alexmat.sa
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-- Copyright (C) in the sense of GPL.
-- 2003/12 twisted Novikov(extended Alexander)ideal. kdm
-- 2002/12 twisted Alexander based on ALEXMAT class. kdm
-- 2002/11 Novikov (extended Alexander) ideal. kdm
-- 2001/12 Grobner bases for Elementary ideals. kdm
-- 2001/02 Sather version. kdm
-- 1998/1 LongInt kdm
-- 1997/1 Theta curve, un-oriented surface in S^4 kdm
-- 1996/10 LINUX version, bug fix for 2-knot kdm
-- 1992/6 Alexander matrix and Alexander polynomial. kdm
-- 1992/6/20: 10para99(parallel of k10c99. 40 crossing) 10sec. kdm
-- 1983 Alexander poly. 1st version Kouji kdm KODAMA
--K. Kodama
class ALEXMAT
class ALEXMAT is
include ALEXMAT_ALG{MAT_POLYS_INTI,POLYS_INTI,INTI};
include ALEXMAT_IDEAL_ALG{MAT_POLYS_INTI,POLYS_INTI,INTI};
include ALEXMAT_IDEAL_INTI_ALG{MAT_POLYS_INTI,POLYS_INTI,INTI};
include C2A_ALG{POLYS_INTI};
APolyOut(p0:CARD, testSym:BOOL,use_conway:BOOL) is
pa:POLYS_INTI;
AlexPoly(p0,out pa);
if use_conway then pc:POLYS_INTI;
apoly2conway(pa, out pc); printConway(pc);
else
printApoly(pa,"Alexander Polynomial");
end;
if testSym then -- Test symmetry using Murasugi's condition.
SYMMET::TestSym(pa);
end;
end;
AlexMat(TCode:TCODE, testSym:BOOL, testGb:BOOL) is
AlexMat(TCode,testSym,testGb,false);
end;
MatReduction:CARD is
-- reduce relation matrix and get ideals.
rDeg:CARD;
MAT_POLYS_INTI_REDUCTION::reduce(inout AMat,out jPivot,out rDeg);
return rDeg;
end;
AlexMat(TCode:TCODE, testSym:BOOL, testGb:BOOL, use_conway:BOOL) is
if testSym then
if TCode.has_band then
#OUT+"[test cyclic] not support knotting surface.\n";
return;
elsif TCode.number_compo/=1.int then
#OUT+"[test cyclic] not support links.\n";
return;
end;
end;
--
num_gen:=TCode.number_gen;
if (num_gen<=1)or(num_gen=TCode.number_compo.card) then
#LOGOUT+"The ";
if TCode[TCode.length].compo=1.int then #LOGOUT+"knot"; else #LOGOUT+"link"; end;
if num_gen=0 then #LOGOUT+" is empty.";
else #LOGOUT+" is Trivial.";
end;
LOGOUT::flush;
return;
end;
POLYS_INTI::init;
MakeMatrix(TCode);
LOGOUT::LogTime; WriteMatrix(0); LOGOUT::flush;
trimMat1(TCode);
rDeg::=MatReduction;
WriteMatrix(rDeg); LOGOUT::flush;
if testGb then AlexanderIdeals(rDeg); end;
LOGOUT::flush;
if ~TCode.has_band then APolyOut(rDeg,testSym,use_conway); end;
#LOGOUT+"\n"; LOGOUT::flush;
AMat:=#(0,0);
end;
end;
class ALEXMAT_TWIST
class ALEXMAT_TWIST is
-- CT: INTI
-- ET: POLYS_INTI
-- 2002/12 twisted Alexander based on ALEXMAT class.
include ALEXMAT_ALG{MAT_POLYS_INTI,POLYS_INTI,INTI};
include ALEXMAT_IDEAL_ALG{MAT_POLYS_INTI,POLYS_INTI,INTI};
include ALEXMAT_IDEAL_INTI_ALG{MAT_POLYS_INTI,POLYS_INTI,INTI};
include ALEXMAT_TWIST_ALG{MAT_POLYS_INTI,POLYS_INTI,INTI};
MatReduction:CARD is
rDeg:CARD;
MAT_POLYS_INTI_REDUCTION::reduce(inout AMat,out jPivot,out rDeg);
return rDeg;
end;
end;
class ALEXMAT_TWIST_FP
class ALEXMAT_TWIST_FP is
-- CT: Finite Field
-- ET: polynomial is principal ideal domain (with GCD algorithm)
-- 2002/12 twisted Alexander based on ALEXMAT class.
include ALEXMAT_ALG{MAT_POLYS_FP,POLYS_FP,FINITE_FIELD};
include ALEXMAT_IDEAL_ALG{MAT_POLYS_FP,POLYS_FP,FINITE_FIELD};
include ALEXMAT_IDEAL_FP_ALG{MAT_POLYS_FP,POLYS_FP,FINITE_FIELD};
include ALEXMAT_TWIST_ALG{MAT_POLYS_FP,POLYS_FP,FINITE_FIELD};
MatReduction:CARD is
rDeg:CARD;
AMat.reduce(out jPivot,out rDeg);
return rDeg;
end;
end;
partial class ALEXMAT_IDEAL_ALG{MT,ET,CT}
partial class ALEXMAT_IDEAL_ALG{MT,ET,CT} is
splitMatrix(p0:CARD, out diagonal_part:ARRAY{ET}, out SAMat:MT) is
-- split Alexander matrix AMat to diagoanl part and other part.
zFlg:BOOL; zFlg:=true; degR::=AMat.nr; degG::=AMat.nc;
if (degR=degG)and(p0=degR-1) then p0:=degR; end;
-- Get diagonal part.
d0::=(p0).min(degR).min(degG);-- length of diagonel part
diagonal_part:=#; -- Number of 0-part is trimN.
loop i::=0.upto!(d0-1); j::=jPivot[i];
if (AMat[i][j].is_one)or(AMat[i][j].negate.is_one) then ;--remove trivial
elsif (AMat[i][j].is_zero) then trimN:=trimN+1;
else
diagonal_part:=diagonal_part.append(AMat[i][j]);
end;
end;
-- Get other part
SAMat:=#(degR-p0,degG-p0);
if (p0<degR)or(p0<degG) then
loop i::=(p0).upto!(degR-1); si::=0.up!;
loop
SAMat[si][0.up!]:=AMat[i][jPivot[(p0).upto!(degG-1)]];
end;
end;
end;
end;
AlexanderIdeals(p0:CARD) is
diagonal_part:ARRAY{ET};
SAMat:MT;
splitMatrix(p0,out diagonal_part,out SAMat);
-- #OUT+"diagonal-0: "; loop trimN.times!; #OUT+"0,"; end; #OUT+"\n";
-- #OUT+"diagonal: "+diagonal_part+"\n";
-- #OUT+"matrix : "+SAMat.str+"\n";
-- diagonal is: trimN times 0 and diagonal_part
loop ideg::=trimN.upto!(trimN+diagonal_part.size+SAMat.nc-1); --upto #_of_generator-1
if AlexIdeal(trimN.int,diagonal_part,SAMat,ideg.int) then break!; end;
-- "true" if ideal is trivial, so higher ideals are all trivial.
end
end;
end;
partial class ALEXMAT_IDEAL_FP_ALG{MT,ET,CT}
partial class ALEXMAT_IDEAL_FP_ALG{MT,ET,CT} is
-- CT: field. So ET is PID(euclidean domain).
AlexIdeal(trimN:INT,diagonal_part:ARRAY{ET},SAMat:MT,ideg:INT):BOOL is
-- When free differential method, we must delete __one__ free meridian generator.
ideg:=ideg-trimN;
if ideg.is_neg then return false; end; -- [ 0 ]
#LOGOUT+"Elementary_ideal_E( "+ideg.str+" ):\n";
ddeg::=diagonal_part.size.int;
degr::=ddeg+SAMat.nr.int; -- number of relator
degc::=ddeg+SAMat.nc.int; -- number of generator
mdeg:INT:=degc-ideg; -- degree of minor.
ap:ET; -- generator of the alexander ideal
if mdeg.is_non_pos then #LOGOUT+"[\n"+"1\n"+"]\n\n"; return true;
elsif degr<mdeg then #LOGOUT+"[\n"+"0\n"+"]\n\n"; return false;
end;
-- separate "mdeg" to diagonal part "mdegd" and other part mdeg-mdegd.
-- (1) 0<=mdegd<=ddeg , (2) 0<=mdeg-mdegd<= min(degr,degc)-ddeg
-- (2) means mdegd<=mdeg and mdeg+ddeg-min(degr,degc)<=mdegd.
ap:=ET::zero;
loop mdegd::=0.int.max(mdeg+ddeg-(degr.min(degc))).upto!(ddeg.min(mdeg));
combi3::=#COMBI_NR_STREAM(ddeg.card,mdegd.card); c3:ARRAY{CARD};
loop while!(combi3.get(out c3));
f3::=ET::one; loop f3:=f3*diagonal_part[c3.elt!(1)-1]; end;
if (mdeg=mdegd) then ap:=ap.gcd(f3);
else
combi1::=#COMBI_NR_STREAM((degr-ddeg).card,(mdeg-mdegd).card); c1:ARRAY{CARD};
loop while!(combi1.get(out c1));
sub1::=c1.create(c1.size-1); loop sub1[0.up!]:=c1.elt!(1)-1; end;
combi2::=#COMBI_NR_STREAM((degc-ddeg).card,(mdeg-mdegd).card); c2:ARRAY{CARD};
loop while!(combi2.get(out c2));
sub2::=c2.create(c2.size-1); loop sub2[0.up!]:=c2.elt!(1)-1; end;
ap:=ap.gcd(SAMat.sub_matrix(sub1,sub2).det*f3);
end;
end;
end;
end;
end;
ap:=ap.shift0;
#LOGOUT+"[\n";
#LOGOUT+ap.str("tex","t",false)+"\n";
#LOGOUT+"]\n";
LOGOUT::flush;
if ap.degree.is_one then return true; end;
return false;
end;
end;
partial class ALEXMAT_IDEAL_INTI_ALG{MT,ET,CT}
partial class ALEXMAT_IDEAL_INTI_ALG{MT,ET,CT} is
-- ET:POLYS_INTI, CT:INTI
ExtendedAlexanderPoly(ap:ET, ideg:INT) is
-- Base in Z[[t]][1/t].
-- Note that it is PID, and monic is invertible.
ap:=ap.reverseDeg; -- The lowest term is strong. c.f. GBASES_INTI_L
if ap.is_zero then ;
elsif (ap.lc.abs)=(1.inti) then ap:=ET::one;
else
loop f::=ap.factorize.elt!;
if (f.lc.abs)=(1.inti) then ap:=ap/f; end;
end;
end;
ap:=ap.reverseDeg;
#LOGOUT+"Novikov(Extended_Alexander)Ideal:"+"A( "+ideg.str+" )= "+ap.str("tex","t",false)+"\n";
LOGOUT::flush;
end;
printAp(ap:ET) is
-- ET: POLYS_INTI
x::=ET::x;
b1:ET:=ap/x; low_c:CT:=b1.low_coeff;
b0::=(ap-b1*x)*x; #LOGOUT+b0.str("tex","(1/t)",true); -- lowest term
if low_c.is_neg then #LOGOUT+b1.str("tex","t",false);
elsif low_c.is_pos then #LOGOUT+"+"+b1.str("tex","t",false);
else -- is_zero
end;
#LOGOUT+"\n";
end;
AlexIdeal(trimN:INT,diagonal_part:ARRAY{ET},SAMat:MT,ideg:INT):BOOL is
-- Assume that ET: POLYS_INTI, CT: INTI
-- When free differential method, we must delete __one__ free meridian generator.
ideg:=ideg-trimN;
if ideg.is_neg then return false; end; -- [ 0 ]
#LOGOUT+"Elementary_ideal_E( "+ideg.str+" ):\n";
ddeg::=diagonal_part.size.int;
degr::=ddeg+SAMat.nr.int; -- number of relator
degc::=ddeg+SAMat.nc.int; -- number of generator
mdeg:INT:=degc-ideg; -- degree of minor.
ap:ET;
if mdeg.is_non_pos then #LOGOUT+"[\n"+"1\n"+"]\n";
ap:=ET::one; ExtendedAlexanderPoly(ap, ideg); return true;
elsif degr<mdeg then #LOGOUT+"[\n"+"0\n"+"]\n";
ap:=ET::zero; ExtendedAlexanderPoly(ap, ideg); return false;
end;
-- separate "mdeg" to diagonal part "mdegd" and other part mdeg-mdegd.
-- (1) 0<=mdegd<=ddeg , (2) 0<=mdeg-mdegd<= min(degr,degc)-ddeg
-- (2) means mdegd<=mdeg and mdeg+ddeg-min(degr,degc)<=mdegd.
gBases:ARRAY{ET}:=#;
loop mdegd::=0.int.max(mdeg+ddeg-(degr.min(degc))).upto!(ddeg.min(mdeg));
combi3::=#COMBI_NR_STREAM(ddeg.card,mdegd.card); c3:ARRAY{CARD};
loop while!(combi3.get(out c3));
f3::=POLYS_INTI::one; loop f3:=f3*diagonal_part[c3.elt!(1)-1]; end;
if (mdeg=mdegd) then gBases:=gBases.append(f3);
else
combi1::=#COMBI_NR_STREAM((degr-ddeg).card,(mdeg-mdegd).card); c1:ARRAY{CARD};
loop while!(combi1.get(out c1));
sub1::=c1.create(c1.size-1); loop sub1[0.up!]:=c1.elt!(1)-1; end;
combi2::=#COMBI_NR_STREAM((degc-ddeg).card,(mdeg-mdegd).card); c2:ARRAY{CARD};
loop while!(combi2.get(out c2));
sub2::=c2.create(c2.size-1); loop sub2[0.up!]:=c2.elt!(1)-1; end;
gBases:=gBases.append(SAMat.sub_matrix(sub1,sub2).det*f3);
end;
end;
end;
end;
end;
--#OUT+"grobner bases:\n"+gBases.str+"\n\n";
gBases:=GBASES_INTI_L::getGBaseIL(gBases,out ap); -- get Grobner bases
--#LOGOUT+gBases.str+"\n\n";
#LOGOUT+"[\n";
if gBases.size=0 then #LOGOUT+"0\n";
else
printAp(ap);
loop b::=gBases.elt!; #LOGOUT+b.str("tex","t",false)+"\n"; end;
end;
#LOGOUT+"]\n";
LOGOUT::flush;
ExtendedAlexanderPoly(ap, ideg);
if ap.is_one then return true; end;
return false;
end;
end;
class ALEXMAT_TWIST_ALG{MT,ET,CT}
class ALEXMAT_TWIST_ALG{MT,ET,CT} is
-- matrix MT
-- element polynomial ET
-- coeffifient of polynomial CT
APolyOut(p0:CARD) is
pa:ET;
AlexPoly(p0,out pa);
printApoly(pa,"Twisted Alexander Polynomial");
end;
AlexMat(TCode:TCODE, Rep:ARRAY{MT},RepR:ARRAY{MT}) is
-- Rep/RepR: Representation / reverse
-- Rep[i] : image of generator x_i.
-- Rep[0] : not used.
#LOGOUT+"\n";
num_gen:=TCode.number_gen;
if (num_gen<=1)or(num_gen=TCode.number_compo.card) then
#LOGOUT+"The ";
if TCode[TCode.length].compo=1.int then #LOGOUT+"knot"; else #LOGOUT+"link"; end;
if num_gen=0 then #LOGOUT+" is empty.";
else #LOGOUT+" is Trivial.";
end;
LOGOUT::flush;
return;
end;
POLYS_INTI::init;
MakeMatrix(TCode,Rep,RepR);
LOGOUT::LogTime; WriteMatrix(0); LOGOUT::flush;
trimMat1(TCode);
rDeg::=MatReduction;
WriteMatrix(rDeg); LOGOUT::flush;
AlexanderIdeals(rDeg);
if ~TCode.has_band then APolyOut(rDeg); end;
#LOGOUT+"\n"; LOGOUT::flush;
AMat:=#(0,0);
end;
end;
class C2A_ALG{ET}
class C2A_ALG{ET} is
-- polynomial ET (assumed that POLYS_INTI)
apoly2conway(apoly:ET, out cpoly:ET) is
a2c(apoly,out cpoly); a1:ET; c2a(cpoly,out a1);
if (apoly/=a1)and(apoly/=-a1) then
raise "Failed to convert between a.poly. and c.poly.\n";
end;
end;
conway2apoly(cpoly:ET, out apoly:ET) is
c2a(cpoly, out apoly); c1:ET; a2c(apoly,out c1);
if (cpoly/=c1)and(cpoly/=-c1) then
raise "Failed to convert between a.poly. and c.poly.\n";
end;
end;
private a2c(apoly:ET, out cpoly:ET) is
deg::=apoly.degree;
CN:ARRAY{ET}:=POLY_COEFF::AllocCN(deg.card);
cpoly:=#; x::=ET::x;
loop z::=deg.downto!((deg+1)/2.int);
cpoly:=cpoly+x^(z*2-deg)*apoly[z.card];
apoly:=apoly-CN[(z*2-deg).card]*x^(deg-z)*apoly[z.card];
end;
end;
private c2a(cpoly:ET, out apoly:ET) is
-- a=c.substitute(x-1/x)
a:ET:=#; x::=ET::x;
cn::=ET::one; cn1::=x^2-1;
loop z::=0.upto!(cpoly.degree.card); a:=a*x+cn*cpoly[z]; cn:=cn*cn1; end;
-- apoly=a.substitute(sqrt("t"))
apoly:=#; loop i::=a.arr.ind!; j::=i/2; apoly[j]:=apoly[j]+a[i]; end;
end;
printConway(cpoly:ET) is
LOGOUT::Title("Conway","Polynomial");
#LOGOUT+cpoly.str("tex","z",true)+"\n";
LOGOUT::flush;
end;
end;
class ALEXMAT_ALG{MT,ET,CT}
class ALEXMAT_ALG{MT,ET,CT} is
-- matrix MT
-- element polynomial ET
-- coeffifient of polynomial CT
shared AMat:MT;
shared jPivot:ARRAY{CARD};
shared trimN:CARD; -- number of free generator
shared trimC:CARD; -- component of over bridge of deleted relation
shared Relator:ARRAY{WORD};
shared g2c:ARRAY{CARD}; -- map: generator to comploent
shared compoN:CARD; -- #of component
shared rep_deg:CARD; -- degree of representation matrix
shared num_gen:CARD; -- number of generatrs
printApoly(apoly:ET, title:STR) is
LOGOUT::Title(title,"");
ps:STR:=""; -- to be "+".
loop
--i::=apoly.degree.card.downto!(0);
i::=0.upto!(apoly.degree.card);
if (apoly[i].inti>=0.inti) then #LOGOUT+ps; end;
#LOGOUT+apoly[i].str;
ps:="+";
end;
#LOGOUT+"\n"; LOGOUT::flush;
end;
WriteMatrix(p0:CARD) is
LOGOUT::Title("Alexander matrix","");
zFlg:BOOL; zFlg:=true; degR::=AMat.nr; degG::=AMat.nc;
--if p0>degR then p0:=degR; end;
--if p0>degG then p0:=degG; end;
-- Alexander matrix is the form of
-- \pmatrix{
-- diagonal & 0 \cr
-- 0 & matrix \cr
-- }
if (degR=degG)and(p0=degR-1) then p0:=degR; end;
-- Write diagonal part.
d0::=(p0).min(degR).min(degG);-- length of diagonel part
loop trimN.times!; zFlg:=false; #LOGOUT+"diagonal part: 0\n"; end;
if d0>0 then
loop i::=0.int.upto!(d0.int-1); j::=jPivot[i];
if (AMat[i][j].is_one.not)and(AMat[i][j].negate.is_one.not) then
zFlg:=false; #LOGOUT+"diagonal part: "+AMat[i][j].str("tex","t",true)+"\n";
end;
end;
end;
-- Write other part
if (p0<degR)or(p0<degG) then
sub1:ARRAY{CARD}:=#(degR-p0); loop sub1[0.up!]:=p0.upto!(degR-1); end;
sub2:ARRAY{CARD}:=#(degG-p0); loop sub2[0.up!]:=jPivot[p0.upto!(degG-1)]; end;
#LOGOUT+AMat.sub_matrix(sub1,sub2).str_pmatrix("t");
if (p0<degR)and(p0<degG) then zFlg:=false; end;
end;
if zFlg then #LOGOUT+"0\n"; end;
LOGOUT::flush;
end;
MakeMatrix(TCode:TCODE,Rep:ARRAY{MT},RepR:ARRAY{MT}) is
rep_deg:=Rep[1].nc;
KNOT_GROUP::get_Relator(TCode, out Relator, out g2c);
--
#OUT+"Relations:\n";
loop i::=Relator.ind!; #OUT+Relator[i].str+"\n"; end;
--
trimN:=0; trimC:=0; compoN:=(TCode[TCode.length].compo).card;
g:CARD:=g2c.size-1; -- # of generators
num_gen:=g;
r:CARD:=Relator.size; -- # of relators
jPivot:=#(g*rep_deg); loop i::=jPivot.ind!; jPivot[i]:=i; end;
AMat:=#(r*rep_deg,g*rep_deg); AMat.clear;
if (r=0)or(g=0) then return; end;
warr:MT:=#(rep_deg,rep_deg);
warr.to_unit;
p_one::=ET::one;
p_x::=ET::x;
loop i:CARD:=0.upto!(r-1);
-- Make relation matrix with Fox's free differential
-- w=(a b a~ c~) or (a b c~ b~)
AMat.plus(warr,i*rep_deg,((Relator[i][0]-1)*rep_deg).card);
warr:=warr*Rep[(Relator[i][0]).card];
AMat.plus(warr*p_x,i*rep_deg,((Relator[i][1]-1)*rep_deg).card);
if Relator[i].w.has_ind(2) then
warr:=warr*Rep[(Relator[i][1]).card]*RepR[(-Relator[i][2]).card];
AMat.minus(warr*p_x, i*rep_deg, ((-Relator[i][2]-1)*rep_deg).card);
end;
if Relator[i].w.has_ind(3) then
warr:=warr*RepR[(-Relator[i][3]).card];
AMat.minus(warr, i*rep_deg, ((-Relator[i][3]-1)*rep_deg).card);
end;
--#OUT+"AMat "+i.int+"-th relation:"+"\n";
end;
end;
MakeMatrix(TCode:TCODE) is
-- non-twisted case is concluded to trivial representation.
rep_deg:=1;
num_gen:=TCode.number_gen;
Rep:ARRAY{MT}:=#(num_gen+1);
RepR:ARRAY{MT}:=#(num_gen+1);
loop i::=1.upto!(num_gen); Rep[i]:=MT::unit(1,1); end;
loop i::=1.upto!(num_gen); RepR[i]:=MT::unit(1,1); end;
MakeMatrix(TCode,Rep,RepR);
end;
trimMat1(TCode:TCODE) is
trimN:=0; -- Set # of "0" element.
delR,delG:CARD;
pt:INT:=0.int; word:WORD;
if TCode.has_band then -- 2-dim knot/link
res::=KNOT_GROUP::getCrossR(TCode, inout pt, out word);
if res and (pt<TCode.bandStart) then ; end;
else -- 1-dim knot/link
-- delete a crossing relation
res::=KNOT_GROUP::getCrossR(TCode, inout pt, out word);
if res and (pt<TCode.bandStart) then
trimN:=rep_deg;
delR:=1; delG:=word[1].card;
trimC:=g2c[delG].card;
loop i::=(delR*rep_deg-1).downto!((delR-1)*rep_deg);
AMat:=AMat.minor_matrix_row(i);
end;
loop i::=(delG*rep_deg-1).downto!((delG-1)*rep_deg);
AMat:=AMat.minor_matrix_column(i);
end;
jPivot:=#(AMat.nc); loop i::=jPivot.ind!; jPivot[i]:=i; end;
end;
end;
end;
AlexPoly(p0:CARD, out apoly:ET) is
apoly:=ET::one;
if AMat.nr/=AMat.nc then
#OUT+"Alexander matrix is not square.\n";
apoly:=ET::zero;
return;
end;
pw:ET; -- work
-- diagonal part
countz::=0; -- count "0"
loop i::=0.upto!(p0-1); pw:=AMat[i,jPivot[i]];
if pw.is_zero then countz:=countz+1; else apoly:=apoly*pw; end;
end;
-- non-diagonal part
if countz+trimN<=rep_deg then
if p0<AMat.nr then
mat::=AMat.copy; jPivotm::=jPivot.copy;
loop i::=0.upto!(p0-1); mat:=mat.minor_matrix(0,jPivotm[i]);
loop j::=i.upto!(jPivotm.size-1);
if jPivotm[j]>jPivotm[i] then jPivotm[j]:=jPivotm[j]-1; end;
end;
end;
apoly:=apoly*mat.det;
end;
else apoly:=ET::zero;
end;
if apoly.is_zero.not then
apoly:=apoly.shift_deg(-(apoly.low_deg));
--if (trimN=1)and(compoN>1) then
-- apoly:=apoly/(#POLYS_INTI(1.inti,1)-1.inti);
--end;
-- if apoly.substitute(#CT(1.int)).inti.is_neg then apoly:=-apoly; end;
end;
end;
end;