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Quellcode-Bibliothek nparith.gi
Sprache: unbekannt
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######################### BEGIN COPYRIGHT MESSAGE #########################
# GBNP - computing Gröbner bases of noncommutative polynomials
# Copyright 2001-2010 by Arjeh M. Cohen, Dié A.H. Gijsbers, Jan Willem
# Knopper, Chris Krook. Address: Discrete Algebra and Geometry (DAM) group
# at the Department of Mathematics and Computer Science of Eindhoven
# University of Technology.
#
# For acknowledgements see the manual. The manual can be found in several
# formats in the doc subdirectory of the GBNP distribution. The
# acknowledgements formatted as text can be found in the file chap0.txt.
#
# GBNP is free software; you can redistribute it and/or modify it under
# the terms of the Lesser GNU General Public License as published by the
# Free Software Foundation (FSF); either version 2.1 of the License, or
# (at your option) any later version. For details, see the file 'LGPL' in
# the doc subdirectory of the GBNP distribution or see the FSF's own site:
# https://www.gnu.org/licenses/lgpl.html
########################## END COPYRIGHT MESSAGE ##########################
### filename = "nparith.gi"
### vs 0.9
### (04/09/23) added LMonNP, LTermNP and LTermsNP
### (21/09/23) added FactorOutGcdNP
### THIS IS PART OF A GAP PACKAGE FOR COMPUTING WITH NON-COMMUTATIVE POLYNOMIALS
#AddNP:=function(u,v,c,d) local ans,hlp; -> now in nparith3.gi
#LtNP:=function(u,v)
#GtNP:=function(u,v)
#LMonNP:=function(pol)
#LMonsNP:=function(lst)
#LTermNP:=function(pol)
#LTermsNP:=function(lst)
#CleanNP:=function(pol) local i,h,l,v,mons,polh,coeffs,ansmons,anscoeffs;
#MkMonicNP:=function(pol)
#FactorOutGcdNP:=function(pol)
#BimulNP:=function(ga,u,dr) local i,ans;
#MulNP:=function(u,v) local ans,i,j;
#GBNP.LTermsTrace:=function(pol) local i;
#GBNP.MkMonicNPTrace:=function(pol) local i,ans,lc;
#GBNP.AddTrace:=function(trace,le,new,ri,scal) local ans,k,j;
#GBNP.TraceNP:=function(trace,G) local i,l,ans,temp;
#GBNP.Traced:=function(G) local ans,pol,i,n;
#GBNP.IsNPMNotZero:=function(np)
#NumAlgGensNP:=function(np)
#NumAlgGensNPList:=function(Lnp)
#NumModGensNP:=function(npm)
#NumModGensNPList:=function(Lnpm)
##################
### AddNP
### <#GAPDoc Label="AddNP">
### <ManSection>
### <Func Name="AddNP" Comm="" Arg="u, v, c, d" />
### <Returns>
### <A>c</A><M>*</M><A>u</A><M>+</M><A>d</A><M>*</M><A>v</A>
### </Returns>
### <Description>
### Computes <A>c</A><M>*</M><A>u</A><M>+</M><A>d</A><M>*</M><A>v</A>
### where <A>u</A> and <A>v</A> are polynomials in NP format
### and <A>c</A> and <A>d</A> are scalars.
### <P/>
### <#Include Label="example-AddNP">
### </Description>
### </ManSection>
### <#/GAPDoc>
### - Computes c*u+d*v where u,v are non-commutative Polynomials
### and c and d are scalars
###
### Arguments:
### u,v - two non-commutative polynomial
### c,d - two scalars
###
### Returns:
### pol - the non-commutative polynomial c*u+d*v
###
### #AddNP uses: LtNP#
### #AddNP is used in: EvalTrace GBNP.GP2NPM GBNP.IsGrobnerBasisTest GBNP.NormalForm2 GBN P.NormalForm2T GBNP.Spoly GBNP.SpolyTrace GBNP.StrongNormalForm2 GBNP.StrongNormalForm2TS GBNP.StrongNormalForm2TSPTS GBNP.StrongNormalForm2Tall GBNP.StrongNormalForm3Dall GBNP.StrongNormalFormTrace2 GBNP.TraceNP StrongNormalFormTraceDiff#
###
InstallGlobalFunction( AddNP, function(u,v,c,d)
local lans,ans,posu,posv,posans,ulen,vlen,co;
ans:=[[],[]];
# don't add the polynomial if the coefficient is zero
if IsZero(c) then
u:=[[],[]];
fi;
if IsZero(d) then
v:=[[],[]];
fi;
lans:=Length(u[1])+Length(v[1]);
posu:=1;
posv:=1;
posans:=1;
ulen:=Length(u[1]);
vlen:=Length(v[1]);
while posans<=lans do
if posu>ulen then
ans[1][posans]:=v[1][posv];
ans[2][posans]:=v[2][posv]*d;
posv:=posv+1;
elif posv>vlen then
ans[1][posans]:=u[1][posu];
ans[2][posans]:=u[2][posu]*c;
posu:=posu+1;
elif LtNP(u[1][posu],v[1][posv]) then
ans[1][posans]:=v[1][posv];
ans[2][posans]:=v[2][posv]*d;
posv:=posv+1;
elif u[1][posu]=v[1][posv] then
co:=u[2][posu]*c+v[2][posv]*d;
if not IsZero(co) then
ans[1][posans]:=v[1][posv];
ans[2][posans]:=co;
lans:=lans-1;
else
lans:=lans-2;
posans:=posans-1; #stay the same...
fi;
posu:=posu+1;
posv:=posv+1;
else # u[1][posu] greater than v[1][posv]
ans[1][posans]:=u[1][posu];
ans[2][posans]:=u[2][posu]*c;
posu:=posu+1;
fi;
posans:=posans+1;
od;
return [ans[1]{[1..lans]},ans[2]{[1..lans]}];
end);
##################
### LtNP
### <#GAPDoc Label="LtNP">
### <ManSection>
### <Func Name="LtNP" Comm="less than" Arg="u, v" />
### <Returns>
### <C>true</C> if <M>u < v</M> and <C>false</C> if <M>u \geq v</M>
### </Returns>
### <Description>
### Less than function for NP monomials, tests whether
### <A>u</A><M><</M><A>v</A>. The ordering is done by degree and then
### lexicographically.
### <P/>
### <#Include Label="example-LtNP">
### </Description>
### </ManSection>
### <#/GAPDoc>
### - 'Less than' function
###
### Arguments:
### u,v - two monomials
###
### Returns:
### true - if u < v
### false - if u >= v
###
### #LtNP uses:#
### #LtNP is used in: AddNP GBNP.IsGrobnerBasisTest GBNP.LeftObsT GBNP.LeftOccur GBNP.Occur GBNP.ReducePol2 GBNP.ReducePolTrace2 GBNP.RightObsT GBNP.RightOccur GBNP.THeapOTBalanceDown GBNP.THeapOTBalanceUp GBNP.THeapOTCheck THeapOT#
###
InstallGlobalFunction(
LtNP,function(u,v)
if Length(u)<Length(v) then
return(true);
elif Length(u)=Length(v) then
return(u<v);
else return(false);
fi;
end);;
##################
### GtNP
### <#GAPDoc Label="GtNP">
### <ManSection>
### <Func Name="GtNP" Comm="greater than" Arg="u, v" />
### <Returns>
### <C>true</C> if <M>u > v</M> and <C>false</C> if <M>u \leq v</M>
### </Returns>
### <Description>
### Greater than function for monomials <A>u</A> and <A>v</A> represented as in Section
### <Ref Sect="NP"/>. It tests whether
### <A>u</A><M>></M><A>v</A>. The ordering is done by degree and then
### lexicographically.
### <P/>
### <#Include Label="example-GtNP">
### </Description>
### </ManSection>
### <#/GAPDoc>
### - 'Greater than' function
###
### Arguments:
### u,v - two monomials
###
### Returns:
### true - if u > v
### false - if u <= v
###
### used in: - CleanNP
### #GtNP uses:#
### #GtNP is used in:#
###
InstallGlobalFunction( GtNP, function(u,v)
if Length(u)>Length(v) then
return(true);
elif Length(u)=Length(v) then
return(u>v);
else
return(false);
fi;
end);;
##################
### LMonNP and LMonsNP
###
### <#GAPDoc Label="LMonsNP">
### <ManSection>
### <Func Name="LMonNP" Comm="returns the leading monomial of a noncommutative polynomial" Arg="Lnp" />
### <Func Name="LMonsNP" Comm="returns the leading monomials of a list of noncommutative polynomials" Arg="Lnp" />
###
### <Returns>
### The leading monomial or a list of leading monomials.
### </Returns>
###
### <Description>
### These functions return the leading monomial of a polynomial
### (resp. the leading monomials of a list of polynomials) in NP format.
### The polynomials of <A>Lnp</A> are required to be clean;
### see Section <Ref Sect="CleanNP"/>.
### <P/>
### <#Include Label="example-LMonsNP">
### </Description>
### </ManSection>
### <#/GAPDoc>
###
### - Returns the leading monomial(s) of a (list of)
### Noncommutative Polynomials.
### The polynomials are ordered such that the leading monomial comes first.
### If the zero polynomial is given as argument then 'fail' is returned.
###
### Arguments:
### pol - list of Noncommutative Polynomials
###
### Returns:
### - a list of leading monomials
###
### #LMonsNP uses:#
### #LMonsNP is used in: BaseQA BaseQM DimQA DimQM GBNP.AllObs GBNP.AllObsTrunc GBNP.CentralT GBNP.CheckHom GBNP.IsGrobnerBasisTest GBNP.NormalForm2 GBNP.ObsTall GBNP.ObsTrunc GBNP.ReducePol2 GBNP.SGrobnerLoops GBNP.SGrobnerTrunc GBNP.StrongNormalForm2 IsGrobnerPair MakeGrobnerPair THeapOT#
###
InstallGlobalFunction( LMonNP, function(pol)
if IsZero(pol) then
return fail;
else
return pol[1][1];
fi;
end);
InstallGlobalFunction(LMonsNP, lst -> List(lst,LMonNP));
##################
### LTermNP and LTermsNP
###
### <#GAPDoc Label="LTermsNP">
### <ManSection>
### <Func Name="LTermNP" Comm="returns the leading term of a noncommutative polynomial" Arg="Lnp" />
### <Func Name="LTermsNP" Comm="returns the leading terms of a list of noncommutative polynomials" Arg="Lnp" />
###
### <Returns>
### The leading term or a list of leading terms.
### </Returns>
###
### <Description>
### These functions return the leading term of a polynomial
### (resp. the leading terms of a list of polynomials) in NP format.
### The polynomials of <A>Lnp</A> are required to be clean;
### see Section <Ref Sect="CleanNP"/>.
### <P/>
### </Description>
### </ManSection>
### <Example>
### gap> p1 := [[[1,1,2],[1]],[6,-7]];;
### gap> p2 := [[[1,2,2],[2]],[8,-9]];;
### gap> Lnp := [p1,p2];;
### gap> LTermNP( p1 );
### [ [ [ 1, 1, 2 ] ], [ 6 ] ]
### gap> LTnp := LTermsNP( Lnp );
### [ [ [ [ 1, 1, 2 ] ], [ 6 ] ], [ [ [ 1, 2, 2 ] ], [ 8 ] ] ]
### gap> PrintNPList( LTnp );
### 6a^2b
### 8ab^2
### </Example>
### <#/GAPDoc>
###
### - Returns the leading term(s) of a (list of)
### Noncommutative Polynomials.
### The polynomials are ordered such that the leading monomial comes first.
### If the zero polynomial is given as argument then 'fail' is returned.
###
### Arguments:
### pol - list of Noncommutative Polynomials
###
### Returns:
### - a list of leading monomials
###
### #LTermNP uses:#
### #LTermNP is used in:#
###
InstallGlobalFunction( LTermNP, function(pol)
if IsZero(pol) then
return fail;
else
return [ [ pol[1][1] ], [ pol[2][1] ] ];
fi;
end);
InstallGlobalFunction(LTermsNP, lst -> List(lst,LTermNP));
##################
### CleanNP
### <#GAPDoc Label="CleanNP">
### <ManSection>
### <Func Name="CleanNP" Comm="clean up for NP" Arg="u" />
### <Returns>
### The cleaned up version of <A>u</A>
### </Returns>
### <Description>
### Given a polynomial in NP format, this function
### collects terms with same monomial, removes trivial monomials,
### and orders the monomials, with biggest one first.
### <P/>
### <#Include Label="example-CleanNP">
### </Description>
### </ManSection>
### <#/GAPDoc>
### - Collects terms with same monomial, removes trivial monomials,
### and orders the monomials, with biggest one first
###
### Arguments:
### pol - non-commutative polynomial
###
### Returns:
### pol - the cleaned version of the non-commutative polynomial
###
### #CleanNP uses:#
### #CleanNP is used in: CheckHomogeneousNPs EvalTrace GBNP.GP2NPM GBNP.IsGrobnerBasisTest GBNP.MakeGrobnerPairMakeMonic GBNP.NPArray2NPM GBNP.ReducePol GBNP.ReducePolTrace GP2NP IsGrobnerPair MulNP StrongNormalFormNP#
###
InstallGlobalFunction( CleanNP, function(pol)
local i,h,l,v,mons,polh,coeffs,ansmons,anscoeffs;
polh:=StructuralCopy(pol);
SortParallel(polh[1],polh[2],GtNP);
mons:=polh[1];
coeffs:=polh[2];
ansmons:=[];
anscoeffs:=[];
l:=Length(pol[1]);
h:=0;
while h<l do
h:=h+1;
i:=h;
v:=coeffs[h];
while i<l and mons[i]=mons[i+1] do
v:=v+coeffs[i+1];
i:=i+1;
od;
if not IsZero(v) then
Add(ansmons,mons[h]);
Add(anscoeffs,v);
fi;
h:=i;
od;
return([ansmons,anscoeffs]);
end);;
##################
### MkMonicNP
### <#GAPDoc Label="MkMonicNP">
### <ManSection>
### <Func Name="MkMonicNP" Comm="makes an NP monic by multiplying with the right scalar." Arg="np" />
### <Returns>
### <A>np</A> made monic
### </Returns>
### <Description>
### This function returns the scalar multiple of a polynomial
### <A>np</A> in NP format that is monic, i.e., has leading coefficient equal to 1.
### <P/>
### <#Include Label="example-MkMonicNP">
### </Description>
### </ManSection>
### <#/GAPDoc>
### - Makes a non-commutative polynomial
### monic by multiplying with the right scalar
###
### Assumptions:
### - The polynomial is cleaned.
### - The leading term comes first.
###
### Arguments:
### pol - non-commutative polynomial
###
### Returns:
### pol - the monic version of the non-commutative polynomial
###
### #MkMonicNP uses:#
### #MkMonicNP is used in: GBNP.CentralT GBNP.IsGrobnerBasisTest GBNP.MakeGrobnerPairMakeMonic GBNP.ObsTall GBNP.ObsTrunc GBNP.ReducePol GBNP.ReducePol2 GBNP.SGrobnerLoops IsGrobnerPair StrongNormalFormNP#
###
InstallGlobalFunction( MkMonicNP, function(pol)
if pol=[[],[]] then
return(pol);
fi;
if IsOne(pol[2][1]) then
return(pol);
fi;
return([pol[1],pol[2]/pol[2][1]]);
end);;
##################
### FactorOutGcdNP
### <#GAPDoc Label="FactorOutGcdNP">
### <ManSection>
### <Func Name="FactorOutGcdNP"
### Comm="divides an NP polynomial with integer coefficients by
### the Gcd of its coefficients." Arg="np" />
### <Returns>
### <A>np</A> with Gcd(coefficients) factored out
### </Returns>
### <Description>
### This function returns the scalar multiple of a polynomial
### <A>np</A> in NP format with integer coefficients
### such that its coefficients have Gcd equal to 1.
### If the coefficients are not all integers then <C>fail</C> is returned.
### <P/>
### <#Include Label="example-FactorOutGcdNP">
### </Description>
### </ManSection>
### <#/GAPDoc>
### - Makes a non-commutative polynomial with integer coeffiecients
### into one whose coefficients have Gcd equal to 1.
###
### Assumptions:
### - The polynomial is cleaned.
### - The leading term comes first.
###
### Arguments:
### pol - non-commutative polynomial
###
### Returns:
### pol - the non-commutative polynomial divided by Gcd(coefficients)
###
### #FactorOutGcdNP uses:#
### #FactorOutGcdNP is used in: #
###
InstallGlobalFunction( FactorOutGcdNP, function(pol)
local g;
if pol=[[],[]] then
return(pol);
fi;
if not ForAll( pol[2], c -> IsInt(c) ) then
return fail;
fi;
g := Gcd( pol[2] );
return([pol[1],pol[2]/g]);
end);;
#################
### BimulNP
### <#GAPDoc Label="BimulNP">
### <ManSection>
### <Func Name="BimulNP" Comm="" Arg="ga, np, dr" />
### <Returns>
### the polynomial <A>ga</A><M>*</M><A>np</A><M>*</M><A>dr</A> in
### NP format
### </Returns>
### <Description>
### When called with a polynomial <A>np</A> and two monomials <A>ga</A>,
### <A>dr</A>,
### the function will return <A>ga</A><M>*</M><A>np</A><M>*</M><A>dr</A>.
### Recall from Section <Ref Sect="NP"/> that monomials are represented as lists.
### <P/>
### <#Include Label="example-BimulNP">
### </Description>
### </ManSection>
### <#/GAPDoc>
### - Multiplication of a polynomial by two monomials:
### one on the left and one on the right
###
### Arguments:
### u - a non-commutative polynomial
### ga,dr - two monomials
###
### Returns:
### pol - the non-commutative polynomial ga*u*dr
###
### #BimulNP uses:#
### #BimulNP is used in: EvalTrace GBNP.NormalForm2 GBNP.NormalForm2T GBNP.Spoly GBNP.SpolyTrace GBNP.StrongNormalForm2 GBNP.StrongNormalForm2TS GBNP.StrongNormalForm2TSPTS GBNP.StrongNormalForm2Tall GBNP.StrongNormalForm3Dall GBNP.StrongNormalFormTrace2 GBNP.TraceNP#
###
InstallGlobalFunction( BimulNP, function(ga,u,dr)
local i,ans;
ans:=[];
for i in u[1] do
Add(ans,Concatenation(ga,i,dr));
od;
return([ans,u[2]]);
end);;
#################
### MulNP
### <#GAPDoc Label="MulNP">
### <ManSection>
### <Func Name="MulNP" Comm="Multiplication of two polynomials" Arg="np1, np2" />
### <Returns>
### <A>np1</A><M>*</M><A>np2</A>
### </Returns>
### <Description>
### When invoked with two polynomials <A>np1</A> and <A>np2</A> in NP format,
### this function will return the product <A>np1</A><M>*</M><A>np2</A>.
### <P/>
### <#Include Label="example-MulNP">
### </Description>
### </ManSection>
### <#/GAPDoc>
### - Multiplication of two non-commutative polynomials
###
### Arguments:
### u, v - two non-commutative polynomial
###
###
### Returns:
### pol - the product u*v
###
### #MulNP uses: CleanNP#
### #MulNP is used in: MulQA MulQM SGrobnerModule#
###
InstallGlobalFunction( MulNP, function(u,v)
local ans,i,j;
ans:=[[],[]];
for i in [1..Length(u[1])] do
for j in [1..Length(v[2])] do
Add(ans[1],Concatenation(u[1][i],v[1][j]));
Add(ans[2],u[2][i]*v[2][j]);
od;
od;
return(CleanNP(ans));
end);;
##################
### GBNP.LTermsTrace
### - Returns the leading terms of a traced list of Noncommutative Polynomials.
### The polynomials are not zero and ordered such that
### the leading monomial comes first.
###
### Arguments:
### pol - traced list of Noncommutative Polynomials.
###
### Returns:
### - a list of leading terms. (monomials)
###
### #GBNP.LTermsTrace uses:#
### #GBNP.LTermsTrace is used in: GBNP.AllObsTrace GBNP.CentralTrace GBNP.ObsTrace GBNP.ReducePolTrace2 GBNP.StrongNormalFormTrace2#
###
GBNP.LTermsTrace:=function(G)
return(List(G,i->i.pol[1][1]));
end;
##################
### GBNP.MkMonicNPTrace
### - Makes a traced non-commutative polynomial
### monic by multiplying with the right scalar
###
### Assumptions:
### - The polynomial is cleaned
### - The leading term comes first.
###
### Arguments:
### pol - traced non-commutative polynomial
###
### Returns:
### pol - the monic version of the traced non-commutative polynomial
###
### #GBNP.MkMonicNPTrace uses:#
### #GBNP.MkMonicNPTrace is used in: GBNP.CentralTrace GBNP.ObsTrace GBNP.ReducePolTrace GBNP.ReducePolTrace2 GBNP.SGrobnerTraceLoop#
###
GBNP.MkMonicNPTrace:=function(mon) local i,ans,lc;
if mon.pol=[[],[]] then return(mon); fi;
if IsOne(mon.pol[2][1]) then return(mon); fi;
ans:=StructuralCopy(mon);
lc:=mon.pol[2][1];
ans.pol[2]:=ans.pol[2]/lc;
for i in [1..Length(ans.trace)] do
ans.trace[i][4]:= ans.trace[i][4]/lc;
od;
return(ans);
end;;
#################
### GBNP.AddTrace
### - Updates the trace of a non-commutative polynomial h when somethings
### is added to it, so h + scal * li G(i) re
###
### Arguments:
### trace - trace of the non-commutative polynomial
### le - left monomial
### new - trace of added non-commutative polynomial
### ri - right monomial
### scal - scalar used in addition
###
### Returns:
### trace - new trace of the non-commutative polynomial after addition
###
### #GBNP.AddTrace uses:#
### #GBNP.AddTrace is used in: GBNP.SpolyTrace GBNP.StrongNormalFormTrace2#
###
GBNP.AddTrace:=function(trace,le,new,ri,scal) local ans,k,j;
ans:=StructuralCopy(new);
k:=Length(new);
for j in [1..k] do
ans[j][1]:=Concatenation(le,ans[j][1]);
ans[j][3]:=Concatenation(ans[j][3],ri);
ans[j][4]:=scal*ans[j][4];
od;
return(Concatenation(trace,ans));
end;;
#################
### GBNP.TraceNP
### - Computes non-commutative polynomial from trace
###
### Arguments:
### trace - trace of a non-commutative polynomial w.r.t. G
### G - set of non-commutative polynomials
###
### Returns:
### pol - the non-commutative polynomial
###
### #GBNP.TraceNP uses: AddNP BimulNP GBNP.TraceNP#
### #GBNP.TraceNP is used in: GBNP.TraceNP#
###
GBNP.TraceNP:=function(trace,G) local i,l,ans,temp;
l:=Length(trace);
i:=1;
ans:=[[],[]];
while i <= l do
temp:=trace[i];
if IsInt(temp[2]) then
ans:=AddNP(ans,BimulNP(temp[1],
G[temp[2]],temp[3]),1,temp[4]);
else
ans:=AddNP(ans,BimulNP(temp[1],GBNP.TraceNP(
temp[2].trace,G),temp[3]),1,temp[4]);
fi;
i:=i+1;
od;
return(ans);
end;;
#################
### GBNP.Traced
### - Computes and adds the trace of every element of a
### set of non-commutative polynomials w.r.t. this set.
###
### Arguments:
### G - set of non-commutative polynomials
###
### Returns:
### G - set of traced non-commutative polynomials w.r.t. G
###
### #GBNP.Traced uses:#
### #GBNP.Traced is used in: GBNP.ReducePolTrace#
###
GBNP.Traced:=function(G) local ans,pol,i,n;
i:=0;
n:=Length(G);
ans:=[];
pol:=rec( pol:=[], trace:=[] );
while i < n do
i:=i+1;
pol.pol:=G[i];
pol.trace:=[[[],i,[],1]]; # jwk - check if this is the right field
Add(ans,StructuralCopy(pol));
od;
return(ans);
end;;
#########################
### GBNP.IsNPMNotZero
### - Checking whether a polynomial is in NPM form or in NP form
###
### Arguments: np - the polynomial to be checked
### Returns: true if np is in NPM form (there is a monomial starting with a
### negative number) and false otherwise (NP form or np = [[],[]] =
### zero)
### #GBNP.IsNPMNotZero uses:#
### #GBNP.IsNPMNotZero is used in: GBNP.NPM2NPArray#
###
GBNP.IsNPMNotZero:=function(np)
if Length(np[1])=0 or # zero
Length(np[1][1])=0 or # NP monomial '1'
np[1][1][1] > 0 then # NP monomial
return false;
fi; # else np[1][1][1] < 0, or np[1][1] is an NPM monomial
return true;
end;
#####################
### GBNP.MaximumZ ###
#####################
###
### Calculate the maximum of a list. If the list is empty, return 0.
###
### Arguments:
### - lst a list
###
### Returns:
### - the maximum element in lst, or 0 if the list is empty
GBNP.MaximumZ:=function(lst)
if Length(lst) > 0 then
return Maximum(lst);
fi;
return 0;
end;
####################
### NumAlgGensNP ###
####################
###
### <#GAPDoc Label="NumAlgGensNP">
### <ManSection>
### <Func Name="NumAlgGensNP" Comm="" Arg="np" />
### <Returns>
### The minimum number <C>t</C> so that <A>np</A> belongs to the free
### algebra on <C>t</C> generators.
### </Returns>
### <Description>
### When called with an NP polynomial <A>np</A>, this function returns
### the minimum number of generators needed for the corresponding algebra to
### contain the <A>np</A>. If <A>np</A> is a polynomial
### without generators, that is, equivalent to <M>0</M> or <M>1</M>, then <C>0</C> is
### returned.
### <P/>
### <#Include Label="example-NumAlgGensNP">
### </Description>
### </ManSection>
### <#/GAPDoc>
### Calculate a minimum of the number of generators of the algebra containing
### np.
###
### Arguments:
### - np an NP polynomial
###
### Returns:
### - the maximum value occurring in the monomials of np (or 0 if np is <0> or <1>)
InstallGlobalFunction(
NumAlgGensNP,function(np)
return Maximum([GBNP.MaximumZ(List(np[1], x -> GBNP.MaximumZ(x))),0]);
end);
########################
### NumAlgGensNPList ###
########################
###
### <#GAPDoc Label="NumAlgGensNPList">
### <ManSection>
### <Func Name="NumAlgGensNPList" Comm="" Arg="Lnp" />
### <Returns>
### The minimum number <C>t</C> so that each polynomial in <A>Lnp</A>
### belongs to the free
### algebra on <C>t</C> generators.
### </Returns>
### <Description>
### When called with a list of NP polynomials <A>Lnp</A>, this function returns
### the minimum number of generators needed for the corresponding algebra to
### contain the NP polynomials in <A>Lnp</A>. If <A>Lnp</A> only contains polynomials
### without generators, that is equivalent to <M>0</M> and <M>1</M>, then <C>0</C> is
### returned.
### <P/>
### <#Include Label="example-NumAlgGensNPList">
### </Description>
### </ManSection>
### <#/GAPDoc>
### Calculate a minimum of the number of generators of the algebra containing
### the polynomials in Lnp.
###
### Arguments:
### - Lnp a list of NP polynomials
###
### Returns:
### - the maximum value occurring in the monomials of Lnp (or 0 if only <0> or <1> occur)
InstallGlobalFunction(
NumAlgGensNPList,function(Lnp)
return GBNP.MaximumZ(List(Lnp, x -> NumAlgGensNP(x)));
end);
############################
### GBNP.NumAlgGensNPmon ###
############################
###
### Calculate the highest algebra generator in an NP monomial
###
### Arguments:
### - np_mon a NP monomial
###
### Returns:
### - the highest algebra generator occurring in np_mon
GBNP.NumAlgGensNPmon:=function(np_mon)
return Maximum([GBNP.MaximumZ(np_mon),0]);
end;
################################
### GBNP.NumAlgGensNPmonList ###
################################
###
### Calculate the highest algebra generator in a list of NP monomials
###
### Arguments:
### - Lnp_mon a list of NP monomials
###
### Returns:
### - the highest algebra generator occurring in Lnp_mon
GBNP.NumAlgGensNPmonList:=function(Lnp_mon)
return Maximum([GBNP.MaximumZ(List(Lnp_mon, x -> GBNP.NumAlgGensNPMon(x))),0]);
end;
#####################
### GBNP.MinimumZ ###
#####################
###
### returns the smallest element in a list (or 0 in case of an empty list)
###
### Arguments:
### - lst a list
###
### Returns:
### - the minimum element in the list lst (or 0 if lst was empty)
GBNP.MinimumZ:=function(lst)
if Length(lst) > 0 then
return Minimum(lst);
fi;
return 0;
end;
####################
### NumModGensNP ###
####################
###
### <#GAPDoc Label="NumModGensNP">
### <ManSection>
### <Func Name="NumModGensNP" Comm="" Arg="npm" />
### <Returns>
### The minimum number <C>mt</C> so that <A>npm</A> belongs to the free
### module on <C>mt</C> generators.
### </Returns>
### <Description>
### When called with a polynomial <A>npm</A> in NPM format, this function returns
### the minimum number of module generators needed for the corresponding
### algebra to contain <A>npm</A>. If <A>npm</A> is an NP
### polynomial that does not contain module generators, then <C>0</C> is
### returned.
### <P/>
### <#Include Label="example-NumModGensNP">
### </Description>
### </ManSection>
### <#/GAPDoc>
### Calculate the minimum of the number of generators needed in the algebra
### containing npm.
###
### Arguments:
### - npm an NP polynomial
###
### Returns:
### - the minimum number of module generators needed to contain npm
InstallGlobalFunction(
NumModGensNP,function(npm)
local ans;
if Length(npm[1]) = 0 then return 0; fi;
if Length(npm[1][1]) = 0 then return 0; fi;
ans := -Minimum([GBNP.MinimumZ(List(npm[1], x -> GBNP.MinimumZ(x))),0]);
return ans;
end);
########################
### NumModGensNPList ###
########################
### <#GAPDoc Label="NumModGensNPList">
### <ManSection>
### <Func Name="NumModGensNPList" Comm="" Arg="Lnpm" />
### <Returns>
### The minimum number <C>mt</C> so that each member of <A>npm</A> belongs
### to the free module on <C>mt</C> generators.
### </Returns>
### <Description>
### When called with a list of polynomials <A>Lnpm</A> in NPM format, this function returns
### the minimum number of module generators needed to contain the
### polynomials in <A>Lnpm</A>. If there are only polynomials in
### <A>Lnpm</A> in NP format, then <C>0</C> is returned.
### <P/>
### <#Include Label="example-NumModGensNPList">
### </Description>
### </ManSection>
### <#/GAPDoc>
###
### Calculate a minimum of the number of generators of the algebra containing
### the polynomials in Lnpm.
###
### Arguments:
### - Lnpm a list of npm polynomials
###
### Returns:
### - the minimum number of module generators needed to contain the
### polynomials in <A>Lnpm</A>. If there are only polynomials in
### <A>Lnpm</A> in NP format, then <M>0</M> is returned.
InstallGlobalFunction(
NumModGensNPList,function(Lnpm)
return GBNP.MaximumZ(List(Lnpm, x -> NumModGensNP(x)));
end);
############################
### GBNP.NumModGensNPmon ###
############################
###
### Calculate the minimum number of module generators of the algebra containing
### an NP monomial.
###
### Arguments:
### - npm_mon a NP monomial
###
### Returns:
### - the minimum number of module generators in the algebra containing npm_mon
GBNP.NumModGensNPmon:=function(npm_mon)
return -Minimum([GBNP.MinimumZ(npm_mon),0]);
end;
################################
### GBNP.NumModGensNPmonList ###
################################
###
### Calculate the minimum number of module generators of the algebra containing
### the NP monomials in a list.
###
### Arguments:
### - Lnpm_mon a list of NP monomials
###
### Returns:
### - the minimum number of module generators in the algebra containing the
### monomials in Lnpm_mon
GBNP.NumModGensNPmonList:=function(Lnpm_mon)
return Maximum([GBNP.MaximumZ(List(Lnpm_mon, x -> GBNP.NumModGensNPMon(x))),0]);
end;
[ 0.30Quellennavigators
Projekt
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2026-04-02
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