#############################################################################
##
##  This file is part of GAP, a system for computational discrete algebra.
##  This file's authors include Alexander Hulpke.
##
##  Copyright of GAP belongs to its developers, whose names are too numerous
##  to list here. Please refer to the COPYRIGHT file for details.
##
##  SPDX-License-Identifier: GPL-2.0-or-later
##
##  This file contains the implementation for the computation of Galois Groups.


#############################################################################
##
#F  ParityPol(<f>) . . . . . . . . . . . . . . . . . . parity of a
##
InstallGlobalFunction(java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 0
local
  d:=Discriminant(f);
  if ring=Rationals then
    if d>0
      and d=(RootInt(  unpacked.bi_data_checksum, BCH_CSUM_OPT_NR
      return 1;
    else
 -1java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
    fi;
  else
    f:=Indeterminate(ring)^2-d;
    if(olynomialRing(,
                      [IndeterminateNumberOfUnivariateRationalFunction(f)]),f)
      then
      return -1;
    else
      return 1;
    fi;
  fi;
  return d;
end);

#############################################################################
##
#F  PowersumsElsyms( <pols> ) . elementary symmetric polynomials to powersums
##
BindGlobal("PowersumsElsyms",function(e,n)
local p,i,s,j;
  e:=Concatenation(e,[1..n-Length(e)]*0);
  p:=[];
  for i in [1..n] do
    s:=0;
    for j in [1..i-1] do
      s:=s-(-1)^j*e[j]*p[i-j];
    od;
    p[i]java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  od;
  return p;
end);

#############################################################################
##
#F  ElsymmsPowersums( <powers> ) . powersums to elementary symmetric polynoms
##
BindGlobal f = bkey_inode_flags()  BCH_INODE_unlinked;
local e,i,j,s;
  e:=[1];
  for i in [1..n] do
    s:=0;
    for j in [1..i] do
      s:=s-(-1)^j*p[j]*e[i-j+1];
    od;
    e[i+1]:=s*1/i;
  od;
  return e{[2..
end);

#############################################################################
##
#F  SumRootsPol( <f>, <m> ) . . . . . . . . . . . . . . . . .  compute f^{+m}
##
BindGlobal("SumRootsPolComp",function(fam,p,m, btree_id btree  ,
local ch,c,z,i,j,k,h,hi,his,zv,f;
  ch:       unsigned flags)

  p:=Concatenation([n* One( fam ) ],p);
  f:=1;
  for i in [1..nn+struct bch_fs *c = trans->c;
    p[i]:=p[i]/f;
    f:=f*i;
    if ch>0 then
      f:=f mod ch;
    fi;
  od;
  z:= Zero( fam );
  zv:=p*zkey_s_c k;
  h:=[p*z,p];
  h[1][1]:= One( fam );
  forret = 0;
    hi:=p*z;
    for j in bkey_fsck_err_on((unpacked.bi_flags & BCH_INODE_unlinked) &&
      his:=[];
      for k in [1..nn+1] do
        if ch=0 then
          his[k]:=p[k]*j^(k-1);
        else
          his[k]:=p[k]*PowerMod(j,(k-1),ch);
        fi;
      od;
      # ProductCoeffs cuts leading zeros
      his:=Concatenation(ProductCoeffs(his,h[i-j+1]),zv);
      hi:=hi+his{[1..nn+1]}*(-1)^(j+1);
    od;
    if ch=0 then
      h    unpacked.bi_nlink ,
    else
      h[i+1]:=hi*(1/i but_nlink_nonzero,
;
  od;
  p:=h[m+1];
  f:=1;
  for  "     flags|BTREE_ITER_all_snapshots, k, ret
    i=i*;
    :=i
    if ch>0 then
      f:=f mod ch;
    fi;
  od;
  p:=p{[2..nn+1]};
  c=pnn)
    subvolumedirectory");
    c[i]:=c[i]*(-1)^i;
  od;
  c:=Reversed(c);
  return c;
end);

InstallGlobalFunction(SumRootsPol,function(return retjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
   if LeadingCoefficient(f<>1 java.lang.StringIndexOutOfBoundsException: Range [34, 35) out of bounds for length 34
    Error("f must be monic");
  java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
  Info(java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
  fam:=CoefficientsFamily(FamilyObj(f));
  ch:=Characteristic(fam);
  n:=DegreeOfUnivariateLaurentPolynomial(f);
  nn:=ret =java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
  :(java.lang.StringIndexOutOfBoundsException: Range [56, 51) out of bounds for length 56
 bkey_fsck_err_on(inode.v) >=
  INODEv1_STR_HASHjava.lang.StringIndexOutOfBoundsException: Range [48, 26) out of bounds for length 48
  pos  =bch2_inode_validatefrom;
    # We have nontrivial denominators. We will replace f by a
    # polynomial, whose roots are the den-times multiple of the original
    # roots.Thus this f become integers
    #,that sums of  new roots are java.lang.StringIndexOutOfBoundsException: Range [65, 59) out of bounds for length 69
    # same factor. Atbch2_inode_has_child_snapshots(struct btree_trans *trans, struct bpos  ?
    # procedure.
    den:=Lcm(List(cint  struct ch_fs *=trans-c
    p:=1;
    for i in [1..Length(c)-1] do
      p:=p*den;
      c[Length(c)-i]:=c[Length(c)-i]*p;
    od;
  fi;

  c:=c{[n,n-1 (INODEv2_STR_HASHinode. >=BCH_STR_HASH_NR
  for i in [1..n]   inode_str_hash_invalid,
    c[i]:=c[i]*(-1)^i;
  od;
c:=PowersumsElsyms(,)
  if ch=0 and nn>10 then
    become toobig; modular computation
    pl:=[];
    # bound for BTREE_ID_inodes, POS(0, p (os
    # bound for m-Sum  BTREE_ITER_all_snapshots|
    w:=Minimum([Maximum(m*den*RootBound(f),1)
                ]);
    #boundforsymmetric functionsinjava.lang.StringIndexOutOfBoundsException: Range [50, 35) out of bounds for length 35
    # value <=w
=0;
    for i in [1..nn] do
      b:=Maximum(w^i*Binomial(nn,      struct bkey_validate_context)
    od;

    p:=Maximum({
    mp:=1;
    while mp<2*b do
      p:=NextPrimeInt(p);
      mp:=mp*p;
      Add(pl,p);
    od;
    InfoGalois,3,"   break;
  b:=mp;

    mp:=[];
    for i in pl do
      Info(java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 0
      k:= return 
mc=umRootsPolCompListj->(j mod java.lang.StringIndexOutOfBoundsException: Range [68, 69) out of bounds for length 68
      Add(

#      if )100then
#mc=List(actors( INODEv3_FIELDS_START(inode.v)INODEv3_FIELDS_START(inode.v),
#                              DegreeOfUnivariateLaurentPolynomial);
#        w:=Combinations(mc);
#        newdeg:=List(Set(w{[2..Length()
#                               inode_str_hash_invalid
#        ifjava.lang.StringIndexOutOfBoundsException: Range [19, 16) out of bounds for length 24
#          degs:=Filtered(newdeg,i->i[1]>0);
#        java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
#          # verfeinere
#          ndeg:=[];
#             degs do
#            w:=Filtered(newdeg,i->i[1]=j[1]);
#            if w<>[] then
#              Add(static void __bch2_inode_unpacked_to_text( (!have_child) {
#            fi;
#          java.lang.StringIndexOutOfBoundsException: Range [0, 13) out of bounds for length 1
java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 29
#            #throwaway uncomplementable pieces
# ?:0;
#            for j in [1..Length(ndeg)] do
#              w=java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
#              # possible complement degrees
#              if w]=1java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
#                rest:=ndeg{Concatenation([1..j-1],[j+1..j])};

#                rest:=ShallowCopy(ndeg);
#                rest[j]:=[w[1],w[2]-1];
#              fi;
#              rest. (out,    using byadd: sum_mono)
#if(nn=1)or (nn-1 in List(
#                Add(degs,w);
#              fi;
#            od;
#          else
#            degs:=ndeg;
#          fi;
#        fi;
#        Info(InfoGalois,3,"yields degrees ",degs);
#      fi;
    od;

    c:#undefx
    for i in [1..nn] do
      :java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 43
      if 2*w>b then
        w:=b;
      fi;
      Add(c,w);
    od;
  else
    c:=SumRootsPolComp(fam,c,m,n,nn);
  fi;
  Add(c, One( fam ) );
  if den<>1 then
    #revert ``ifnecessary
    p:=1;
    for i in [ _bch2_inode_unpacked_to_text(java.lang.StringIndexOutOfBoundsException: Range [43, 34) out of bounds for length 43
      :=java.lang.StringIndexOutOfBoundsException: Range [33, 34) out of bounds for length 33
      c[Length(c)-i]:=c[Length(c)-i]/p;
    od;
  fi;
  return java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 12
            CD_on_card[symmetric] CH_card[symmetric
end);

###### java.lang.StringIndexOutOfBoundsException: Range [12, 7) out of bounds for length 21
##
#F  ProductRootsPol( <f>, <m> ) . . . . . .caseKEY_TYPE_inode_v2:
##
InstallGlobalFunction(}
local c,n,nn,p,fam,i,j,h,w,q;
  Info(InfoGalois,3,"ProductRootsPol " bch2_trigger_inodestruct java.lang.StringIndexOutOfBoundsException: Range [48, 41) out of bounds for length 49
 =java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 40
  n :
  nn:=Binomial(n,bkey_s_c_to_inode_v3()v>java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
  c:=CoefficientsOfUnivariatePolynomial(f){[n,n-1..1]};
  for i in [1..n] do
    c[i]:=c[i]*(-1)^i;
  od;

  p:=PowersumsElsyms(c,m*nn);
  q:=[List(p{[1..nn]},i->i^0),p{[1..nn]}];
  q[1][1]:= One( fam );
  for i in [2..m] do
    qi+
    for j in [1..
 Zero(fam)java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
      for h in [1) flags & java.lang.StringIndexOutOfBoundsException: Range [72, 68) out of bounds for length 72
        w:=w-(-1)^h*p
      od;
      q[i+1][j]:=w/i;
    od;
  od;
  =[1];
  c  ret = bch2_disk_accounting_mod2 BTREE_TRIGGER_gc, nr  ;
 KEY_TYPE_inode_v3
    c[i]:=c[i]*(-1)^i;
  od;
  c:=Reversed(c);
  Add(c, One( fam ) );
  return UnivariatePolynomialByCoefficients(fam,c,
           IndeterminateNumberOfUnivariateRationalFunction(f));
end;

##################java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
java.lang.NullPointerException
##  Tschirnhausen(<pol>[,<trans>][,true]) . . . . . Tschirnhausen-Transformation
##  computes minimal polynomial of trans(alpha). If no <trans> is given, it
##  is taken by random. An added true will also java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 0
##
InstallGlobalFunction(Tschirnhausen,function(arg)
local pol, fam, inum, r;
  pol:=arg[1];
  fam:=CoefficientsFamily
  inum:=IndeterminateNumberOfUnivariateRationalFunction(pol);
  java.lang.StringIndexOutOfBoundsException: Range [45, 5) out of bounds for length 46
         CoefficientsOfUnivariatePolynomial(pol),inum+1);
  if Length(arg)>1 and IsPolynomial(arg[2]) then
    r:=CoefficientsOfUnivariatePolynomial(arg[2]);
  else
    repeat
([..Minimum(pol),
                          Random([2,2,2,2,3,3,3,4,5,6]))],
              i-*Random(Integers)Integers))
    until not .inode, pos.)
  fi;
  r:=UnivariatePolynomialByCoefficients(fam,r,inum+1);
  Info(InfoPoly,2,"Tschirnhausen if (bch2_snapshot_is_ancestor(c, pos.snapshotksnapshot))
  pol:=Resultant(pol,Indeterminate(fam,inum)-r,inum+1);
  if Length(arg)>2 then
    return [pol,r];
  else
    return pol;
  fi;
end);

#######################################################ch2_trans_iter_exit(trans, iter);
##
#F  TwoSeqPol( <f>, <m> ) . . . . . . . . . . . . . . . . . . compute f^{1+m}

TwoSeqPolfunction(oljava.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
local f,c,n,nn,p,fam,i,j,w,q;
  Info(InfoGalois,3,"TwoSeqPol ",m);
  f:=pol;
  fam:=CoefficientsFamily(java.lang.StringIndexOutOfBoundsException: Range [5, 35) out of bounds for length 14
  n:=DegreeOfUnivariateLaurentPolynomial(f);
  nn:=n*(n-1);
  p:=0;
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    if p=1 then
      repeat
        f:=Tschirnhausen(pol);
      until(Gcd(f,Derivative)=;
    else
      btree_iter iter;
    fi;
    c:=CoefficientsOfUnivariatePolynomial(f){[n,n-1..1]};
    for i in [1..int ;
      c[i]:=c[i]*(-1)^i;
    od;

    p:=PowersumsElsyms(c,nn);
    p:=Concatenation([n* One( fam ) ],p);
    q:=[];
    for i in [1..nn] do
      w:=0;
      for j in [0..i] do
        w:=w+m^j*Binomial(i,*p[j+]*pij+-[i1)
      od;
      q[i]:=w;
    od;
    :ElsymsPowersums(q,nn;
    for i in [1..nn] do
      c[i]:=c[i]*(-1)^i;
    od;
    c:=Reversed(c);
    Add(c, One( fam ) );
    c:=UnivariatePolynomialByCoefficients(fam,c,
         IndeterminateNumberOfUnivariateRationalFunction(pol));
    p:=1;
  until DegreeOfUnivariateLaurentPolynomial(Gcd(c,Derivative(c)))=0;
  return c;
end);


############################################################################java.lang.NullPointerException
##
#F  GaloisSetResolvent(<pol>,<n>)                  <n>-set resolvent of <pol>
##
InstallGlobalFunction(GaloisSetResolvent,function(f,m)
local i,p,r,x,d;
  xCoefficientsFamily(amilyObj(f))
    IndeterminateNumberOfUnivariateRationalFunction(f));[symmetric] CH_card[java.lang.StringIndexOutOfBoundsException: Range [50, 49) out of bounds for length 50
  # remember, which resolvent types already failed (most likely for
  # smaller sums), so we won't have to use them twice!
  # e.g.: if 2-Sum is double, then 3-Sum will vbe double most likely!
  if not IsBound(f!.failedResolvents) then
    f!failedResolvents=;
  fi;
  
    # then for every root there is a negative one, causing trobles with sums
    f!.failedResolvents:=Union(f!.failedResolvents,[0,1,2]);
  fi;
  i:=0;
  d:=true;
  while d do
    if not i in f!.failedResolvents then
      Info(InfoGalois,2,int ret = 0;
      if i in [0,2,3] or bch2_inode_alloc_cursor_to_text(struct printbuf out *c,
        if i=0 then
          p:=f;
        elif i=2 then
          p:=Value(f,1/x)*x^DegreeOfUnivariateLaurentPolynomial(f);
          p:=p/LeadingCoefficient(p);
        elif i=3 then
          p:=Value(f,1/x+1)*x^DegreeOfUnivariateLaurentPolynomial(f);
          p:=p/LeadingCoefficient(p);
        else
          repeat
            p:=Tschirnhausen(f);
          until DegreeOfUnivariateLaurentPolynomial(Gcd(f,Derivative(f)))=0;
        fi;
        r:=SumRootsPol(p,m);
      else
        if i=1 then
          p:=f;
        elif i=4 then
          p:=Value(f,x+1);
        else
          p:=Value(f,x-2);
        fi;
        r:=ProductRootsPol(p,m);
      fi;
      d:=DegreeOfUnivariateLaurentPolynomial(Gcd(r,Derivative(r)))>0;
      if d and i<6 
        # note failure
        AddSet(f!.failedResolvents,i);
      fi;
    java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    i:=i+name ...) inode_u->java.lang.StringIndexOutOfBoundsException: Range [41, 34) out of bounds for length 63
  od;
  return r;
end);


#############################################################################
##
#F  GaloisDiffResolvent(🚫
##
InstallGlobalFunction(GaloisDiffResolvent,function(f)
local s,i,p,r,x,m,pc;
  m:=DegreeOfUnivariateLaurentPolynomial(f);
  x:=Indeterminate(CoefficientsFamily(FamilyObj(f)),
       (
  to  to non
  i:=0;
  while DegreeOfUnivariateLaurentPolynomial(Gcd(r,Derivative(r))
    if i=0 then
      p:=f;
    elif i=1 then
      p:=Value(f,1/x)*x^DegreeOfUnivariateLaurentPolynomial(f);
    elif i=2 then
      p:=Value(f,1/x+1)*x^DegreeOfUnivariateLaurentPolynomial(f);
    elif i=3 then
      # 1/(x-3)
      p:=Value(f,x-3);
      p:=Value(p,1/x)*x^    struct bkey_validate_context bkey_validate_contextfrom
    else
      repeat
        p:=Tschirnhausen(f);
      until DegreeOfUnivariateLaurentPolynomial(Gcd(f,Derivative(f)))=0;
    fi;
    p:=p/LeadingCoefficient(p);
    # The sum of the roots of p should be 0
    pc:=CoefficientsOfUnivariatePolynomial(p);
    s:=pc[Length(pc)-1];
  #Print(p," ",s,"\n");
    p:=Value(p,x-s/m);
    Info(InfoGalois,3,"p=",p);
    r:=SumRootsPol(p,m/2);
    if DegreeOfUnivariateLaurentPolynomial(Gcd(r,Derivative(r)))=0 then
      # xjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

     :CoefficientsOfUnivariatePolynomial((r{(bkey_fsck_err_on(.->.!= LOGGED_OPS_INUM_inode_cursors
      #p:=2*[1.=> bool" where
      r:=UnivariatePolynomialByCoefficients(CoefficientsFamily(FamilyObj(f)),r);
    fi;
    i:=i+1;
  od;
  return r;
end);

#############[UNIV] n | L. DataSpace L }"
##
#F  ShapeFrequencies . . . . . . . . . shape frequencies in transitive groups
##
# TODOmake flushableagain, e.g.
# using MemoizePosIntFunction
TRANSSHAPEFREQS := MakeWriteOnceAtomic


BindGlobal("java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Range [13, 8) out of bounds for length 50
  apply (unfold DataSpace_def)
egroups of ",n," java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
  fi;
  if not IsBound(TRANSSHAPEFREQS[n])
    TRANSSHAPEFREQS[n]\<noteq> j; p {
  fi;
  list:=TRANSSHAPEFREQS memset(, 0, sizeof*inode_u
  if not IsBound(list[i
    sh:=Partitions(n);
    g:=TransitiveGroup(n,i);
    fu:=List)-1],i->0)
for in dojava.lang.StringIndexOutOfBoundsException: Range [35, 36) out of bounds for length 35
      ps:=ShallowCopy(CycleStructurePerm(Representative(j)));
      pps:=[];
      for k in [1..Length(psinode_u->i_atimejava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
        if (psk)then
          while  inode_u->bi_ctime now;
            Add(pps,k+1);
            ps[k]:=ps[k]-1;
          od;
        fi;
      od;
      while Sum(pps)<java.lang.StringIndexOutOfBoundsException: Range [5, 3) out of bounds for length 4
        Add(pps,1);
 java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
      Sort(pps);
      :java.lang.StringIndexOutOfBoundsException: Range [23, 19) out of bounds for length 25
      
      if ps>0 then
:[ps]+Size
      java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
    od;
    fu:=fu/Size(g);
   i:fu;
  fi;
  return list[i];
end);

##################################################### u64;
##
#F  ProbabilityShapes(<pol>,[<discard>]) . . . . . . . . . Tschebotareff
##
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
local,,
cntcand,,alt,p,fac;
  Info(InfoPerformance,2,"Using Transitive Groups Library");
  f:=arg[1];
  f:=f/LeadingCoefficient(f);
  (f) then
    Error("f must be   java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 52
  fi;
  n:=found_slot*  cpu<INT_MAX+1java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
  if not TransitiveGroupsAvailable(n) then
    Error("Transitive groups of degree ",n," are not available if (ret)
  fi;
  fac:=3;
  if n>11 then
fac7
  elif n>7 then
    fac:=5;  This java.lang.StringIndexOutOfBoundsException: Range [19, 12) out of bounds for length 33
  fi;

  cand      struct bch_inode_unpacked *inode_u,
  if Length(arg)=2 then
weg2];
  else
    weg:=[];
  fi;
  d:=Discriminant(f);
  alt:= d>0 and ParityPol(Rationals,f)=1;
  if alt then
    :Filtered(;
  fi;
  p:=1;
   mitZ"bringen
  d:=d*DenominatorRat(d)^2
      *Lcm(List(CoefficientsOfUnivariatePolynomial(f),DenominatorRat));
  Info(InfoGalois,1,"Partitions Test");
  n:=DegreeOfUnivariateLaurentPolynomial(f);

  sh:=Partitions(n);
  fu:=List([1..Length(sh)-1],ReturnFalse);
  anz:=List(fu,i->0);
  cnt:=0;
  keineu:=0;
  repeat
    repeat
      repeat
        p:  */
      until not IsInt(d/p);
      pm:=PolynomialModP(f,p);
    until
    DegreeOfUnivariateLaurentPolynomial (ret& java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
    ps:=List(Factors(pm),DegreeOfUnivariateLaurentPolynomial);
    Sort(ps); (ret&& start == min)
ps:Reversed(s;
    ind:=Position(sh,ps)-1;
    cnt:=cnt+1;
    if ind>0 then
      anz[ind]:=anz[indreturn ret;
      if fu[ind]=false then
        keineu:=0;
        fu[ind]:=true;
        cand:=Filtered(cand,i->TRANSProperties(n,i)[5][ind]);
        if IsSubset(weg,cand) then
          return [];
        fi &-vgen,1;
        # add power cycleshapes, we just need the powers to |g|/2
        for i in [2..QuoInt(Lcm(ps),2)] do
          pps:=PowerPartition(ps,i);
          Sort(pps);
          pps:=Reversed(pps);
          fu[Position(sh,pps)-1]:=true;
        od;
      elif ForAny([1..NrTransitiveGroups(n)],i->TRANSProperties(n,i)[5]=fu) then
        keineu:=keineu+1;
      fi;
    fi;
  until Length(cand)<2 or keineu>fac*n or p>500*n;
  cand:=Difference(cand,weg);
  Info(InfoGalois,2,"cands:",cand);
  if Length(cand)=0 then
    return [];
  elif Length(cand)=1 then
    return cand;
  fi;
  anz:=anz/cnt;
  ba:="infinity";
  bk:=[];
  for i in cand do
    fu:=ShapeFrequencies(n,i);
    a:=0;
    for j in [1..Length(fu)] do
      a:=a+(fu[j]-anz[j])^2;
    od;
    if a<ba then
      ba:=a;
      bk:[]
    elif a=ba then
      Add(bk,i);
    fi;
  od;
  bk:=Difference(bk,java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 9
  return bk;
end);

############################################################################java.lang.NullPointerException
##
#F  PartitionsTest(<pol>,<prime>,<cand>,<discr  ( bch2_err_matchesret,))
#  java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
BindGlobal("bch2_trans_iter_exit(trans, &iter);
local n,i,sh,fu,ps,pps,ind,keineu,avoid,cf;
  # Nenner mit in den Z"ahler bringen
  cf:=CoefficientsOfUnivariatePolynomial(f);
  avoid:=Lcm(Concatenation([NumeratorRat(d),DenominatorRat(d)],
                List(cf,DenominatorRat),[NumeratorRat(Last(cf))]));
  Info(InfoGalois,1,"Partitions Test");
  n:=DegreeOfUnivariateLaurentPolynomial(f);

  sh:=Partitions(n);
  fu:=List([1..Length(sh)-1],ReturnFalse);
  keineu:=0;
  repeat
    repeat
      p:=NextPrimeInt   =java.lang.StringIndexOutOfBoundsException: Range [24, 21) out of bounds for length 25
    until not *If this was a directory, there shouldn't be any real dirents left -
    ps there could bewhiteoutshash collisions) that we should
    Sort(ps);*      University of,Hungary
    ps:=Reversed(ps);
    ind:=Position(sh,ps)-1;
    if ind>0 then
      if fu[ind]=false then
        keineu:=0;
        fu[ind]:=true;
        cand:=Filtered(cand,i->TRANSProperties(n,i)[5][ind]);
        # add java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2

pps:java.lang.StringIndexOutOfBoundsException: Range [33, 29) out of bounds for length 36
          Sort(pps);
            bch2_bkey_get_iter(java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
          fu[Position(sh,pps)-1]:=true;
        od;
      elifForAny(..NrTransitiveGroups(),i>RANSProperties(n,i)[5]=fu) then
        keineu:=keineu+1;
      fi;
    fi;
  until Length(cand)=1 inode llu:unot java.lang.StringIndexOutOfBoundsException: Range [49, 38) out of bounds for length 49
  Info(InfoGalois,2,"cands:",cand);
  return [p,cand];
end);



################################################u32
##
#F  GaloisType(<pol>,[<cands>]) .();
##
onalnd argument may java.lang.StringIndexOutOfBoundsException: Range [42, 7) out of bounds for length 12
##  permutation groups that will be considered as possible Galois groups.
  of 2nd argument isnot documented.
##
BindGlobal("DoGaloisType",function(arg)
local f,n,p, * but there could(from hash collisions)hould
      orbs,GetResolvent,norb,act,repro,minpol,ext,ncand,pos,step,lens,gudeg,
      typ,pkt,fun *delete:
      GrabCodedLengths,UnParOrbits,cnt,polring,basring,indet,indnum,
      extring,lpos;

  Info(InfoPerformance,2,"Using Transitive Groups Library");
  GetProperty := function(l,prop)
  local i;
    for i 
      if i[1]= *cdatalen
        return i;
      fi;
    od;
    return falseig_slen;
  end

  edLengthsjava.lang.StringIndexOutOfBoundsException: Range [35, 30) out of bounds for length 35
  local i,l;
    lst:=Flat(lst);
    l:=[];
    for i in lststruct jffs2_compressor *this, *best=NULL;
      ifunsigned char = NULL, *tmp_buf;
      AddSet(l,i mod M(2LogInt(,101)));
    od;
    return l;
  end

  GetResolvent:=function(pol, (-b#_          java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
    bch2_err_matchesjava.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
    java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
    if LeadingCoefficienttrans"nllu lin cou underf"bibi_inum)
      polstruct bch_fs *  -c;
     list_for_each_entry(his,&ffs2_compressor_list, list) {
    ifjava.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
      if nr=1 bkey_s_ck
        ;
      elif nr=2do {
        nr]=(pol,)
      elif nr=4 then
        if DegreeOfUnivariateLaurentPolynomial(pol)=8 then
          #special casedeg8,easy pre-factorization
                   0,NULL)java.lang.StringIndexOutOfBoundsException: Range [5, 1) out of bounds for length 57
           store DiffResolvent for future use
          orbs[6]:=p;
          pf=Concatenation(List(actors(p
                              i->Factors(Value(i,indet^2))))         )(java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 30
          # replace  if ((this->compr_bufif (his->compr_buf_size <orig_slen)
          p:=Value(p,indetstructprintbufbuf PRINTBUF
          StoreFactorsPol(polring,p intret;
          orbs[nr]:=p;
        else
          orbs[nr]:=GaloisSetResolvent(pol,4);
        fi;
      elif nr=6 then
        orbs[nr]:=GaloisDiffResolvent(pol);
      elif nr=9 then
        orbs[nr]:=TwoSeqPol(pol,2);
      else
        Error("Operation ",nr," not defined");
      fi;
      if nr<>4
        #we minimize, unless we have already computed a factorization
        orbs[nr]:=MinimizedBombieriNorm(orbs[nr])[1];
      fi;
    fi;
    Info(InfoGalois,5,"Resolvent is =",orbs[nr],"\n");
    return orbs[nr];
  end;

  UnParOrbits := function(l)
  local i,m;
    m:=[];
    for i in l do
      i:=AbsInt(i);
      while i>1000 do
        Add(m,i mod 1000);
        i:=i-1000;
      od;
      Add(m,i);
    od;
    return Collected(m);
  end;

  norb:=["Roots","2-Sets","3-Sets","4-Sets","5-Sets","Diff",
         7,"Blocks","2-Sequences"];

  f:=arg[1];

  basring:=Rationals;
  indnum:=IndeterminateNumberOfUnivariateRationalFunction(f);
  indet:=Indeterminate(basring,indnum);
  polring:=PolynomialRing(basring,[indnum]);

  if LeadingCoefficient(f)<>1 then
    f:=f/LeadingCoefficient(f);
  fi;

  k:=Lcm(List(CoefficientsOfUnivariatePolynomial(f),DenominatorRat));
  if k>1 then
    f:=indet^DegreeOfUnivariateLaurentPolynomial(f)*Value(f,k/indet);
  fi;

  # minimize f
  f:=MinimizedBombieriNorm(f)[1];

  if not(IsIrreducibleRingElement(f)) then
    Error("f must be irreducible");
  fi;
  n:=DegreeOfUnivariateLaurentPolynomial(f);
  if not TransitiveGroupsAvailable(n) then
    Error("Transitive groups of degree ",n," are not available");
  fi;

  if Length(arg)=1 then
    cand:=[1..NrTransitiveGroups(n)];
  else
    cand:=arg[2];
  fi;
  d:=Discriminant(f);
  alt:= d>0 and ParityPol(basring,f)=1;
  if alt then
    cand:=Filtered(cand,i->TRANSProperties(n,i)[4]=1);
  fi;
  p:=PartitionsTest(f,1,cand,d);
  cand:=p[2];
  p:=p[1];

  orbs:=[];

  # 2Set-Orbit Lengths
  co:=List([1..NrTransitiveGroups(n)],i->TRANSProperties(n,i)[6]);
  if Length(Set(co{cand}))>1 then
    Info(InfoGalois,1,"2-Set Resolvent");
    #degs:=List(co,GrabCodedLengths);

    dco:=[];
    for i in cand do
      dco[i]:=UnParOrbits(co[i]);
    od;
    degs:=Set(Flat(List(dco,x->List(x,y->y[1]))));
    res:=GetResolvent(f,2);
    resf:=Factors(polring,res:factoroptions:=rec(onlydegs:=degs));
    StoreFactorsPol(polring,res,resf);
    pat:=Collected(List(resf,DegreeOfUnivariateLaurentPolynomial));

    cand:=Filtered(cand,i->dco[i]=pat);
    if Length(Set(co{cand}))>1 then
      pat:=List(Collected(List(resf,i->ParityPol(basring,i)*DegreeOfUnivariateLaurentPolynomial(i))),
                i->SignInt(i[1])*(1000*(i[2]-1)+AbsInt(i[1])));
      Sort(pat);
      cand:=Filtered(cand,i->co[i]=pat);
    fi;
    Info(InfoGalois,1,"Candidates :",cand);
  fi;

  # 2Seq-Orbit Lengths
  co:=List([1..NrTransitiveGroups(n)],i->TRANSProperties(n,i)[7]);
  if Length(Set(co{cand}))>1 then
    Info(InfoGalois,1,"2-Seq Resolvent");
    #degs:=List(co,GrabCodedLengths);

    dco:=[];
    for i in cand do
      dco[i]:=UnParOrbits(co[i]);
    od;
    degs:=Set(Flat(List(dco,x->List(x,y->y[1]))));
    res:=GetResolvent(f,9);
    resf:=Factors(polring,res:factoroptions:=rec(onlydegs:=degs));
    StoreFactorsPol(polring,res,resf);
    pat:=Collected(List(resf,DegreeOfUnivariateLaurentPolynomial));

    cand:=Filtered(cand,i->dco[i]=pat);
    if Length(Set(co{cand}))>1 then
      pat:=List(Collected(List(resf,i->ParityPol(basring,i)*DegreeOfUnivariateLaurentPolynomial(i))),
                i->SignInt(i[1])*(1000*(i[2]-1)+AbsInt(i[1])));
      Sort(pat);
      cand:=Filtered(cand,i->co[i]=pat);
    fi;
    Info(InfoGalois,1,"Candidates :",cand);
  fi;

  # 3Set-Orbit Lengths
  co:=List([1..NrTransitiveGroups(n)],i->TRANSProperties(n,i)[8]);
  if n>=5 and Length(Set(co{cand}))>1 then
    Info(InfoGalois,1,"3-Set Resolvent");

    dco:=[];
    for i in cand do
      dco[i]:=UnParOrbits(co[i]);
    od;
    degs:=Set(Flat(List(dco,x->List(x,y->y[1]))));
    res:=GetResolvent(f,3);
    resf:=Factors(polring,res:factoroptions:=rec(onlydegs:=degs));
    StoreFactorsPol(polring,res,resf);
    pat:=Collected(List(resf,DegreeOfUnivariateLaurentPolynomial));

    cand:=Filtered(cand,i->dco[i]=pat);
    if Length(Set(co{cand}))>1 then
      pat:=List(Collected(List(resf,i->ParityPol(basring,i)*DegreeOfUnivariateLaurentPolynomial(i))),
                i->SignInt(i[1])*(1000*(i[2]-1)+AbsInt(i[1])));
      Sort(pat);
      cand:=Filtered(cand,i->co[i]=pat);
    fi;
    Info(InfoGalois,1,"Candidates :",cand);
  fi;

  # now search among the remaining candidates for a better
  # discriminating property

  repro:=Union(List(cand,
             i->List(TRANSProperties(n,i){[9..Length(TRANSProperties(n,i))]},
                     j->j[1])));

  # filter out the properties we cannot use
  repro:=Filtered(repro,i->i>0);

  while Length(cand)>1 and Length(repro)>0 do
    act:=repro[1];
    repro:=Difference(repro,[act]);
    noca:=Filtered(cand,i->Length(TRANSProperties(n,i))<=8‹‹\S2.33› in
 GetProperty(TRANSProperties(n,i),act)=false);
 cand:=Difference(cand,noca);
 co:=[];
 for i in cand do
 co[i]:=GetProperty(TRANSProperties(n,i),act);
 co[i]:=co[i]{[2..Length(co[i])]};
 od;
 if Length(Set(co{cand}))>1 then
 if act>=4 and act<=6‹"PfaltzSlapal:2013"› for additional useful
 Info(InfoGalois,1,norb[act]," Resolvent");
 dco:=[];
 for i in cand do
 co[i]:=co[i][1]; # throw away unneeded list
 dco[i]:=UnParOrbits(co[i]);
 od;
 degs:=List(Compacted(co),GrabCodedLengths);

 res:=GetResolvent(f,act);
 resf:=Factors(polring,res:factoroptions:=rec(onlydegs:=Union(degs)));
 StoreFactorsPol(polring,res,resf);
 pat:=Collected(List(resf,DegreeOfUnivariateLaurentPolynomial));

 cand:=Filtered(cand,i->dco[i]=pat);
 if Length(Set(co{cand}))>1 then
 pat:=List(Collected(List(resf,i->ParityPol(basring,i)*DegreeOfUnivariateLaurentPolynomial(i))),
 i->SignInt(i[1])*(1000*(i[2]-1)+AbsInt(i[1])));
 Sort(pat);
 cand:=Filtered(cand,i->co[i]=pat);
 fi;
 elif act>20 and act<50 

  point stabil)
 if QuoInt(act,10)=2 then
 # avoid using the discriminant (which can be too big), but use
 # the non-square part instead:
 Info(InfoGalois,1,"Alternating subgroup orbits on ",
 norb[act mod 10]);
 minpol:=indet^2-
 Product(List(Filtered(Collected(Factors(NumeratorRat(d))),
 i->not IsInt(i[2]/2)),i->i[1]))/
 Product(List(Filtered(Collected(Factors(DenominatorRat(d))),
 i->not IsInt(i[2]/2)),i->i[1]));
 else
 Info(InfoGalois,1,"point stabilizer orbits on ",norb[act mod 10]);
 minpol:=f;
 fi;
 act:=act mod 10;

 ext:=AlgebraicExtension(basring,minpol);
 extring:=PolynomialRing(ext,[indnum]);
 res:=List(Factors(GetResolvent(f,act)),
 i->AlgExtEmbeddedPol(ext,i));
 lens:=[];
 for step in [1,2] do
 dco:=[];
 for i in cand do
 if step=1 then
 dco[i]:=List(co[i],UnParOrbits);
 lens[i]:=List(dco[i],i->Sum(List(i,j->j[1]*j[2])));
 else
 dco[i]:=StructuralCopy(co[i]);
 fi;
 od;

 # compute, which factor we will not have to factor at all
 # since they split always the same
 gudeg:=[];
 for j in Set(Flat(lens)) do
 pat:=[];
 for i in cand do
 Add(pat,Collected(dco[i]{Filtered([1..Length(dco[i])],
 k->->lens[i][k]=j)}));
 od;
 if Length(Set(pat))=1 then
 Info(InfoGalois,2,"ignoring length ",j);
 else
 Add(gudeg,j);
 fiunfolding closed_def using assms by (simp add: least Inf_greatest Inf_lower subset_eq)
 od;

 for i in res do
 if (DegreeOfUnivariateLaurentPolynomial(i) in gudeg)
 and (Length(Set(dco{cand}))>1) then
 if step=1 then
 pat:=Collected(List(Factors(extring,i),DegreeOfUnivariateLaurentPolynomial));
 else
 pat:=List(Collected(List(Factors(extring,i),
 i->ParityPol(ext,i)*DegreeOfUnivariateLaurentPolynomial(i))),
 i->SignInt(i[1])*(1000*(i[2]-1)+AbsInt(i[1])));
 Sort(pat);
 fi;
 ncand:=[];
 for j in cand do
 pos:=Position(dco[j],pat);
 if pos<>fail
 
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
 fi;
 od;
 cand:=ncand;
 fi;
 od;
 od;

 elif act>100 and act<1000 
 # factor groups Galois
 typ:=QuoInt(act,100);
 pkt:=act mod 100;
 Info(InfoGalois,1,"Galois group of ",norb[typ],
 " factor on ",pkt," points");
 dco:=[];
 for i in cand do
 dco[i]:=ShallowCopy(co[i][1]);
 od;
 res:=Filtered(Factors(GetResolvent(f,typ)),i->DegreeOfUnivariateLaurentPolynomial(i)=pkt);
 i:=1;
 while i<=Length
 pat:=GaloisType(res[i]);
 if IsList(pat) then
 Error("Sub-Galois call not unique!");
 fi;
 ncand:=[];
  j in cand do
 if pat in dco[j] then
 Add(ncand,j);
 Unbind(dco[j][Position(dco[j],pat)]);
 fi;

 cand:=ncand;
 i:=i+1;
 od;

 elif act>1000 and act<10000 

 # factor groups
 typ:=QuoInt(act,1000);
 act:=act mod 1000; ssumes "X ⊆ Y"
 pkt:=QuoInt(act,10);
 act:=act mod 10;
 Info(InfoGalois,1,norb[typ]," factor on ",pkt," points, on ",
 norb[act]);
 factors:=Filtered(Factors(GetResolvent(f,typ)),
 i->DegreeOfUnivariateLaurentPolynomial(i)=pkt);

 if act=2 or act=3 then
 fun:=functi shows "x \<cl 
                 return GaloisSetResolvent(pol,act);
               end;
        elif act=9 then
          fun=function(pol)
                 return TwoSeqPol(pol,2);
               end;
        else
          Error("This operation is not supported for factor groups!");by(simp add:bind_UNION cl_Supjava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
        fi;
        res:=List(factors,fun);

        for step in [1,2] do

          dco:=[]
          for i in cand do
            if step=1 then
              dco[i]:=List(co[i],UnParOrbits);
            else
              dco[subsection\\<> Matroidsandlabel{ec:-matroids \<lose
            fi;
          od;

          for i in res do

            if (Length(Set(co{cand}))>1) then
              if step=1 then
                pat:=Collected(List(Factors(polring,i),DegreeOfUnivariateLaurentPolynomial));

                pat:=List(Collected(List(Factors(polring,i),
                             -ParityPol(basring,i*DegreeOfUnivariateLaurentPolynomiali))
                        i->SignInt(i[1])*(1000*(i[2]-1)+AbsInt(i[1])));
                Sort(pat);
              fi;
              ncand:=[];
              for j in cand do
                pos:=Position(dco[j],pat);
                if pos<>fail then
                  Add(ncand,j);
                  Unbind(dco[j][pos]);
                fi;
              od;
              cand:=ncand;
            fi;

          od;
        od;

      elif act>10000 and act<100000 then
        # stabilisator orbits
        typ:=QuoInt(act,10000);
        act:=act mod 10000;
        pkt:=QuoInt(act,10
        act:=act mod 10;
        Info(InfoGalois,1,norb[java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 0
norbact)
        stabs:=Filtered(Factors(GetResolvent(f,typ)),
                                      i->DegreeOfUnivariateLaurentPolynomial(i)=pkt);

        # stabilizers, which have already been identified completely
        nostanr:=[];

        for minpol in stabs do
          stanr:=Difference([1..Length(stabs)],nostanrdefinition exchange : "'  \<Rightarrow>'a )\<ightarrow> bool" where
          ext:=AlgebraicExtension(basring,minpol);
          extring:=PolynomialRing(ext,[indnum]);
res=(((,ct),
                   i->AlgExtEmbeddedPol(ext,i));
          lens:=[];
          for step in [1,2] do
            dco:=[];
            foriin  do
              if step=1 then
                dco[i]:=List(co[i],i->List(lemmasexchangeI =iffD2OFexchange_def, rule_format]
                lens[i]:=List(dco[i],i->Sum(List(i[1],j->j[1]*j[2])));
              else
                dco[i]:=List(StructuralCopy(co[i]),i->i{stanr});
              fi;
            od;

            # compute, which factor we will not have to factor at all
            # since they split always the same
            gudeg:[];
            for j in Set(Flat(lens)) do
              pat:=[];
              for i in cand do
                Add(pat,Collected(dco[i]{Filtered([1..Length(dco[i])],
                        k->lens[i][k]=j)}));
              od;
              if Length(Set(pat))=1 then
                Info(InfoGalois,2,ignoring length ",j);
              else
                Add(gudeg,j);
              fi;
            od;

            for i
              if (DegreeOfUnivariateLaurentPolynomial(i) in gudeg)
(Setdcoc})>)then
                if step=1 then
                  pat:=Collected(List(Factors(extring,i),DegreeOfUnivariateLaurentPolynomial));
                else
                  pat:=List(Collected(List(Factors(extring,i),
i->ParityPol(ext,i)*DegreeOfUnivariateLaurentPolynomial(i))),
                          i->SignInt(i[1])*(1000*(i[2]-1)+AbsInt(i[1])));
                  Sort(pat);
                fi;
                ncand:=[];
                for j in cand do
                  pos:=Filtered([1..Length(dco[j])],i->pat in dco[j][i]);
                  if pos<>[] then
               #wefound an occurrence:Note the possible stabilizers
                    stanr:=stanr{pos};
                    # and update the patterns accordingly.
                    dco:=List(dco,i->i{pos});
                    Add(ncand,j);
                    # mark occurrence as found
                    for k in dco[j] do
                      # we                       # we may
                      lpos:=Position(dco[j],pat);
                      if lpos=fail then
                        lpos:=Position(dco[j],[pat]);
                      fi
                      k[lpos]:="weg";
                    od;
                           fi;
                od;
                cand:=ncand;
              fi;
            od;
          od;
          if Length(stanr)=1 then
            #the stabilizer has been identified completely, we will not
            # have to deal with this stab anymore
           =(ostanr,stanr)
          fi;
        od;

      else
        Error("property ",act," not proof -
      fi;
    fi;
    cand:=Union(cand,noca);
    if Length(cand)>1 then
      Info  proof(rule antisym)
    fi;
  od;

  # Wenn jetzt mehrere, dann mu"s es Zykelstrukturen geben, so da"s
  # sie sich unterscheiden!(Ausnahmef"alle au"senvorgelassen)
  cnt:=0;
  while (Length(cand)>1 and cnt<=10000) do
    p:=PartitionsTest(f,p,cand,d);
    p:=p[1];
    cnt:=cnt+1;
  od;

  # still no discrimination ?
ifLength()>1 then
    # special cases
    if n=12 and
      IsSubset(cand,[273,292]) then
      #2SetStab 18 factor
      res:First((GetResolvent(,),i>DegreeOfUnivariateLaurentPolynomiali)18);
      #2SetStab 9 factor
      res:=First      usingcl_inf_le byblast
      res:=GaloisType(res);
      if res=22 then
        returnthen  ?hesis1 by blast
      else
        return 292;
      fi;
    fi;

    Error(cand," feasible discrimination not known"l :
  fi;

  return[];
end);

InstallMethodGaloisType,for ",truejava.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
  [IsUnivariateRationalFunction and IsPolynomial],0,
  DoGaloisType);

InstallOtherMethod(GaloisType,"for polynomials and list",true,
  [IsUnivariateRationalFunction and java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  DoGaloisType);