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Added Documentation for DecomPoly and NormalizerViaRadical #2360

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Apr 17, 2018
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Added Documentation for DecomPoly and NormalizerViaRadical #2360

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107dbd5
Added documentation for `NormalizerViaRadical`.
hulpke Apr 12, 2018
444be92
Added documentation (and minor fixes) for DecomPoly.
hulpke Apr 11, 2018
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9 changes: 9 additions & 0 deletions doc/ref/algfld.xml
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Expand Up @@ -95,6 +95,15 @@ gap> m^2;

<#Include Label="IsAlgebraicElement">

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Finding Subfields">
<Heading>Finding Subfields</Heading>

<#Include Label="IdealDecompositionsOfPolynomial">

</Section>
</Chapter>

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9 changes: 8 additions & 1 deletion doc/ref/groups.xml
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Expand Up @@ -587,7 +587,6 @@ see <Cite Key="BJR87"/>.

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Tests for the Availability of Methods">
<Heading>Tests for the Availability of Methods</Heading>
Expand All @@ -601,6 +600,14 @@ see <Cite Key="BJR87"/>.
<#Include Label="CanComputeIsSubset">
<#Include Label="KnowsHowToDecompose">

</Section>

<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Specific functions for Normalizer calculation">
<Heading>Specific functions for Normalizer calculation</Heading>

<#Include Label="NormalizerViaRadical">

</Section>
</Chapter>

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42 changes: 42 additions & 0 deletions lib/algfld.gd
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Expand Up @@ -188,3 +188,45 @@ DeclareGlobalFunction("AlgExtEmbeddedPol");

DeclareGlobalFunction("AlgExtSquareHensel");

#############################################################################
##
#F IdealDecompositionsOfPolynomial( <f> [:"onlyone"] ) finds ideal decompositions of rational f
##
## <#GAPDoc Label="IdealDecompositionsOfPolynomial">
## <ManSection>
## <Func Name="IdealDecompositionsOfPolynomial" Arg='pol'/>
##
## <Description>
## Let <M>f</M> be a univariate, rational, irreducible, polynomial. A
## pair <M>g</M>,<M>h</M> of polynomials of degree strictly
## smaller than that of <M>f</M>, such that <M>f(x)|g(h(x))</M> is
## called an ideal decomposition. In the context of field
## extensions, if <M>\alpha</M> is a root of <M>f</M> in a suitable extension
## and <M>Q</M> the field of rational numbers. Such decompositions correspond
## to (proper) subfields <M>Q\lt Q(\beta)\lt Q(\alpha)</M>, where <M>g</M> is the
## minimal polynomial of <M>\beta</M>.
## This function determines such decompositions up to equality of the subfields
## <M>Q(\beta)</M>, thus determining subfields of a given algebraic extension.
## It returns a list of pairs <M>[g,h]</M> (and an empty list if no such
## decomposition exists). If the option <A>onlyone</A> is given it returns at
## most one such decomposition (and performs faster).
## <Example><![CDATA[
## gap> x:=X(Rationals,"x");;pol:=x^8-24*x^6+144*x^4-288*x^2+144;;
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Offtopic: the reference manual says this on X:

X is simply a synonym for Indeterminate. However, we do not recommend to use this synonym which is supported only for the backwards compatibility.

Now, don't get me wrong, I am not suggesting you should change the examples; rather, I always use X myself, so I am wondering if this last sentence in the manual should just be deleted?

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Yes. (At the time when this was written there was a push to make X obsolete as operation.)

## gap> l:=IdealDecompositionsOfPolynomial(pol);
## [ [ x^2+72*x+144, x^6-20*x^4+60*x^2-36 ],
## [ x^2-48*x+144, x^6-21*x^4+84*x^2-48 ],
## [ x^2+288*x+17280, x^6-24*x^4+132*x^2-288 ],
## [ x^4-24*x^3+144*x^2-288*x+144, x^2 ] ]
## gap> List(l,x->Value(x[1],x[2])/pol);
## [ x^4-16*x^2-8, x^4-18*x^2+33, x^4-24*x^2+120, 1 ]
## gap> IdealDecompositionsOfPolynomial(pol:onlyone);
## [ [ x^2+72*x+144, x^6-20*x^4+60*x^2-36 ] ]
## ]]></Example>
## In this example the given polynomial is regular with Galois group
## <M>Q_8</M>, as expected we get four proper subfields.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("IdealDecompositionsOfPolynomial");
DeclareSynonym("DecomPoly",IdealDecompositionsOfPolynomial);
29 changes: 13 additions & 16 deletions lib/algfld.gi
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Expand Up @@ -172,7 +172,8 @@ if Length(extra)>0 and IsString(extra[1]) then
SetCharacteristic(fam,Characteristic(f));
fam!.indeterminateName:=nam;
colf:=CollectionsFamily(fam);
e:=Objectify(NewType(colf,IsAlgebraicExtensionDefaultRep),
e:=Objectify(NewType(colf,
IsAlgebraicExtensionDefaultRep and IsAlgebraicExtension),
rec());

fam!.wholeField:=e;
Expand Down Expand Up @@ -2193,13 +2194,14 @@ end);

#############################################################################
##
#F DecomPoly( <f> [,"all"] ) finds (all) ideal decompositions of rational f
#F IdealDecompositionsOfPolynomial( <f> [,"onlyone"] ) finds ideal decompositions of rational f
## This is equivalent to finding subfields of K(alpha).
##
DecomPoly := function(arg)
local f,n,e,ff,p,ffp,ffd,roots,allroots,nowroots,fm,fft,comb,combi,k,h,i,j,
gut,avoid,blocks,g,m,decom,z,R,scale,allowed,hp,hpc,a,kfam;
f:=arg[1];
InstallGlobalFunction(IdealDecompositionsOfPolynomial,function(f)
local n,e,ff,p,ffp,ffd,roots,allroots,nowroots,fm,fft,comb,combi,k,h,i,j,
gut,avoid,blocks,g,m,decom,z,R,scale,allowed,hp,hpc,a,kfam,only;

only:=ValueOption("onlyone")=true;
n:=DegreeOfUnivariateLaurentPolynomial(f);
if IsPrime(n) then
return [];
Expand Down Expand Up @@ -2255,9 +2257,9 @@ local f,n,e,ff,p,ffp,ffd,roots,allroots,nowroots,fm,fft,comb,combi,k,h,i,j,
fft:=ff{combi};
ffp:=List(fft,i->AlgebraicPolynomialModP(kfam,i,fm[1],p));
roots:=Filtered(fm,i->ForAny(ffp,j->Value(j,i)=Zero(k)));
if Length(roots)<>Sum(ffd{combi}) then
Error("serious error");
fi;
if Length(roots)<>Sum(ffd{combi}) then
Error("serious error");
fi;
allroots:=Union(roots,[fm[1]]);
gut:=true;
j:=1;
Expand All @@ -2270,9 +2272,7 @@ local f,n,e,ff,p,ffp,ffd,roots,allroots,nowroots,fm,fft,comb,combi,k,h,i,j,
if gut then
Info(InfoPoly,2,"block found");
Add(blocks,combi);
if Length(arg)>1 then
gut:=false;
fi;
if only<>true then gut:=false; fi;
fi;
h:=h+1;
od;
Expand Down Expand Up @@ -2328,15 +2328,12 @@ local f,n,e,ff,p,ffp,ffd,roots,allroots,nowroots,fm,fft,comb,combi,k,h,i,j,
#h:=Value(h,X(Rationals)*z);
Add(decom,[g,h]);
od;
if Length(arg)=1 then
decom:=decom[1];
fi;
return decom;
else
Info(InfoPoly,2,"primitive");
return [];
fi;
end;
end);

#############################################################################
##
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32 changes: 32 additions & 0 deletions lib/grp.gd
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Expand Up @@ -4528,6 +4528,38 @@ DeclareGlobalFunction("GroupEnumeratorByClosure");
DeclareOperation( "LowIndexSubgroups",
[ IsGroup, IsPosInt ] );

#############################################################################
##
#F NormalizerViaRadical(<G>,<S>)
##
## <#GAPDoc Label="NormalizerViaRadical">
## <ManSection>
## <Func Name="NormalizerViaRadical" Arg='G,S'/>
##
## <Description>
## This function implements a particular approach, following the
## SolvableRadical paradigm, for calculating the
## normalizer of a subgroup <A>S</A> in <A>G</A>. It is at the moment
## provided only as a separate function, and not as method for the operation
## <C>Normalizer</C>, as it can often be slower than other built-in routines.
## In certain hard cases (non-solvable groups with nontrivial radical), however
## its performance is substantially superior.
## The function thus is provided as a
## non-automated tool for advanced users.
## <Example><![CDATA[
## gap> g:=TransitiveGroup(30,2030);;
## gap> s:=SylowSubgroup(g,5);;
## gap> Size(NormalizerViaRadical(g,s));
## 28800
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Sounds good to me.

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This example of course is slower than the backtrack Normalizer. I have yet to find an example that is while being short enough for the manual.

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That's OK, I think (though we might want to mention that the given example is not actually one where NormalizerViaRadical is faster?)

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The function thus provided as a
there is an is missing here.

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Added a line about example and inserted is.

## ]]></Example>
## Note that this example only demonstrates usage, but that in this case
## in fact the ordinary <C>Normalizer</C> routine performs faster.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("NormalizerViaRadical");

#############################################################################
##
#E
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2 changes: 1 addition & 1 deletion lib/norad.gi
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Expand Up @@ -378,7 +378,7 @@ local expandvec,bassrc,basimg,transl,getbasimg,gettransl,myact2;
end);

# main normalizer routine
BindGlobal("NormalizerViaRadical",function(G,U)
InstallGlobalFunction(NormalizerViaRadical,function(G,U)
local sus,ser,len,factorhom,uf,n,d,up,mran,nran,mpcgs,pcgs,pcisom,nf,ng,np,sub,
central,f,ngm,npm,no2pcgs,part,stb,mods,famo,part0,nopcgs,uff,ufg,prev,
famo2,ufr,dims,vecs,ovecs,vecsz,l,prop,properties,clusters,clusterspaces,
Expand Down