Map], [Map]
DirectProduct(G, H) : GrpPerm, GrpPerm -> GrpPerm, [ Hom(Grp) ], [ Hom(Grp) ]
DirectProduct(A,B) : Prj,Prj -> PrjProd,SeqEnum
DirectProduct(A,B) : Sch,Sch -> Sch,SeqEnum
DirectProduct(R, S) : SgpFP, SgpFP -> SgpFP
DirectProduct(Q) : [ Grp ] -> Grp
DirectProduct(Q) : [ GrpFP ] -> GrpFP
DirectProduct(Q) : [ GrpMat ] -> GrpMat
DirectProduct(Q) : [ GrpPerm ] -> GrpPerm, [ Hom(Grp) ], [ Hom(Grp) ]
DirectProduct(Q) : [GrpPC] -> GrpPC, [ Map ], [ Map ]
FreeProduct(G, H) : GrpFP, GrpFP -> GrpFP
FreeProduct(R, S) : SgpFP, SgpFP -> SgpFP
FreeProduct(Q) : [ GrpFP ] -> GrpFP
InnerProduct(x, y) : AlgChtrElt, AlgChtrElt -> FldCycElt
InnerProduct(a, b) : AlgGenElt, AlgGenElt -> RngElt
InnerProduct(v, w) : LatElt, LatElt -> RngElt
InnerProduct(x,y) : ModBrdtElt, ModBrdtElt -> RngElt
InnerProductMatrix(L) : Lat -> AlgMatElt
InnerProductMatrix(M) : ModBrdt -> AlgMatElt
KroneckerProduct(A, B) : Mtrx, Mtrx -> Mtrx
LexProduct(G, H) : GrphDir, GrphDir -> GrphDir
PrimitiveWreathProduct(G, H) : GrpPerm, GrpPerm -> GrpPerm
PrimitiveWreathProduct(Q) : [ GrpPerm ] -> GrpPerm
ProductProjectiveSpace(k,N) : Rng,SeqEnum -> PrjScrl
ProductRepresentation(a) : FldFunGElt -> [FldFunGElt], [RngIntElt]
ProductRepresentation(a) : RngFunOrdElt -> [RngElt], [RngIntElt]
ProductRepresentation(Q, S) : [FldFunGElt], [RngIntElt] -> FldFunGElt
TensorProduct(A, B) : AlgBas, AlgBas-> AlgBas
TensorProduct(A, B) : AlgMat, AlgMat -> AlgMat
TensorProduct(a, b) : AlgMatElt, AlgMatElt -> AlgMatElt
TensorProduct(G, H) : GrphDir, GrphDir -> GrphDir
TensorProduct(L, M) : Lat, Lat -> Lat
TensorProduct(U, V) : ModTupFld, ModTupFld -> FldElt
TensorProduct(u, v) : ModTupFldElt, ModTupFldElt -> FldElt
TensorProduct(M, N) : ModTupRng, ModTupRng -> ModTupRng
TensorWreathProduct(G, H) : GrpMat, GrpPerm -> GrpMat
WordProduct( F, w1, w2 ) : GrpFP, GrpFPElt, GrpFPElt -> GrpFPElt
WreathProduct(G, H) : GrpMat, GrpPerm -> GrpMat
WreathProduct(G, H) : GrpPC, GrpPC -> GrpPC
WreathProduct(G, H, f) : GrpPC, GrpPC, Map -> GrpPC
WreathProduct(G, H) : GrpPerm, GrpPerm -> GrpPerm, SeqEnum[Map], Map, Map
WreathProduct(Q) : [ GrpPerm ] -> GrpPerm

product

KSpace(K, n, F) : Fld, RngIntElt, Mtrx -> ModTupFld
Construction of a Vector Space with Inner Product Matrix (VECTOR SPACES)
Inner Products (FREE MODULES)
Operators (OVERVIEW)
Tensor Products (MATRIX GROUPS)
The Cartesian Product Constructors (SETS)
TUPLES AND CARTESIAN PRODUCTS
Unions and Products of Graphs (GRAPHS)

ProductProjectiveSpace

ProductProjectiveSpace(k,N) : Rng,SeqEnum -> PrjScrl

ProductRepresentation

ProductRepresentation(a) : FldFunGElt -> [FldFunGElt], [RngIntElt]
ProductRepresentation(a) : RngFunOrdElt -> [RngElt], [RngIntElt]
ProductRepresentation(Q, S) : [FldFunGElt], [RngIntElt] -> FldFunGElt

Products

BasisProducts(A) : AlgGen -> [[ AlgGenElt ]]
AlgMat_Products (Example H72E5)
GrpPerm_Products (Example H20E8)

products

Direct Products and Wreath Products (PERMUTATION GROUPS)
Tensor Products of K[G]-Modules (MODULES OVER A MATRIX ALGEBRA)

Progression

Seq_Progression (Example H8E1)
Set_Progression (Example H7E5)

progression

Sequences (OVERVIEW)
Sets (OVERVIEW)
The Arithmetic Progression Constructors (SEQUENCES)
The Arithmetic Progression Constructors (SETS)

Proj

Proj(R) : AlgAff -> Sch,Prj
ProjectiveSpace(R) : RngMPol -> Prj

proj-cl-commutes

Crv_proj-cl-commutes (Example H82E7)

Projection

Projection(X,Y) : Prj,Prj -> MapSch
ProjectionChains(X,DB) : VSrfK3,SeqEnum -> SeqEnum
ProjectionFromNonsingularPoint(X,p) : Sch,Pt -> Sch,MapSch,Sch

ProjectionChains

ProjectionChains(X,DB) : VSrfK3,SeqEnum -> SeqEnum

ProjectionFromNonsingularPoint

ProjectionFromNonsingularPoint(X,p) : Sch,Pt -> Sch,MapSch,Sch

Projective

projective

Combinatorial and Geometrical Structures (OVERVIEW)
Indecomposable Projective Modules (BASIC ALGEBRAS)
Projective Automorphisms (SCHEMES)
Projective Covers (BASIC ALGEBRAS)
The Connection between Projective and Affine Planes (FINITE PLANES)

projective-affine

The Connection between Projective and Affine Planes (FINITE PLANES)

projective-automorphism-group

Scheme_projective-automorphism-group (Example H81E26)

projective-automorphisms

Projective Automorphisms (SCHEMES)

projective-closure

Scheme_projective-closure (Example H81E10)

projective-covers

Projective Covers (BASIC ALGEBRAS)

ProjectiveClosure

ProjectiveClosure(f) : MapSch -> MapSch
ProjectiveClosure(A): Sch -> Sch
ProjectiveClosure(C) : Sch -> Sch
ProjectiveClosure(X) : Sch -> Sch

ProjectiveClosureMap

PCMap(A) : Aff -> MapSch
ProjectiveClosureMap(A) : Aff -> MapSch

ProjectiveCover

ProjectiveCover(M) : ModAlg -> ModAlg, ModMatFldElt, SeqEnum[ModMatFldElt], SeqEnum[ModMatFldElt], SeqEnum[RngIntElt]

ProjectiveCurve

ProjectiveCurve(Div) : DivCrv -> Crv
Curve(Div) : DivCrv -> Crv
Curve(D) : DivCrvElt -> Crv
Curve(F) : FldFun -> Crv
Curve(P) : PlcCrv -> Crv
Curve(P) : PlcCrvElt -> Crv
ProjectiveCurve(F) : FldFun -> Crv

ProjectiveEmbedding

ProjectiveEmbedding(P) : PlaneAff -> PlaneProj, PlanePtSet, PlaneLnSet, Map

ProjectiveGammaLinearGroup

PGammaL(arguments)
ProjectiveGammaLinearGroup(arguments)

ProjectiveGammaUnitaryGroup

PGammaU(arguments)
ProjectiveGammaUnitaryGroup(arguments)

ProjectiveGeneralLinearGroup

PGL(arguments)
ProjectiveGeneralLinearGroup(arguments)

ProjectiveGeneralOrthogonalGroup

ProjectiveGeneralOrthogonalGroup(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}
PGO(arguments)

ProjectiveGeneralOrthogonalGroupMinus

ProjectiveGeneralOrthogonalGroupMinus(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}
PGOMinus(arguments)

ProjectiveGeneralOrthogonalGroupPlus

ProjectiveGeneralOrthogonalGroupPlus(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}
PGOPlus(arguments)

ProjectiveGeneralUnitaryGroup

PGU(arguments)
ProjectiveGeneralUnitaryGroup(arguments)

ProjectiveModule

ProjectiveModule(B, i) : AlgBas, RngIntElt -> ModRng
ProjectiveModule(B, S) : AlgBas, SeqEnum[RngIntElt] -> ModAlg, SeqEnum, SeqEnum

ProjectiveOmega

POmega(arguments)
ProjectiveOmega(arguments)

ProjectiveOmegaMinus

POmegaMinus(arguments)
ProjectiveOmegaMinus(arguments)

ProjectiveOmegaPlus

POmegaPlus(arguments)
ProjectiveOmegaPlus(arguments)

ProjectiveOrder

ProjectiveOrder(a) : AlgMatElt -> RngIntElt
ProjectiveOrder(A) : AlgMatElt -> RngIntElt, RngElt

ProjectivePlane

ProjectivePlane(k) : Rng -> Prj
ProjectiveSpace(k,2) : Rng,RngIntElt -> Prj

ProjectiveResolution

ProjectiveResolution(M, n) : ModAlg, RngIntElt -> ModCpx, ModMatFldElt

Projectives

NumberOfProjectives(A) : AlgBas -> RngIntElt

ProjectiveSigmaLinearGroup

PSigmaL(arguments)
ProjectiveSigmaLinearGroup(arguments)

ProjectiveSigmaSymplecticGroup

PSigmaSp(arguments)
ProjectiveSigmaSymplecticGroup(arguments)

ProjectiveSigmaUnitaryGroup

PSigmaU(arguments)
ProjectiveSigmaUnitaryGroup(arguments)

ProjectiveSpace

ProjectivePlane(k) : Rng -> Prj
ProjectiveSpace(k,2) : Rng,RngIntElt -> Prj
ProjectiveSpace(k,n) : Rng,RngIntElt -> Prj
ProjectiveSpace(R) : RngMPol -> Prj

ProjectiveSpecialLinearGroup

PSL(arguments)
ProjectiveSpecialLinearGroup(arguments)

ProjectiveSpecialOrthogonalGroup

ProjectiveSpecialOrthogonalGroup(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}
PSO(arguments)

ProjectiveSpecialOrthogonalGroupMinus

ProjectiveSpecialOrthogonalGroupMinus(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}
PSOMinus(arguments)

ProjectiveSpecialOrthogonalGroupPlus

ProjectiveSpecialOrthogonalGroupPlus(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}
PSOPlus(arguments)

ProjectiveSpecialUnitaryGroup

PSU(arguments)
ProjectiveSpecialUnitaryGroup(arguments)

ProjectiveSuzukiGroup

PSz(arguments)
ProjectiveSuzukiGroup(arguments)

ProjectiveSymplecticGroup

PSp(arguments)
ProjectiveSymplecticGroup(arguments)

Projectivity

Projectivity(A,M) : Aff,Mtrx -> MapAutSch

projectivity

Scheme_projectivity (Example H81E25)

ProjPl

Combinatorial and Geometrical Structures (OVERVIEW)

Prompt

GetIgnorePrompt() : -> BoolElt
SetIgnorePrompt(b) : BoolElt ->
SetPrompt(s) : MonStgElt ->

prompt

Prompt (OVERVIEW)

PRoot

HasPRoot(L) : RngLoc -> BoolElt

prop

Properties (LOCAL RINGS AND FIELDS)
Properties (p-ADIC RINGS AND FIELDS)

prop-generic

Properties (LOCAL RINGS AND FIELDS)
Properties (p-ADIC RINGS AND FIELDS)

Proper

IsProper(I) : RngMPol -> BoolElt
IsProper(I) : RngMPolRes -> BoolElt
IsProperChainMap(f) : MapChn -> BoolElt

proper

The Number Field Sieve (RING OF INTEGERS)

Properties

RootDtm_Properties (Example H35E8)

properties

Abstract Properties of a Group (PERMUTATION GROUPS)
Basic Group Properties (FINITE SOLUBLE GROUPS)
Basic Group Properties (p-GROUPS)
Basic Invariants of a Matrix Group (MATRIX GROUPS)
Determinant and Other Properties (MATRICES)
Elementary Properties of a Subgroup (PERMUTATION GROUPS)
Elementary Properties of Subgroups (MATRIX GROUPS)
Geometrical Properties (SCHEMES)
Minimal and Characteristic Polynomials and Eigenvalues (MATRICES)
Properties (COXETER GROUPS)
Properties of AG--Codes (LINEAR CODES OVER FINITE FIELDS)
Properties of Elements (FINITE SOLUBLE GROUPS)
Properties of Incidence Geometries and Coset Geometries (INCIDENCE GEOMETRY)
Properties of Invariant Rings (INVARIANT RINGS OF FINITE GROUPS)
Properties of Lattices (LATTICES)
Properties of Module Elements (MODULES OVER A MATRIX ALGEBRA)
Properties of root data (ROOT DATA FOR LIE THEORY)
Properties of Subgroups (FINITE SOLUBLE GROUPS)
Properties of Vectors (FREE MODULES)

properties-root-datum

Properties of root data (ROOT DATA FOR LIE THEORY)

properties-subgroup

Elementary Properties of a Subgroup (PERMUTATION GROUPS)

propertiess

Properties of groups of Lie type (GROUPS OF LIE TYPE)

property

Properties (ALGEBRAICALLY CLOSED FIELDS)

Proportional

IsProportional(X, k) : Mtrx, RngIntElt -> BoolElt, Tup

Prune

Prune(A) : FldAC ->
Prune(~S) : List ->
Prune(S) : List -> List
Prune(C) : ModCpx -> ModCpx
Prune(C,n) : ModCpx, RngIngElt -> ModCpx
Prune(~S) : SeqEnum ->
Prune(~T) : Tup ->
Prune(T) : Tup -> Tup

pSelmer

pSelmerGroup(O, p, S) : RngOrd P, prime p, { RngOrdIdl } -> G, m

pSelmerGroup

pSelmerGroup(O, p, S) : RngOrd P, prime p, { RngOrdIdl } -> G, m

Pseudo

PseudoAdd(P1, P2, P3) : SrfKumPt, SrfKumPt, SrfKumPt -> SrfKumPt
PseudoAddMultiple(P1, P2, P3, n) : SrfKumPt, SrfKumPt, SrfKumPt, RngIntElt -> SrfKumPt
PseudoBasis(M) : ModOrd -> SeqEnum
PseudoRemainder(f, g) : RngUPolElt, RngUPolElt -> RngUPolElt

pseudo

PSEUDO-RANDOM BIT SEQUENCES

pseudo-random-sequences

PSEUDO-RANDOM BIT SEQUENCES

PseudoAdd

PseudoAdd(P1, P2, P3) : SrfKumPt, SrfKumPt, SrfKumPt -> SrfKumPt

PseudoAddMultiple

PseudoAddMultiple(P1, P2, P3, n) : SrfKumPt, SrfKumPt, SrfKumPt, RngIntElt -> SrfKumPt

PseudoBasis

PseudoBasis(M) : ModOrd -> SeqEnum

PseudoRemainder

PseudoRemainder(f, g) : RngUPolElt, RngUPolElt -> RngUPolElt

Psi

IsogenyMapPsi(I) : Map -> RngUPolElt
IsogenyMapPsiMulti(I) : Map -> RngUPolElt
IsogenyMapPsiSquared(I) : Map -> RngUPolElt
LogDerivative(s) : FldPrElt -> FldPrElt

PSigma

PSigmaL(arguments)
ProjectiveSigmaLinearGroup(arguments)
ProjectiveSigmaSymplecticGroup(arguments)
ProjectiveSigmaUnitaryGroup(arguments)

PSigmaL

PSigmaL(arguments)
ProjectiveSigmaLinearGroup(arguments)

PSigmaSp

PSigmaSp(arguments)
ProjectiveSigmaSymplecticGroup(arguments)

PSigmaU

PSigmaU(arguments)
ProjectiveSigmaUnitaryGroup(arguments)

PSL

PSL(arguments)
ProjectiveSpecialLinearGroup(arguments)

PSL2

PSL2(R) : Rng -> GrpPSL2

PSO

ProjectiveSpecialOrthogonalGroup(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}
PSO(arguments)

PSOMinus

ProjectiveSpecialOrthogonalGroupMinus(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}
PSOMinus(arguments)

PSOPlus

ProjectiveSpecialOrthogonalGroupPlus(n, q) : RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt @}
PSOPlus(arguments)

PSp

PSp(arguments)
ProjectiveSymplecticGroup(arguments)

PSU

PSU(arguments)
ProjectiveSpecialUnitaryGroup(arguments)

pSylowComputation

GrpAbGen_pSylowComputation (Example H27E5)

PSz

PSz(arguments)
ProjectiveSuzukiGroup(arguments)

ptbools

Tests for Points and Faces (NEWTON POLYGONS)

pts-blks-ops

Design_pts-blks-ops (Example H94E8)

Puiseux

DuvalPuiseuxExpansion(f, n) : RngUPolElt, RngIntElt -> SeqEnum
ParametrizationToPuiseux(T) : Tup -> SeqEnum
PuiseuxExpansion(f, n) : RngUPolElt, RngIntElt -> SeqEnum[RngSerPuisElt]
PuiseuxExponents(p) : RngSerElt -> SeqEnum
PuiseuxExponentsCommon(p, q) : RngSerElt, RngSerElt -> SeqEnum
PuiseuxSeriesRing(R) : Rng -> RngSerPuis
PuiseuxToParametrization(S) : RngSerElt -> Tup
FldAC_Puiseux (Example H56E3)

puiseux

Puiseux Series (POWER, LAURENT AND PUISEUX SERIES)

PuiseuxExpansion

PuiseuxExpansion(f, n) : RngUPolElt, RngIntElt -> SeqEnum[RngSerPuisElt]

PuiseuxExponents

PuiseuxExponents(p) : RngSerElt -> SeqEnum

PuiseuxExponentsCommon

PuiseuxExponentsCommon(p, q) : RngSerElt, RngSerElt -> SeqEnum

PuiseuxSeriesRing

PuiseuxSeriesRing(R) : Rng -> RngSerPuis

PuiseuxToParametrization

PuiseuxToParametrization(S) : RngSerElt -> Tup

pull_back

Pull-Back of an Element (GENERIC ABELIAN GROUPS)

Pullback

Pullback(f,P) : MapCrvHyp, PtHyp -> PtHyp
P @@ f : PtHyp, MapCrvHyp -> PtHyp
Pullback(f,L) : AmbProjMap,LinSys -> LinSys
Pullback(N, f1, M1, f2, M2) : ModAlg, ModMatFldElt, ModAlg, ModMatFldElt, ModAlg -> ModAlg, ModMatFldElt, ModMatFldElt

Puncture

PunctureCode(C, i) : Code, RngIntElt -> Code
PunctureCode(C, i) : Code, RngIntElt -> Code
PunctureCode(C, S) : Code, { RngIntElt } -> Code
PunctureCode(C, S) : Code, { RngIntElt } -> Code

PunctureCode

PunctureCode(C, i) : Code, RngIntElt -> Code
PunctureCode(C, i) : Code, RngIntElt -> Code
PunctureCode(C, S) : Code, { RngIntElt } -> Code
PunctureCode(C, S) : Code, { RngIntElt } -> Code

Punctures

NumberOfPunctures(C): Crv -> RngIntElt

Pure

PureBraidGroup( W ) : GrpCox -> GrpFP, Map
PureBraidGroup( F ) : GrpFP -> GrpFP, Map
PureLattice(L) : Lat -> Lat

PureBraidGroup

PureBraidGroup( W ) : GrpCox -> GrpFP, Map
PureBraidGroup( F ) : GrpFP -> GrpFP, Map

PureLattice

PureLattice(L) : Lat -> Lat

Push

IndentPush() : ->
PushThroughIsogeny(I, v) : Map, RngUPolElt -> RngUPolElt

Pushout

Pushout(M, f1, N1, f2, N2) : ModAlg, ModMatFldElt, ModAlg, ModMatFldElt, ModAlg -> ModAlg, ModMatFldElt, ModMatFldElt

PushThroughIsogeny

PushThroughIsogeny(I, v) : Map, RngUPolElt -> RngUPolElt

Put

Put(F, S) : File, MonStgElt ->

Puts

Puts(F, S) : File, MonStgElt ->

[____] [____] [_____] [____] [__] [Index] [Root]