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#############################################################################
##
#W ringpoly.gi GAP Library Frank Celler
##
#Y Copyright (C) 1996, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany
#Y (C) 1999 School Math and Comp. Sci., University of St. Andrews, Scotland
#Y Copyright (C) 2002 The GAP Group
##
## This file contains the methods for attributes, properties and operations
## for polynomial rings.
##
Revision.ringpoly_gi :=
"@(#)$Id: ringpoly.gi,v 4.41.2.5 2008/04/15 10:02:11 stefan Exp $";
#############################################################################
##
#M GiveNumbersNIndeterminates(<ratfunfam>,<count>,<names>,<avoid>)
BindGlobal("GiveNumbersNIndeterminates",function(rfam,cnt,nam,avoid)
local idn,i,nbound;
avoid:=List(avoid,IndeterminateNumberOfLaurentPolynomial);
idn:=[];
i:=1;
while Length(idn)<cnt do
nbound:=IsBound(nam[Length(idn)+1]);
# skip unwanted indeterminates
while (i in avoid) or (nbound and HasIndeterminateName(rfam,i)) do
i:=i+1;
od;
Add(idn,i);
if nbound then
SetIndeterminateName(rfam,i,nam[Length(idn)]);
fi;
i:=i+1;
od;
return idn;
end);
#############################################################################
##
#M PolynomialRing( <ring>, <rank> ) . . . full polynomial ring over a ring
##
#T polynomial rings should be special cases of free magma rings! one needs
#T to set an underlying magma with one, and modify the type to be
#T AlgebraWithOne and FreeMagmaRingWithOne. (for example, ring generators in
#T the case of polynomial rings over finite fields are then automatically
#T computable ...)
##
#############################################################################
InstallMethod( PolynomialRing,"indetlist", true, [ IsRing, IsList ],
# force higher ranking than following (string) method
1,
function( r, n )
local efam, rfun, zero, one, ind, i, type, prng;
if IsPolynomialFunctionCollection(n) and ForAll(n,IsLaurentPolynomial) then
n:=List(n,IndeterminateNumberOfLaurentPolynomial);
fi;
if IsEmpty(n) or not IsInt(n[1]) then
TryNextMethod();
fi;
# get the elements family of the ring
efam := ElementsFamily( FamilyObj(r) );
# get the rational functions of the elements family
rfun := RationalFunctionsFamily(efam);
# cache univariate rings - they might be created often
if not IsBound(r!.univariateRings) then
r!.univariateRings:=[];
fi;
if Length(n)=1
# some bozo might put in a ridiculous number
and n[1]<10000
# only cache for the prime field
and IsField(r)
and IsBound(r!.univariateRings[n[1]]) then
return r!.univariateRings[n[1]];
fi;
# first the indeterminates
zero := Zero(r);
one := One(r);
ind := [];
for i in n do
Add( ind, UnivariatePolynomialByCoefficients(efam,[zero,one],i) );
od;
# construct a polynomial ring
type := IsPolynomialRing and IsAttributeStoringRep and IsFreeLeftModule;
# over a field the ring should be an algebra with one.
if HasIsField(r) and IsField(r) then
type:=type and IsAlgebraWithOne;
fi;
if Length(n) = 1 and HasIsField(r) and IsField(r) then
type := type and IsUnivariatePolynomialRing and IsEuclideanRing;
#and IsAlgebraWithOne; # done above already
elif Length(n) = 1 and IsRingWithOne(r) then
type := type and IsUnivariatePolynomialRing and IsFLMLORWithOne;
elif Length(n) = 1 then
type := type and IsUnivariatePolynomialRing;
fi;
# Polynomial rings over commutative rings are themselves commutative.
if HasIsCommutative( r ) and IsCommutative( r ) then
type:= type and IsCommutative;
fi;
# Polynomial rings over commutative rings are themselves commutative.
if HasIsAssociative( r ) and IsAssociative( r ) then
type:= type and IsAssociative;
fi;
# set categories to allow method selection according to base ring
if HasIsField(r) and IsField(r) then
if IsFinite(r) then
type := type and IsFiniteFieldPolynomialRing;
elif IsRationals(r) then
type := type and IsRationalsPolynomialRing;
elif # catch algebraic extensions
IsIdenticalObj(One(r),1) and IsAbelianNumberField( r ) then
type:= type and IsAbelianNumberFieldPolynomialRing;
elif IsAlgebraicExtension(r) then
type:= type and IsAlgebraicExtensionPolynomialRing;
fi;
fi;
prng := Objectify( NewType( CollectionsFamily(rfun), type ), rec() );
# set the left acting domain
SetLeftActingDomain( prng, r );
# set the indeterminates
SetIndeterminatesOfPolynomialRing( prng, ind );
# set known properties
SetIsFinite( prng, false );
SetIsFiniteDimensional( prng, false );
SetSize( prng, infinity );
# set the coefficients ring
SetCoefficientsRing( prng, r );
# set one and zero
SetOne( prng, ind[1]^0 );
SetZero( prng, ind[1]*Zero(r) );
# set the generators left operator ring-with-one if the rank is one
if IsRingWithOne(r) then
SetGeneratorsOfLeftOperatorRingWithOne( prng, ind );
fi;
if Length(n)=1 and n[1]<10000
# only cache for the prime field
and IsField(r) then
r!.univariateRings[n[1]]:=prng;
fi;
# and return
return prng;
end );
InstallMethod( PolynomialRing,"names",true, [ IsRing, IsList ], 0,
function( r, nam )
if not IsString(nam[1]) then
TryNextMethod();
fi;
return PolynomialRing( r, GiveNumbersNIndeterminates(
RationalFunctionsFamily(ElementsFamily(FamilyObj(r))),
Length(nam),nam,[]));
end );
InstallMethod( PolynomialRing,"rank",true, [ IsRing, IsPosInt ], 0,
function( r, n )
return PolynomialRing( r, [ 1 .. n ] );
end );
InstallOtherMethod( PolynomialRing,"rank,avoid",true,
[ IsRing, IsPosInt,IsList ], 0,
function( r, n,a )
return PolynomialRing( r, GiveNumbersNIndeterminates(
RationalFunctionsFamily(ElementsFamily(FamilyObj(r))),n,[],a));
end );
InstallOtherMethod(PolynomialRing,"names,avoid",true,[IsRing,IsList,IsList],0,
function( r, nam,a )
return PolynomialRing( r, GiveNumbersNIndeterminates(
RationalFunctionsFamily(ElementsFamily(FamilyObj(r))),
Length(nam),nam,a));
end );
#############################################################################
InstallOtherMethod( PolynomialRing,
true,
[ IsRing ],
0,
function( r )
return PolynomialRing(r,[1]);
end );
#############################################################################
##
#M UnivariatePolynomialRing( <ring> ) . . full polynomial ring over a ring
##
InstallMethod( UnivariatePolynomialRing,"indet 1", true, [ IsRing ], 0,
function( r )
return PolynomialRing( r, [1] );
end );
InstallOtherMethod(UnivariatePolynomialRing,"indet number",true,
[ IsRing,IsPosInt ], 0,
function( r,n )
return PolynomialRing( r, [n] );
end );
InstallOtherMethod(UnivariatePolynomialRing,"name",true,
[ IsRing,IsString], 0,
function( r,n )
if not IsString(n) then
TryNextMethod();
fi;
return PolynomialRing( r, GiveNumbersNIndeterminates(
RationalFunctionsFamily(ElementsFamily(FamilyObj(r))),1,[n],[]));
end);
InstallOtherMethod(UnivariatePolynomialRing,"avoid",true,
[ IsRing,IsList], 0,
function( r,a )
if not IsRationalFunction(a[1]) then
TryNextMethod();
fi;
return PolynomialRing( r, GiveNumbersNIndeterminates(
RationalFunctionsFamily(ElementsFamily(FamilyObj(r))),1,[],a));
end);
InstallOtherMethod(UnivariatePolynomialRing,"name,avoid",true,
[ IsRing,IsString,IsList], 0,
function( r,n,a )
if not IsString(n[1]) then
TryNextMethod();
fi;
return PolynomialRing( r, GiveNumbersNIndeterminates(
RationalFunctionsFamily(ElementsFamily(FamilyObj(r))),1,[n],a));
end);
#############################################################################
##
#M ViewString( <pring> ) . . . . . . . . . . . . . . . for a polynomial ring
##
InstallMethod( ViewString,
"for a polynomial ring", true, [ IsPolynomialRing ], 0,
R -> Concatenation(String(LeftActingDomain(R)),
Filtered(String(IndeterminatesOfPolynomialRing(R)),
ch -> ch <> ' ')) );
#############################################################################
##
#M ViewObj( <pring> ) . . . . . . . . . . . . . . . . for a polynomial ring
##
InstallMethod( ViewObj,
"for a polynomial ring", true, [ IsPolynomialRing ],
# override the higher ranking FLMLOR method
RankFilter(IsFLMLOR),
function( R )
Print(ViewString(R));
end );
#############################################################################
##
#M String( <pring> ) . . . . . . . . . . . . . . . . . for a polynomial ring
##
InstallMethod( String,
"for a polynomial ring", true, [ IsPolynomialRing ],
RankFilter(IsFLMLOR),
R -> Concatenation("PolynomialRing( ",
String(LeftActingDomain(R)),", ",
String(IndeterminatesOfPolynomialRing(R)),
" )") );
#############################################################################
##
#M PrintObj( <pring> )
##
InstallMethod( PrintObj,
"for a polynomial ring",
true,
[ IsPolynomialRing ],
# override the higher ranking FLMLOR method
RankFilter(IsFLMLOR),
function( obj )
local i,f;
Print( "PolynomialRing( ", LeftActingDomain( obj ), ", [");
f:=false;
for i in IndeterminatesOfPolynomialRing(obj) do
if f then Print(", ");fi;
Print("\"",i,"\"");
f:=true;
od;
Print("] )" );
end );
#############################################################################
##
#M Indeterminate( <ring>,<nr> )
##
InstallMethod( Indeterminate,"number", true, [ IsRing,IsPosInt ],0,
function( r,n )
return UnivariatePolynomialByCoefficients(ElementsFamily(FamilyObj(r)),
[Zero(r),One(r)],n);
end);
InstallOtherMethod(Indeterminate,"fam,number",true,[IsFamily,IsPosInt],0,
function(fam,n)
return UnivariatePolynomialByCoefficients(fam,[Zero(fam),One(fam)],n);
end);
InstallOtherMethod( Indeterminate,"number 1", true, [ IsRing ],0,
function( r )
return UnivariatePolynomialByCoefficients(ElementsFamily(FamilyObj(r)),
[Zero(r),One(r)],1);
end);
InstallOtherMethod( Indeterminate,"number, avoid", true, [ IsRing,IsList ],0,
function( r,a )
if not IsRationalFunction(a[1]) then
TryNextMethod();
fi;
r:=ElementsFamily(FamilyObj(r));
return UnivariatePolynomialByCoefficients(r,[Zero(r),One(r)],
GiveNumbersNIndeterminates(RationalFunctionsFamily(r),1,[],a)[1]);
end);
InstallOtherMethod( Indeterminate,"number, name", true, [ IsRing,IsString ],0,
function( r,n )
if not IsString(n) then
TryNextMethod();
fi;
r:=ElementsFamily(FamilyObj(r));
return UnivariatePolynomialByCoefficients(r,[Zero(r),One(r)],
GiveNumbersNIndeterminates(RationalFunctionsFamily(r),1,[n],[])[1]);
end);
InstallOtherMethod( Indeterminate,"number, name, avoid",true,
[ IsRing,IsString,IsList ],0,
function( r,n,a )
if not IsString(n) then
TryNextMethod();
fi;
r:=ElementsFamily(FamilyObj(r));
return UnivariatePolynomialByCoefficients(r,[Zero(r),One(r)],
GiveNumbersNIndeterminates(RationalFunctionsFamily(r),1,[n],a)[1]);
end);
#############################################################################
##
#M \. Access to indeterminates
##
InstallMethod(\.,"pring indeterminates",true,[IsPolynomialRing,IsPosInt],
function(r,n)
local v, fam, a, i;
v:=IndeterminatesOfPolynomialRing(r);
n:=NameRNam(n);
if ForAll(n,i->i in CHARS_DIGITS) then
# number
n:=Int(n);
if Length(v)>=n then
return v[n];
fi;
else
fam:=ElementsFamily(FamilyObj(r));
for i in v do
a:=IndeterminateNumberOfLaurentPolynomial(i);
if HasIndeterminateName(fam,a) and IndeterminateName(fam,a)=n then
return i;
fi;
od;
fi;
TryNextMethod();
end);
#############################################################################
##
#M <poly> in <polyring>
##
InstallMethod( \in,
"polynomial in polynomial ring",
IsElmsColls,
[ IsPolynomialFunction,
IsPolynomialRing ],
0,
function( p, R )
local ext, crng, inds, exp, i;
# <p> must at least be a polynomial
if not IsPolynomial(p) then
return false;
fi;
# get the external representation
ext := ExtRepPolynomialRatFun(p);
# and the indeterminates and coefficients ring of <R>
crng := CoefficientsRing(R);
inds := Set( List( IndeterminatesOfPolynomialRing(R),
x -> ExtRepPolynomialRatFun(x)[1][1] ) );
# first check the indeterminates
for exp in ext{[ 1, 3 .. Length(ext)-1 ]} do
for i in exp{[ 1, 3 .. Length(exp)-1 ]} do
if not i in inds then
return false;
fi;
od;
od;
# then the coefficients
for i in ext{[ 2, 4 .. Length(ext) ]} do
if not i in crng then
return false;
fi;
od;
return true;
end );
#############################################################################
##
#M DefaultRingByGenerators( <gens> ) . . . . ring containing a collection
##
InstallMethod( DefaultRingByGenerators,
true,
[ IsRationalFunctionCollection ],
0,
function( ogens )
local gens,ind, cfs, g, ext, exp, i,univ;
if not ForAll( ogens, IsPolynomial ) then
TryNextMethod();
fi;
# the indices of the non-constant functions that have an indeterminate
# number
g:=Filtered([1..Length(ogens)],
i->HasIndeterminateNumberOfUnivariateRationalFunction(ogens[i]) and
HasCoefficientsOfLaurentPolynomial(ogens[i]));
univ:=Filtered(ogens{g},
i->DegreeOfUnivariateLaurentPolynomial(i)>-1 and
DegreeOfUnivariateLaurentPolynomial(i)<infinity);
gens:=ogens{Difference([1..Length(ogens)],g)};
# univariate indeterminates set
ind := Set(List(univ,IndeterminateNumberOfUnivariateRationalFunction));
cfs := []; # univariate coefficients set
for g in univ do
UniteSet(cfs,CoefficientsOfUnivariateLaurentPolynomial(g)[1]);
od;
# the nonunivariate ones
for g in gens do
ext := ExtRepPolynomialRatFun(g);
for exp in ext{[ 1, 3 .. Length(ext)-1 ]} do
for i in exp{[ 1, 3 .. Length(exp)-1 ]} do
AddSet( ind, i );
od;
od;
for i in ext{[ 2, 4 .. Length(ext) ]} do
Add( cfs, i );
od;
od;
if Length(cfs)=0 then
# special case for zero polynomial
Add(cfs,Zero(CoefficientsFamily(FamilyObj(ogens[1]))));
fi;
if Length(ind)=0 then
# this can only happen if the polynomials are constant. Enforce Index 1
return PolynomialRing( DefaultField(cfs), [1] );
else
return PolynomialRing( DefaultField(cfs), ind );
fi;
end );
#############################################################################
##
#M MinimalPolynomial( <ring>, <elm> )
##
InstallOtherMethod( MinimalPolynomial,"supply indeterminate 1",
[ IsRing, IsMultiplicativeElement and IsAdditiveElement ],
function(r,e)
return MinimalPolynomial(r,e,1);
end);
#############################################################################
##
#M StandardAssociate( <pring>, <upol> )
##
InstallMethod(StandardAssociate,"normalize leading coefficient",IsCollsElms,
[IsPolynomialRing, IsPolynomial],0,
function(R,f)
local c;
c:=LeadingCoefficient(f);
return f*StandardAssociate(CoefficientsRing(R),c)/c;
end);
#############################################################################
##
#E
|