;---------------------------------------*******************************************-----------------------------
;------ Fermat functions to factor multivariate polynomials.  May 30, 2023.  Robert H. Lewis, rlewis@fordham.edu
;; Revsisons to May 1, 2024.
;; Requires Fermat 7.0 or later.

;; Revised December 2023, January 2024, February 11, February 27, March 3, April 16, May 2. Needs 7.5a to work for polymodding.

;; Jan 2024:  added function Max, Sqfree test, and using better selection of primes when polymodding.

;; Algorithm of Daqing Wan, Factoring multivariate polynomials over large finite fields.  Mathematics of Computation volume 54.
;; number 190, April 1990, pages 755-770.  The paper is written for the bivarite case, but can pretty easily be extended to
;; multivariate, as here.

;; Readin this file after reading your own code.

;; One factor is produced, named 'fct'.

;; To invoke over Z:  Use FCTZ. See documentation below. FCTZ works by dropping in and out of modular mode.  

;; To invoke over Z/p: Use SPF.  See documentation below.  p is assumed to be "reasonably large", say > 150.  Perhaps
;; more realistically, > 1000.  There are a few increment commands, i:+, so mod p, cycling around is possible. There should be no
;; problem if p > 100, say.

;; Mod p the factor is not reduced to have leading coefficient 1. If your poly 'arg' has one or two variables,
;; use Factor(arg, [q]). If your poly has contents, a content will be returned.

;; It's assumed that &a = 1 at start.
;; To reduce name conflicts, many of the variables (other than parameters) created and deleted here end with "_", as
;; r_, s_, aa_, n_, etc.  Longer names do not, such as k_used, lastvar, orig_k, etc.  Check for conflicts with your own names,
;; including function names.  If you have a global variable name identical to one of these, it will be deleted at the
;; end of the factoring.  If you have a poly var the same name as one of the parameters of a function here, that will be bad,
;; as poly var names have precedence.  A future version of Fermat may change that.

;; This has not been tested for more than eight variables.  Should work fine.  Of course, ideally, the more vars, the smaller
;; the argument poly should be.

;; The program is quite a bit more complex than the algorithm sketched in Wan's paper, for two main reasons.
;; First, a lot of time can be saved by looking for the "best" pair of variables to be the main pair, denoted x and y in Wan's
;; paper.  We want a pair that will produce relatively small polys later when the argument must be made monic. Hence the array [co].
;; Secondly, Functions G5 and H5 (and G3, H3) implement the sequence of polys denoted g_i and h_j on page 760.  We go to a lot of
;; effort to not recompute values; hence the complexity here, [g], [h], prevr, prevs, etc.

;; The code contains various explanatory messages that I found useful in debugging, such as "SUCCESS" or "FAILURE".
;; Delete them if you desire.  Lower case "fail" messages are not serious and could be removed.
;; Similarly, there are a couple places where &_P is used to interrupt after a failure.
;;
;; When I say "polymodding" I mean modding out by a polynomial to create a field.  See the Fermat manual if this is new to you.
;; These procedures will not work if you modded out by a polynomial and did not create a field.
;;
;; Update: this version of factr removes most of the messages referred to in the previous paragraphs.
;; This version March 3, 2024. May 1, 2024. June 4, 2024

;; BUG discovered, fixed April 2024.  maxn_ is set to 20 arbitrarily in FCT, which is the mod p function.  But FCTZ could start with a larger number of vars
;; and then hit that limit when it goes mod p and calls FCT. That's bad. FCTZ should set maxn_ for its problem.  Fixed below.

;; Sep 19, 2024:  fixed typo with D5 -> D_5.


;;====================================================================================================
;;====================================================================================================

;; added April 2024. Whether this file is read in modularmode or not, this should work.  Spelling changed to maxn_, Jun 2024.
maxn_ := 0;

;; First we have a group of functions that works quickly mod p with 3 or 4 poly vars when polymodding.
;; Assume the 3 or 4 vars are the highest precedence.  This group ends around line 500.

;; Is there a fourth variable?
Func Exists4 =
  if Polymod then
    if Flevel > 4 then
      Return(1)
    else
      Return(0)
    fi
  else
    if Numvars > 3 then
      Return(1)
    else
      Return(0)
    fi
  fi.;

;; homogenize to put in y (shorthand for Var(1)).
Func Homogy3(s) =  Numer(s#(Var(2) = Var(2)/Var(1))).;

;; Factor two var homogeneous.  Don't forget the Content.
;; Create global array [a].
Func Fth(za; z2,i) =
  Totdeg(za,[b_]);
  if b_[1] <> b_[2] then
    !!('in Ft, input is not homogeneous, return 0');
    Return(0)
  fi;
  z2 := Content(za);
  if Deg(z2,2) > 0 then
    Factor(z2, [c_])
  else
    Array c_[1]
  fi;
  z2 := za#(Var(2) = 1);
  Factor(z2, [a]);
  for i = 1, Rows[a] do
    a[i] := a[i]#(Var(1) = Var(1)/Var(2))*Var(2)^Deg(a[i]);
    a[i] := a[i] / Nlcoef(a[i])
  od;
  if Deg[c_] > 1 then
    [v_] := [a]_[c_];
    @[a];
    Rname[v_] := '[a]'
  fi;
  @([c_], [b_]);
  Return(Rows[a]).;

;; Compute homogeneous part of largest degree - k.  Use [var]. NB: enter positive k.
;; 7651 is random.
;; keep two top vars.  substitute out all others
Func Homzw3(za, k, s) =`
  if Exists4 then
    Totdeg((za#(Var(4) = 7651))#(Var(3)=Rand), [aa_])
  else
    Totdeg(za#(Var(3)=Rand), [aa_])
  fi; `
  s := WDeg(za, [var], aa_[1]-k);`
  @[aa_];`
  s.;

;; make monic in y = Var(1).  15 is just a "guaranteed" large enough value.  b0 is global.
Func Monicy3(uu,i,ww) =
  ww := Homzw3(uu);
  { ww contains no w or z }
  b0 := 0;
  for i = 1, 15 do
    if ww#(i,1) <> 0 then
      b0 := i;
      &>
    fi
  od;
  if b0 = 0 then
    !!('FAIL monic y ', Var(1));
    Return(0)
  fi;
  uu#(Var(2) = Var(2) + b0*Var(1))/ww#(b0,1).;

;; make monic in x = Var(2).  15 is just a "guaranteed" large enough value.  bb is global.
Func Monicx3(uu,i,ww) =
  ww := Homzw3(uu);
  bb := 0;
  for i = 1, 15 do
    if ww#(1,i,1) <> 0 then
      bb := i;
      &>
    fi
  od;
  if bb = 0 then
    !!('FAIL monic x ', Var(2));
    Return(0)
  fi;
  uu#(Var(1) = Var(1) + bb*Var(2))/ww#(1,bb,1).;

;; Do steps 2, 3, 4 on page 763 of Wan paper.  assume za square free.  a_a is global.
;; Feb 2022:  add uu and vv, as in dioq.
Func Mkgood(za; uu, vv, i) =
  { Step 2: u = f*, normalized w.r.t. y. Actually, should first check this is necessary. }
  uu := Monicy3(za);
  { Step 3: Don't do Dixon twice. That's a huge waste of time. Just try a = 1, 2, etc. }
  vv := Deriv(uu, Var(2), 1);
  a_a := 0;
  for i = 1, 15 do
    if GCD(uu#(Var(2)=i), vv#(Var(2)=i)) then
      a_a := i;
      &>
    fi
  od;
  if a_a = 0 then
    !!('FAIL mkgood. Not square free?');
    Return(0)
  fi;
  Totdeg(uu, [b_]);
  a_a := 1/a_a;
  vv := Sigma<i=0,b_[1]> [ WDeg(uu, [vv], i)*(a_a*Var(2) - a_a)^(b_[1] - i) ];
  @[b_];
  { Put it back!!}
  a_a := 1/a_a;
  Monicx3(vv).;

;; assume degf, r_ and s_ given as globals.  also f, gr, hs, u_ and v_. Use global [aaf]. hfktest is reduced.
Func G3(m; k, i, t, j, gval, hval, hfkt, hfktest, usedold, dum) =
  j := m + degf + 1 - r_;
  if j = 0 then
    !!'failure 0 in G';
    Return(mm)
  fi;
  if g[j] = mm then
    k := r_ - m;
    if r_=prevr and s_=prevs and k=prevk then
      if prevt = mm then
        { make new t value }
        for i = 1, k-1 do
          hval := H3(s_-k+i);
          if hval = mm then
            !!'failure 1 in G';
            Return(mm)
          fi;
          gval := G3(r_-i);
          if gval = mm then
            Return(mm)
          fi;
          t := t + gval*hval
        od;
        prevt := t;
        prevs := s_;
        prevr := r_;
        prevk := k
      else
         if hfktest < 2 then
           t := prevt
         fi;
         usedold := 1;
      fi
    else
      { make new t value }
      for i = 1, k-1 do
        hval := H3(s_-k+i);
        if hval = mm then
          !!'failure 1 in G';
          Return(mm)
        fi;
        gval := G3(r_-i);
        if gval = mm then
          Return(mm)
        fi;
        t := t + gval*hval
      od;
      prevt := t;
      prevs := s_;
      prevr := r_;
      prevk := k
    fi;
    if hfzerg < 2 and hfk[k] = mm then
      hfk[k] := WDeg(fmk, [var], aaf[1]-k)#(1)
    fi;
    if hfzerg >= 2 then
      hfktest := 0
    else
      if usedold then
        if Exists4 then
          hfktest := ( (hfk[k]#(Var(4)=2003))#(Var(3)=181) - (t#(Var(4)=2003))#(Var(3)=181) ) | gr
        else
          hfktest := ( hfk[k]#(Var(3)=181) - t#(Var(3)=181) ) | gr
        fi
      else
         hfktest := 1
      fi
    fi;
    if hfktest = 0 then
      hfzerg :+ ;
      g[j] := 0
    else
      hfkt := hfk[k] - t;
      if Deg(^hfkt, 2) >= Deg(gr, 2) then
        hfkt := hfkt | gr
      fi;
      if hfkt = 0 then
        hfzerg :+
      fi;
      &(#=20);
      dum := v_*hfkt;
      &(#=80);
      dum :=  dum | gr;
      g[j] := dum;
    fi;
    g[j]
  else
    g[j]
  fi.;


;; degf, r_ and s_ are globals.   also gr, hs, mm, u_ and v_.  Note Homz.  Use global [aaf]
Func H3(m; k, i, t, j, dum) =
  j := m + degf + 1 - s_;
  if j = 0 then
    !!'failure 0 in H';
    Return(mm)
  fi;
  if hfzerg >= 4889 then
    h[j] := 0;
    Return(0);
  fi;
  if h[j] = mm then
    k := s_ - m;
    if r_=prevr and s_=prevs and k=prevk then
      if prevt = mm then
        { make new t value }
        for i = 1, k-1 do
          hval := H3(s_-k+i);
          if hval = mm then
            Return(mm)
          fi;
          gval := G3(r_-i);
          if gval = mm then
            Return(mm)
          fi;
          t := t + gval*hval
        od;
        prevt := t;
        prevs := s_;
        prevr := r_;
        prevk := k
      else
        t := prevt
      fi
    else
      { make new t value }
      for i = 1, k-1 do
        hval := H3(s_-k+i);
        if hval = mm then
          Return(mm)
        fi;
        gval := G3(r_-i);
        if gval = mm then
          Return(mm)
        fi;
        t := t + gval*hval
      od;
      prevt := t;
      prevs := s_;
      prevr := r_;
      prevk := k
    fi;
    if hfk[k] = mm then
      hfk[k] := WDeg(fmk, [var], aaf[1]-k)#(1)
    fi;
    hfkt2 := hfk[k] - t;
    if Deg(^hfkt2,2) >= Deg(hs,2) then
      hfkt2 := hfkt2 | hs
    fi;
    if hfkt2 = 0 then
      hfzerh :+
    fi;
    &(#=20);
    dum := u_*hfkt2;
    &(#=80);
    dum :=  dum | hs;
    h[j] := dum;
    if Deg(h[j],2) > j then
      Return(mm)
    fi;
    h[j]
  else
    h[j]
  fi.;

;; mm = impossible value for Z/p.
;; creates (or uses) globals fn, mm, r_, s_, degf, gr, hs, u_, v_.
;; same as in dioq.  Note gr and hs involve only x (Var(2)).
Func Setup3(d) =
  r_ := Deg(d);
  s_ := degf - r_;
  gr := d#(1);
  hs := fn#(1)/gr;
  EGCD(gr,hs,u_,v_);
  { all of these polys only contain x and gr*u + hs*v = 1. }
  [g] := mm;
  [h] := mm;
  g[degf+1] := gr;
  h[degf+1] := hs.;
  
;; Uses r_, degf.
Func Fact3(d; i, deg) =
   Setup3(d);
  deg := r_-1;
  for i = 0, deg do
    G3(deg);
    if degf = i then
      Return(0)
    fi;
    if Deg(g[degf-i], 2) > deg then
      Return(0)
    fi;
    deg :-
  od;
  G3(-1);
  { added this Feb 2022, bug fix: }
  if degf = r_ then
    !!'failure 0';
    Return(0)
  fi;
  if g[degf-r_] <> 0 then
    Return(0)
  fi;
  deg := 0;
  for i = degf-r_+1, degf+1 do
    deg := deg + Homogy3(g[i])
  od;
  deg.;

;; secondary driver.  for three or four vars. Create global arrays aaf,g,h,hfk.  Use array a.
;;  Create aaf once and for all. Creat global degf
;; use global fmk instead of argument f.
Func Sdriv(i,j,dd,ff,gg,ij,limit) =
  if Exists4 then
    Totdeg((fmk#(Var(4) = 7651))#(Var(3)=2003), [aaf])
  else
    Totdeg(fmk#(Var(3)=2003), [aaf])
  fi;
  fn := WDeg(fmk, [var], aaf[1]);
  if Deg(^fn, 3) > 0 then
    !!'Error fn has third var ';
  fi;
  if Exists4 then
    if Deg(^fn, 4) > 0 then
      !!'Error fn has fourth var '
    fi
  fi;
  degf := Deg(fn);
  Array g[degf+1], h[degf+1], hfk[degf+1];
  [hfk] := mm;
  { Factor homogeneous fn.  Craete [a] }
  Fth(fn);
  dd := Rows[a];
  { Put larger polys first }
  for i = 1, dd \ 2 do
    j := a[i];
    a[i] := a[dd+1-i];
    a[dd+1-i] := j;
    j := a[i,2];
    a[i,2] := a[dd+1-i,2];
    a[dd+1-i,2] := j
  od;
  Array sub[Rows[a]];
  half := _(Rows[a]\2 + 1);
  limit := Sigma<i = 1, half> [ Bin(Rows[a],i) ];
  incr := 0;
  for ij = 1, limit do
    Nextsub(half);
    dd := 1;
    for j = 1, Deg[sub] do
      if sub[j] then
        dd := dd*a[j]^a[j,2]
      fi
    od;
    ff := Fact3(dd);
    if ff <> 0 then
      @([sub],[g],[h],[hfk]);
      fmk := 1;
      Return(ff)
    fi
  od;
  fmk := 1;
  @( [sub], incr, [g],[h],[hfk]).;

;;  main driver. Apr 2024, changed name to D_5.
Func D_5(h; deg, rzx, rzy, rwx, rwy,hzz) =
  { impossible value: }
  mm := _(Modulus+1);  Array var[2];
  if Exists4 then
    Array vv[4]
  else
    Array vv[3]
  fi;
  var[1] := Var(1);
  var[2] := Var(2);
  vv[1] := Var(1);
  vv[2] := Var(2);
  vv[3] := Var(3);
  hfzerg := 0;
  hfzerh := 0;
  prevt := mm;
  prevr := 0;
  prevs := 0;
  prevk := 0;
  prevt := mm;
  fct := 1;
  rzx := Rand;
  rzy := Rand;
  hzz := h#(Var(3) = Var(3) + rzy*Var(1) + rzx*Var(2));
  if Exists4 then
    vv[4] := Var(4);
    rwx := Rand;
    rwy := Rand;
    hzz := hzz#(Var(4) = Var(4) + rwy*Var(1) + rwx*Var(2))
  fi;
  fmk := Mkgood(hzz);
  if fmk = 0 then
    !!'FAIL D_5';
    Return(0)
  fi;
  fct := Sdriv;
  if fct = 0 then
    @([a],[aaf],[vv],[var]);
    @(hfzerg,hfzerh,prevt,prevr,prevs,prevk,prevt,fmk,fn,gr,incr,mm);
    @(degf, a_a, b0, bb, hfkt2, half, hs, r_, s_, u_, v_);
    Return(0)
  fi;
  { return to factors of h. coef here in three vars isn't right, just forget the coef. }
  deg := Deg(fct);
  if Exists4 then
    fct := Numer(fct#(Var(1) = (Var(1) - bb*Var(2) + b0*bb*Var(1))/(Var(2) - bb*Var(1) - a_a), (Var(2) - bb*Var(1))/(Var(2) - bb*Var(1) - a_a), (Var(3) - rzy*Var(1) - rzx*Var(2))/(Var(2) - bb*Var(1) - a_a), (Var(4) - rwy*Var(1) - rwx*Var(2))/(Var(2) - bb*Var(1) - a_a)))
  else
    fct := Numer(fct#(Var(1) = (Var(1) - bb*Var(2) + b0*bb*Var(1))/(Var(2) - bb*Var(1) - a_a), (Var(2) - bb*Var(1))/(Var(2) - bb*Var(1) - a_a), (Var(3) - rzy*Var(1) - rzx*Var(2))/(Var(2) - bb*Var(1) - a_a)) )
  fi;
  @([a],[aaf],[vv],[var]);
  @(hfzerg,hfzerh,prevt,prevr,prevs,prevk,prevt,fmk,fn,gr,incr,mm);
  @(degf, a_a, b0, bb, hfkt2, half, hs, r_, s_, u_, v_);
  fct.;

;; end of group to do 3 or 4 poly vars with polymodding mod p.

;;====================================================================================================
;;====================================================================================================


;; Return j = number of 1s found in binary rep of n. binary rep of n goes into array sub, "backwards".
;; identical to dioq
Func Base2(n; i,j) =
  while n > 0 do
    i :+;
    if n | 2 then
      sub[i] := 1;
      n :- ;
      j :+
    else
      sub[i] := 0
    fi;
    n := n/2
  od;
  i :+ ;
  while i <= Deg[sub] do
    sub[i] := 0;
    i :+
  od;
  j.;

;; Assume array sub[Rows[a]] exists, has previous value.  Use global incr.
;; identical to dioq
Func Nextsub(half; j) =
  j := half + 1;
  while j > half do
    incr :+ ;
    j := Base2(incr)
  od.;

;; uses globals fir and sec, indices of highest and second highest vars that occurs in the argument ar.
Func Set_var(i) =
  var[1] := Var(fir);
  var[2] := Var(sec).;

;; Fill vlist, array of inidices of vars present.  Return number of such vars.
Func Find_list(za; i,j, flevel) =
  if Flevel = 0 then
    flevel := Numvars + 1
  else
    flevel := Flevel
  fi;
  { maxn_ = maximum number of variables we can work with, set in Initialize.  probably 20. }
  { Feb 14, 2024:  bug fix, was i > flevel }
  for i = 1, maxn_ do
    if i > Numvars or i >= flevel then
      &>
    fi;
    if Deg(^za, i) > 0 then
      j :+;
      vlist[j] := i
    fi
  od;
  j.;

;; homogenize to put in y (fir).
;; identical to dioq
Func Homogy(s) =  Numer(s#(Var(sec) = Var(sec)/Var(fir))).;

;; Compute homogeneous part of largest degree - k.  Use [var]. NB: enter positive k. [No, k can be 0 also.]
;; This proc ignores the z and w [third and fourth var], as if z is in the ground field. 7651 is random.
;; Update: ignore all vars but fir and sec.  "third and fourth" is obsolete.

;; new Homzw.  Can no longer use [vlist].  Must use the rows of [vl], which are not in increasing order.  [var], fir, and sec have been set.
;; Dec 2022: Note that the random values plugged in to za are there just to compute the matrix [a_a] of degrees. They do not survive into the
;; answer s.  s is part of the original za.
;; Mar 2024:  Every invocation here has k = 0.
Func Homzw(za, k, i, s) =
  Array eh[lastvar];
  [eh] := [Rand];
  eh[fir] := Var(fir);
  eh[sec] := Var(sec);
  { Substitute out all other vars using [eh]. }
  Totdeg(za#(Var(1) = [eh]), [a_a]);
  s := WDeg(za, [var], a_a[1]-k);
  { Mar 2024: made comment @([eh], [a_a]) }
  s.;

;; make monic in y = Var(fir).  15 is just a "guaranteed" large enough value.  b0 is global.
;; note the Homzw(u) here.
;; Nov 2022: can't we speed up by not copying arg?  Mar 2024
Func Monicy(uu; i, ww, vv, deg) =
  ww := Homzw(uu);
  { ww contains no vars at levels below sec, i.e., it has only levels fir and sec. }
  b0_ := 0;
  Array ee_[lastvar];
  [ee_] := 1;
  for i = 1, 15 do
    ee_[fir] := i;
    vv := ww#(Var(1) = [ee_]);
    if vv <> 0 then
      b0_ := i;
      &>
    fi
  od;
  @[ee_];
  if b0_ = 0 then
    !!('FAIL monic y ', Var(fir));
    Return(0)
  fi;
  deg := Deg(^uu,sec);
  Sigma<i=0,deg> [ (Var(sec) + b0_*Var(fir))^i * Coef(^uu,Var(sec),i) ]/vv.;


;; make monic in x.  15 is just a "guaranteed" large enough value.  bb is global.
;; note the Homzw(vu) here.
;; Nov 2022: can't we speed up by not copying arg?
;; Dec 2022: yes speedup.  Don't copy arg. change name.  use vu created by Mkgood_arg. Create global bb and monic version of vu,
;; named fmk by the caller.  Mar 2024.
Func Monicx_vu(skip, i, ww, vv, deg) =
  ww := Homzw(vu);
  bb_ := 0;
  Array ee_[lastvar];
  [ee_] := 1;
  for i = 1, 15 do
    if i = skip then
      &]
    fi;
    ee_[sec] := i;
    vv := ww#(Var(1) = [ee_]);
    if vv <> 0 then
      bb_ := i;
      &>
    fi
  od;
  @[ee_];
  if bb_ = 0 then
    !!('FAIL monic x');
    Return(0)
  fi;
  deg := Deg(^vu,fir);
  Sigma<i=0,deg> [ (Var(fir) + bb_*Var(sec))^i * Coef(^vu,Var(fir),i) ]/vv.;


;; Do steps 2, 3, 4 on page 763 of Wan's paper.  Assume argument square free.  aa_ is global.
;; new version, no argument - it's arg  Nov 2022. added k
;; Dec 2022: remove vv as local var.  This proc creates global vu instead.
;; One of main points of this is to create fmk.
;; Dec 2022:  The input is arg, which was created from the original poly with rx,ry.
;; vu is created here and destroyed at end.
;; Mar 2023, make bb11 local.
Func Mkgood_arg(uu, i, k, vu, zero,bb11) =
  { Step 2: u = f*, normalized w.r.t. y. Actually, should first check this is necessary. }
  if Equal(arg,zero) then
    !!'0 in Mkgood_arg';
    &_P
  fi;
  uu := Monicy(arg);
  while not Denom(^uu) do
    i := Level(Denom(^uu));
    rx[i] := Rand;
    ry[i] := Rand;
    arg := arg#(Var(vlist[i]) = Var(vlist[i]) + ry[i]*Var(fir) + rx[i]*Var(sec));
    uu := Monicy(arg);
  od;
  { Step 3: Don't do Dixon twice. That's a huge waste of time. Just try a = 1, 2, etc. }
  vu := Deriv(uu, Var(sec), 1);
  aa_ := 0;
  { 15 is a heuristic. }
  for i = 1, 15 do
    if GCD(uu#(Var(sec)=i), vu#(Var(sec)=i)) then
      aa_ := i;
      &>
    fi
  od;
  if aa_ = 0 then
    !!('FAIL Mkgood_arg. Not square free?');
    Return(0)
  fi;
  Totdeg(uu, [b_]);
  aa_ := 1/aa_;
  { vu is f^* . Note the [v] here.  Don't quite understand. But [var] is awful. Apparently here we do consider z as a var, not ground ring. }
  { OK, Explanation:  [v] has x, y, and z. Suppose b[1] = 9.  vu must contain some monomials of total degree 9 in x and y only, no z, because
    earlier we did the subs z = z+y.  Yes there may also be some terms like z x y^7, but those will later be considered to have degree 8, as z
    will then be part of the ground ring. Now look at the monomials of degree 8.  There may be some like x^2 y^6. In the command below they
    are multiplied by x (x-a actually). Those terms will "move up" to degree 9.  Fine.  But any monomial of degree 8 with a z in it, like z x y^6
    will become z x^2 y^6, which is not really of degree 9 once z is considered part of the ground ring.  Same idea with monomials of degree 7
    multiplied by x^2, and so on. So in vu below the highest monomials will have no z.  Not true if [var] is used. }
  { So Yes, it's got to be [v] here. }
  { Feb 2024: the above discussion is obsolete, as we now use array [h] and Mons( , , 3). }
  { Jan 2023. use new Mons( , , 3) option.  Feb 2024: Mons(, , 3) often fails when polymodding in versions before 7.5. }
  Array h[b_[1]+1];
  Mons(^uu,[a]);
  Mons(^uu,[c],3);
  [h] := 0;
  bb11 := b_[1];
  &(a=0);
  for i = 0,Deg[c]-1 do
    h[c[i]] := *h[c[i]] + a[i]
  od;
  vu := Sigma<i=0,bb11> [ h[i]*(aa_*Var(sec) - aa_)^(bb11 - i) ];
  &(a=1);
  @([a], [h], [c]);
  @[b_];
  { Put it back!!}
  aa_ := 1/aa_;
  fmk := Monicx_vu(0).;

;; Nov 2022. Compare r_, s_, k with the two saved previous values. k is local var from G or H.
Func SeenBefore =
  if r_ = prevr and s_ = prevs and k = prevk then
    Return(1)
  fi;
  if r_ = pprevr and s_ = pprevs and k = pprevk then
    Return(2)
  fi;
  Return(0).;

;; mm = impossible value for Z/p.
;; Nov 2022. We have a previous r,s,k. Does that previous t = mm? "found" means it is the most recent previous.
Func Prev_is_mm(found) =
  if found then
    Return(Equal(prevt,mm))
  else
    Return(Equal(pprevt,mm))
  fi.;

;; Nov 2022. We just created a new t and need to store it and maybe the r,s,k. "found" means it is the most recent previous.
Func Store_rsk(found) =
  { if found there is nothing to do. We'll store the new t later. }
  if found = 2 then
    Swap(prevr, pprevr);
    Swap(prevs, pprevs);
    Swap(prevk, pprevk);
    Swap(prevt, pprevt)
  fi.;

Func Remove_rsk =
  Swap(prevr, pprevr);
  Swap(prevs, pprevs);
  Swap(prevk, pprevk);
  Swap(prevt, pprevt);
  pprevr := mm;
  pprevs := mm;
  pprevk := mm;
  pprevt := mm.;

;; assume n_, r_ and s_ given as globals.  also f, gr_, hs_.  Replace Homzw. Use global [aaf]. hfktest is reduced.

;;============= G and H
;; Functions G and H implement the sequence of polys described on page 760.  We go to a lot of effort to not recompute
;; values; hence the complexity here, prevr, prevs, etc.
Func G5(m; k, i, t, j, hfkt, hfktest, usedold, dum, found, move_t) =
  { Nov 2022.  Use arrays [g] and [h] to store values of G5(j) and H5(j).  ALso store values of the poly called t here. }
  gcount :+ ;
  move_t := 0;
  j := m + degf + 1 - r_;
  if j = 0 then
    Return(mm)
  fi;
  if g[j] = mm then
    k := r_ - m;
    { if r=prevr and s=prevs and k=prevk then }
    found := SeenBefore;
    if found > 0 then
      { Yes we have seen it before. }
      prev_cg :+;
      if Prev_is_mm(found) then
        { make new t value }
        for i = 1, k-1 do
          hval := H5(s_- k+i);
          if hval = mm then
            Return(mm)
          fi;
          gval := G5(r_-i);
          if gval = mm then
            Return(mm)
          fi;
          t := t + gval*hval
        od;
        { later set (move) t to prevt   prevt := t }
        move_t := 1;
        Store_rsk(found)
      else
         if hfktest < 2 then
           { This is vacuously true.  Strange.  Bug?  hfktest is used below. Nov 2022 }
           Store_rsk(found);
           { Maybe this can be more efficient in future. do not later move t.  }
           t := prevt
         fi;
         usedold := 1;
      fi
    else
      { not found; make new t value }
      for i = 1, k-1 do
        hval := H5(s_- k+i);
        if hval = mm then
          Return(mm)
        fi;
        gval := G5(r_-i);
        if gval = mm then
          Return(mm)
        fi;
        t := t + gval*hval
      od;
      Store_rsk(2);
      prevr := r_;
      prevs := s_;
      prevk := k;
      {move new t later to prevt}
      move_t := 1
    fi;

    { We have a new t, if we need it, and have updated the storage prev's. Nov 2022. }
    if hfzerg < 2 and hfk[k] = mm then
      {Oct 2022 use fmc not fmk }
      hfk[k] := WDeg(fmc, [var], aaf[1]-k)#(Var(fir) = 1);
    fi;
    if hfzerg >= 2 then
      hfktest := 0
    else
      if usedold then
         hfktest := ( hfk[k]#(Var(fir)=[e_]) - t#(Var(fir)=[e_]) ) | gr_;
      else
         hfktest := 1
      fi
    fi;
    if hfktest = 0 then
      hfzerg :+ ;
      g[j] := 0
    else
      hfkt := hfk[k] - t;
      if Deg(^hfkt,sec) >= Deg(gr_,sec) then
        hfkt := hfkt | gr_
      fi;
      if hfkt = 0 then
        { this happens occassionaly. Nov 2022 }
        hfzerg :+
      fi;
      { Set parameters for multiplication: }
      &(# = 20);
      dum := v4v*hfkt;
      &(# = 80);
      g[j] := dum | gr_;
    fi;
    if move_t then
      Move(t, prevt);
    fi;
    g[j]

  else
    { g[j] != mm, so g[j] is known already }
    reuseg :+;
    g[j]
  fi.;


;; degf, r_ and s_ are globals.   also f, gr, hs, mm, u and v.  Note Homz.  Use global [aaf]
;; hfktest added Mar 2022.  Apr 2023, remove fail.
Func H5(m; k, i, t, j, hfkt, hfktest, dum, found, move_t, hfkt2) =
  hcount :+ ;
  move_t := 0;
  j := m + degf + 1 - s_;
  if j = 0 then
    Return(mm)
  fi;
  { hfzerg is rarely more than 2. }
  if hfzerg >= _10 then
    h[j] := 0;
    Return(0);
  fi;

  if h[j] = mm then
    k := s_ - m;
    found := SeenBefore;
    if found > 0 then
      { Yes we have seen it before. }
      prev_ch :+;
      if Prev_is_mm(found) then
        { make new t value }
        for i = 1, k-1 do
          hval := H5(s_ - k + i);
          if hval = mm then
            Return(mm)
          fi;
          gval := G5(r_ - i);
          if gval = mm then
            Return(mm)
          fi;
          t := t + gval*hval
        od;
        { later set (move) t to prevt   prevt := t }
        move_t := 1;
        Store_rsk(found)
      else
         Store_rsk(found);
         { Maybe this can be more efficient in future. do not later move t.  }
         t := prevt
      fi
    else
      { not found; make new t value }
      for i = 1, k-1 do
        hval := H5(s_ - k + i);
        if hval = mm then
          Return(mm)
        fi;
        gval := G5(r_ - i);
        if gval = mm then
          Return(mm)
        fi;
        t := t + gval*hval
      od;
      Store_rsk(2);
      prevr := r_;
      prevs := s_;
      prevk := k;
      {move new t later to prevt}
      move_t := 1
    fi;

    { We have a new t, if we need it, and have updated the storage prev's. Nov 2022. }
    if hfk[k] = mm then
      { Oct 2022 use fmc not fmk  }
      hfk[k] := WDeg(fmc, [var], aaf[1]-k)#(Var(fir) = 1)
    fi;
    hfkt2 := hfk[k] - t;
    if Deg(^hfkt2, sec) >= Deg(hs_, sec) then
      hfkt2 := hfkt2 | hs_
    fi;
    &(#=20);
    dum := u4u*hfkt2;
    &(# = 80);
    h[j] := dum | hs_;
    if Deg(h[j],sec) > j then
      Return(mm)
    fi;
    if move_t then
      Move(t, prevt);
    fi;
    h[j]

  else
    reuseh :+ ;
    h[j]
  fi.;

;; mm = impossible value for Z/p.
;; creates (or uses) globals fn, mm, r_, s_, n_, gr_, hs_, u4y, v4v, degf.
;; same as in dioq.  Note gr_ and hs_ involve only Var(sec).
;; modified Mar 2023 to return 0 if hs_ is not a poly.
Func Setup_(d) =
  r_ := Deg(d, fir);
  s_ := degf - r_;
  gr_ := d#(Var(fir) = 1);
  hs_ := fn#(Var(fir) = 1)/gr_;
  if not Denom(hs_) then
    Return(0)
  fi;
  EGCD(gr_,hs_,u4u,v4v);
  { all of these polys only contain Var(sec) and gr*uu + hs*vv = 1. }
  [g] := mm;
  [h] := mm;
  g[degf+1] := gr_;
  h[degf+1] := hs_;
  Return(1).;

;; Uses r_, degf, n_.
;; identical to that in dioq.  Oct 2022  gcount and hcount count number of calls.
Func Fact(d; i, deg) =
  if Setup_(d) = 0 then
    if Verbose then
      !!('Setup_ returned 0, hs not poly')
    fi;
    Return(0)
  fi;
  fmc := WDeg(fmk, [var], aaf[1] - r_ - 1, +);
  gcount := 0;
  hcount := 0;
  deg := r_ - 1;
  for i = 0, deg do
    G5(deg);
    if degf = i then
      if Verbose then
        !!('failure 0 Fact ', degf, i);
      fi;
      fail := _(fail+1) ;
      Remove_rsk;
      Return(0)
    fi;
    if Deg(g[degf-i],sec) > deg then
      if Verbose then
        !!('failure 1 Fact ', degf, Deg(g[degf-i], sec), deg, i, sec);
      fi;
      g[degf-i] := mm;
      fail := _(fail+1) ;
      Remove_rsk;
      Return(0)
    fi;
    deg :-
  od;
  G5(-1);
  if degf = r_ then
    fail := _(fail+1) ;
    Return(0)
  fi;
  if g[degf - r_] <> 0 then
    fail := _(fail+1);
    Return(0)
  fi;
  deg := 0;
  for i = degf - r_ + 1, degf+1 do
    deg := deg + Homogy(g[i])
  od;
  deg.;


;;============= Initializing:
Func Initial(again) =
  if again = 0 then
    &(H = 1);
    k_used := 0;
    orig_k := 0;
    failurr := 0;
    do_all := False;
    fail := 0;
  fi;
  { maximum number of variables we can work with. maxn_ <> 0 means it was set by FCTZ.  April 2024 }
  if maxn_ = 0 then
    maxn_ := 20
  fi;
  { fixed syntax error below, Mar 17 2024: }
  if Numvars > maxn_ then
    !!('WARNING.  ERROR.', Numvars, ' is bigger than ', maxn_);
    &_P
  fi;
  { impossible value : }
  mm := _(Modulus+1);
  prevr := 0;
  prevs := 0;
  prevk := 0;
  prevt := mm;
  ;; Nov 2022
  pprevr := 0;
  pprevs := 0;
  pprevk := 0;
  pprevt := mm;
  gcount := 0;
  hcount := 0;
  prev_cg := 0;
  prev_ch := 0;
  recheck := 0;
  archeck := 0;
  { Apr 2023 }
  fmc := 0;
  gval := 0;
  hval := 0;
  r_ := 0;
  s_ := 0;
  gr_ := 0;
  hs_ := 0;
  u4u := 0;
  v4v := 0.;


;;======= May 2023. new driver to call FCT repeatedly with different permutations of the vars.
;; returns 1 = success, answer is fct.  0 = failure, fct is also 0.
;; February 13, 2024:  BUG fix, add i and j to list of parameters.
;; Returning 1 means it worked.  The answer is in fct.
Func SPF(argg; answer,i,j,firstvar,lastvar) =
  if Polymod and Flevel = 0 then
    !!'Error, if polymodding, must have a field.';
    Return(-1)
  fi;
  n_ := 0;
  if Numb(^argg) then
    Return(0)
  fi;
  if Flevel > 0 then
    if Level(^argg) >= Flevel then
      Return(0)
    fi
  fi;
  answer := FCT(argg);

  { irreducible: }
  if answer then
    @(n_, [ovlist]);
    fct := 1;
    Return(1)
  fi;
  { not 0 means answer has been found, is in fct: }
  if answer <> 0 then
    @(n_, orig_last, orig_k, k_used, do_all, [ovlist]);
    Return(1)
  fi;

  { Use SDR to permute vars. orig_last is just n_.  [ans] contains 1,2,3...n.  [ansinv] is the inverse of [ans]. }
  Array ans[1, orig_last], ansinv[1, orig_last], subs[lastvar - firstvar + 1], subs2[lastvar - firstvar + 1];
  for i = 0, lastvar - firstvar do
    subs[i+1] := Var(firstvar + i)
  od;
  [ans] := 0;
  { 6 better be enough! }
  for i = 1, 6 do
    SDR(orig_last);
    { ans now contains a derangement of 1,2,3,...,n_ }
    { Set up the substitutions array }
    [subs2] := [subs];
    for j = 1, orig_last do
      subs2[ovlist[j] - firstvar + 1] := Var(ovlist[ans[j]])
    od;
    { Effect the variable substitution. }
    arp2 := argg#(Var(firstvar) = [subs2]);
    k_used := 0;
    orig_k := 0;
    failurr := 0;
    fail := 0;
    Cancel;
    FCT(arp2, 1);
    if fct = 0 or fct = 1 then
      &]
    fi;
    { Invert [ans].  }
    for j = 1, orig_last do
      ansinv[ans[j]] := j
    od;
    { Use [ansinv] to put [subs2] in correct form to put fct back in the original vars. }
    for j = 1, orig_last do
      subs2[ovlist[j] - firstvar + 1] := Var(ovlist[ansinv[j]])
    od;
    fct := fct#(Var(firstvar) = [subs2]);
    if fct <> 1 and Divides(fct, argg) then
      @([ans], [ansinv], n_, [subs], [subs2], [ovlist], orig_k, k_used, orig_last, orig_k, k_used);
      Return(1)
    fi;
  od;
  { Try FCT once more with do_all }
  do_all := True;
  @([ans], [ansinv], [subs], [subs2]);
  { Feb 2024, third arg }
  FCT(argg, 1, 1);
  if Numb(fct) then
    !!'TOTAL FAILURE';
    fct := 0;
    @(n_, orig_k, k_used, orig_last, orig_k, [ovlist], k_used, do_all, [ovlist]);
    Return(0)
  fi;
  @(n_, orig_k, k_used, orig_last, orig_k, [ovlist], k_used, do_all);
  1.;


;; May 2023: this is the driver for one poly with one arrangement of vars.  It can be used as the top driver.
;; input poly is myar.  Output is:  0 = failure; 1 = irreducible; 2 = success, nontrivial poly = the answer, fct,
;; meaning the degree is positive in at least one of the vars occurring in myar.  The global fct is set to the answer.
;; Note that FCT(arg, 1) can only return 0 or 2.
;; FCT calls a number of procs, last is Fin_.  If things fail, Fin_ will call FCT recursively, in which case again > 0.
;; again is 0 only at the very start of the program.  All subsequent calls to FCT must have again > 0.
Func FCT(myar, again; i, arsimp, answ, argh, homo, my_n) =
  if not Modmode then
    !!'Error, must be in modular mode.';
    Return(-1)
  fi;
  if again = 0 then
    { Do all this only on first call of a problem }
    i := Irred(^myar);
    if i then
      fct := 1;
      Return(1)
    else
      { This will take care of the case where myar has contents }
      { Bug fix, Jan 2024: need to check not Numb }
      if not Numb(i) then
        if Divides(i, myar) then
          fct := i;
          Return(2)
        fi
      fi
    fi;
    { else continue the algorithm.  There has to be a nontrivial factor. }
    { Apr 2023:  So at the end, irreducible is not an acceptable answer! }
    Sqfree(^myar, [q]);
    if Rows[q] then
      if q[2] > 1 then
        fct := q[1];
        @[q];
        Return(2)
      else
        { typical case. continue main algorithm. }
        @[q]
      fi
    else
      fct := q[1];
      @[q];
      Return(2)
    fi
  fi;

  Initial(again);
  Array var[2], vlist[1,maxn_];
  { Mar 2024: }
  if Polymod then
    Array vv[Flevel-1]
  else
    Array vv[Numvars]
  fi;
  { fill [vv].  first Numvars spots are assigned.  }
  for i = 1, Deg[vv] do
    vv[i] := Var(i)
  od;
  { n is number of vars present, important global }
  n_ := Find_list(myar);
  { Feb 2024: keep the original }
  [ovlist] := [vlist[1, 1~n_]];
  firstvar := vlist[1];
  lastvar := vlist[n_];
  { check for homogneous.  Nov 2023 }
  if again = 0 then
    { Feb 2024, changed from orig_last := lastvar; that was a bug. }
    orig_last := n_;
    if n_ > 2 then
      Totdeg(^myar, [a]);
      if a[1] = a[2] then
        { it's homogeneous }
        my_n := n_;
        argh := myar#(Var(lastvar) = 1);
        @([a], [var], [vv]);
        homo := FCT(argh);
        for i = 1, my_n-1 do
          fct := fct#(Var(vlist[i]) = Var(vlist[i])/Var(vlist[my_n]))
        od;
        fct := Numer(fct);
        @([vlist]);
        Return(homo)
      fi;
      @[a]
    fi
  fi;
  
  { Use the fast simple routines for 3 or 4 vars   Mar 2024 }
  if n_ < 5 and Polymod then
    { Convert to use highest three vars }
    Array conv[Flevel-1];
    [conv] := 1;
    for i = 1, n_ do
      conv[ovlist[i]] := Var(i)
    od;
    myar := myar#(Var(1) = [conv]);
    { Sep 2024:  fixed bug, next line was D5 }
    D_5(myar);
    [conv] := 1;
    for i = 1, n_ do
      conv[i] := Var(ovlist[i])
    od;
    fct := fct#(Var(1) = [conv]);
    @([conv], [var], [vv],  [vlist]);
    Return(2)
  fi;

  answ := Pre_(0);
  if answ then
    @([var], [vv], [vlist]);
    Return(1);
  fi;
  { Apr 2023, failure, quit. }
  if answ = 2 then
    @([var], [vv], [vlist]);
    Return(0);
  fi;
  { Feb 2024:  'First' returning 1 is failure.  }
  if First(myar) then
    @([var], [vv], [vlist]);
    Return(0)
  fi;
  Next_;
  { Next_ has created faf, the factor of the poly formed from myar by the normalizing substitutions. }
  { faf = 0 means failure }
  { Fin_ returns either 0 (failure) or 2 (success). answer is in fct a nontrivial poly. }
  Fin_.;
  
;; Create the next row in array [vl].  Assume n_ > 3.
Func Set_next(i,j) =
  j := vl[i+1,i];
  vl[i+1,i] := vl[i+1,3];
  vl[i+1,3] := j;
  if i >= 3 then
    j := vl[i+1,1];
    vl[i+1,1] := vl[i+1,4];
    vl[i+1,4] := j;
  fi;
  if i = 4 then
    j := vl[i+1,2];
    vl[i+1,2] := vl[i+1,3];
    vl[i+1,3] := j;
  fi.;
    
;; special case n_ = 3.
Func Next_perm(i) =
  if i then
    vl[i+1,1] := vl[1,1];
    vl[i+1,2] := vl[1,3];
    vl[i+1,3] := vl[1,2];
    Return(0)
  fi;
  if i = 2 then
    vl[i+1,1] := vl[1,2];
    vl[i+1,2] := vl[1,3];
    vl[i+1,3] := vl[1,1];
    Return(0)
  fi;
  if i = 3 then
    vl[i+1,1] := vl[1,2];
    vl[i+1,2] := vl[1,1];
    vl[i+1,3] := vl[1,3];
    Return(0)
  fi;
  if i = 4 then
    vl[i+1,1] := vl[1,3];
    vl[i+1,2] := vl[1,1];
    vl[i+1,3] := vl[1,2];
    Return(0)
  fi.;

;; Apr 2023.  In case we have to run the algorithm again:
;; preconditioning. Very important stuff here! fir and sec mark the two vars we keep; others are substituted with random values.
;; Dec 2022: global ar is the poly we are trying to factor.  No randomization here.
;; Return early with 1 in simple cases, else return 0 at end.
;; Produce co, fir, sec, [vlist].  More explanation below.
;; Feb 2024: added colsvl.
Func Pre_(i,j,small, k, low, m, deg, next, pn, prev, pp, conum, colsvl) =
  if n_ = 0 then
    fct := 1;
    Return(1)
  fi;
  if n_ = 1 or n_ = 2 then
    Factor(myar, [a]);
    fct := a[Rows[a]];
    Return(1)
  fi;
  { Check three (or more) variations of fir and sec.  Try to find "best." Must make sure each of the first two
    vars is in spot three once. }

  { Feb 2024 }
  conum := 5;
  colsvl := 4;
  if n_ > 4 then
    colsvl := n_
  fi;

  { We need to find fir and sec, the levels of the two "highest" vars that will dominate the computation.  We want a choice that has
    a good leading coefficient; i.e. relatively small, but not too small.  This leading coeff, co, will duplicate and reproduce itself later. }
  { [co] = conum (three or five) versions of polynomial "co", coefficient.  [vl] is three or more versions of vlist. [zer] is to note
    degree of third var in [co]; low is lowest such degree and m is where it is.  [size] is the number of terms of each version of co.
    The point is to find a var that should not be in the top two. }
  { Mar 2023: change array size, add second column }
  { Feb 2024:  change cols of vl to at least 4. }
  Array co[conum], vl[conum, colsvl], zer[conum], size[conum,2];
  { the versions of vlist are all the same in the beginning. }
  
  if n_ > 3 then
    [vl[1,]] := [vlist[1,1~n_]]
  else
    [vl[1,1~n_]] := [vlist[1,1~n_]];
    vl[1,4] := vlist[2]
  fi;
  { Feb 2024: randomly permute [vl]  Mar 2024: disable }
  if n_ > 3 then
  {  j := Rand|n_ + 1;
    Swap(vl[1,1], vl[1,j]);
    j := Rand|n_ + 1;
    Swap(vl[1,2], vl[1,j]) }
    i := 1
  else
    {Swap(vl[1,1], vl[1,2]);
    if Rand|2 then
      Swap(vl[1,2], vl[1,3]);
    fi}
    j := j
  fi;
  for i = 2, conum do
    [vl[i,]] := [vl[1,]]
  od;
  
  { Find which of the conum choices gives smallest co, but don't choose it if that co has only one term. That's a mess later. }
  { k and small mark the smallest.  k is the row in [vl] that we choose for smallest. fir and sec are the first two entries there.}
  { Use next for the next smallest, prev for previous smallest. pn marks next, pp marks prev (if found). }
  { Mar 2023: this is all problematic. What if modulus is small? OTOH, Terms does not mod out its value. Note _4999 below. }
  { Apr 2023: this loop is a bit of kludge.  By rights conum should be larger and most permutations tried.  }
  for i = 1, conum do
    fir := vl[i,1];
    sec := vl[i,2];
    Set_var;
    { co depends on Var(fir) and Var(sec). }
    co[i] := Homzw(myar);
    size[i] := Terms(^co[i]);
    size[i,2] := i;
    if small = 0 or size[i] < small then
      if i >= 3 then
        { last one is smallest. save previous smallest }
        prev := small;
        pp := k
      fi;
      k := i;
      small := size[i];
    fi;
    zer[i] := Deg(^co[i], vl[i,3]);
    if i = 1 then
      low := zer[i];
      m := i;
    else
      if zer[i] < low then
        low := zer[i];
        m := i
      fi
    fi;
    { find second smallest }
    if i = 2 then
      if k then
        next := size[i];
        pn := i
      else
        next := size[1];
        pn := 1
      fi
    fi;
    if i >= 3 then
      if pp > 0 then
        next := prev;
        pn := pp;
        pp := 0;
      else
        if size[i] < next then
          next := size[i];
          pn := i
        fi
      fi
    fi;
    { Mar 2023: was next := 49999. But we're in mod mode! }
    if next then
      next := _4999;
    fi;
    if i = conum then
      &>
    fi;
    { get ready for next pass. 3 is a special case. Feb 2024 }
    if n_ > 3 then
      Set_next(i)
    else
      Next_perm(i)
    fi
  od;

  { See diary. this was tweaked, October 2022.  Bug fix also.  }
  { Mar 2023:  The problem with all of these statements is that small = size[k] might STILL be too small. }
  { Mar 2023: Note size could end up being 2.  Is that OK? }
  { Mar 2023, bubble sort [size]. }
  for i = 1, conum-1 do
    for j = 1, conum-i do
      if size[j] > size[j+1] then
        deg := size[j];
        size[j] := size[j+1];
        size[j+1] := deg;
        deg := size[j,2];
        size[j,2] := size[j+1,2];
        size[j+1,2] := deg
      fi
    od
  od;
  { Take the one in the middle. }
  small := size[3];
  k := size[3,2];
  fir := vl[k,1];
  sec := vl[k,2];
  { But let's check one more thing. }
  deg := Deg(^myar, vl[k,3]);
  

  Set_var;
  { Change vlist! }
  @[vlist];
  [vlist] := [vl[k,]];
  { Apr 2023 }
  if k_used = 0 then
    orig_k := k
  else
    k := k_used + 1;
    if k = orig_k then
      { This shouldn't happen due to stmts elsewhere }
      !!(' All entries in array [co] have been tried. ', k);
      fct := 1;
      Return(2);
    fi;
    if k > Deg[co] then
      k := 1
    fi;
    fir := vl[k,1];
    sec := vl[k,2];
    deg := Deg(^myar, vl[k,3]);
    
    Set_var;
    @[vlist];
    [vlist] := [vl[k,]]
  fi;
  k_used := k;
  co := co[k];
  
  if Verbose then
    !!('co is ', Var(fir), Var(sec), co, k, small);
  fi;
  fct := 1;
  Array rx[n_], ry[n_];
  Return(0).;



;;;; =========== First
;;;==================================================================;
;;  At start arg is the original poly we are factoring.
;; We use co, fir, sec, [vlist], [rx], [ry], num_skip.
;; Main task:  produce fmk, monic versions of original poly w.r.t. fir.
;; here num_skip is global.  Cont_ uses it.
Func First(arg; i, j, deg, k) =
  [rx] := [RRand];
  [ry] := [RRand];
  { fail is used by Fact. }
  { July 2022: This loop is OK. We know fir = vlist[1] and sec = vlist[2]. }
  num_skip := 0;
  { Convert arg to a poly whose leading coef w.r.t fir contains only sec.  This is done by substitution with [rx], [ry],
    producing arg. }
  { This loop is a huge bottleneck!. vars present in co are substituted out with random matrices rx, ry. }
  { first_time := 1; }
  { May 2023, check for small poly. }
  Totdeg(^arg, [as]);
  { Note shortcut version of function Terms. }
  if Terms(^arg, 100) < _15 and as[1] < _9 then
    do_all := True;
  fi;
  @[as];
  { May 2023. added do_all. do_all is good for 'small' polys. Sometimes necessary!! }
  for i = 3, n_ do
    if do_all or Deg(^co, vlist[i]) > 0 then
      deg := 0;
      for j = 1, lastvar do
        if j = vlist[i] then
          &]
        fi;
        if Deg(^arg,j) > deg then
          k := j;
          deg := Deg(^arg,k)
        fi
      od;
      { k := 4; }
      k := vlist[i];
      deg := Deg(^arg,k);
      if k = fir or k = sec then
        !!('error k ', i);
        &_P
      fi;
      arg := Sigma<j=0,deg> [ (Var(k) + ry[i]*Var(fir) + rx[i]*Var(sec))^j * Coef(^arg,Var(k),j) ]
    else
      num_skip :+ ;
      rx[i] := 0;
      ry[i] := 0
    fi
  od;
  do_all := False;

  { Dec 2022: check that no absent var has been introduced into highest coefficient.  Happens rarely. }
  if num_skip > 0 then
    archeck := Homzw(arg);
    recheck := 0;
    for i = 3, n_ do
      if Deg(^archeck, vlist[i]) > 0 and rx[i] = 0 then
        { Dec 2022 Apparently rare. But happened in FCT(fw) in fact9. }
        { Mar 2023. change these fixed choices.  Below too. }
        rx[i] := Rand;
        ry[i] := Rand;;
        recheck := 1;
        {arg := arg#(Var(vlist[i]) = Var(vlist[i]) + ry[i]*Var(fir) + rx[i]*Var(sec))  new Oct 2022: }
        k := vlist[i];
        deg := Deg(^arg,k);
        arg  := Sigma<j=0,deg> [ (Var(k) + ry[i]*Var(fir) + rx[i]*Var(sec))^j * Coef(^arg,Var(k),j) ]
      fi
    od;

    { Nov 2022.  Do we have to do this loop yet again, perhaps? }
    { Dec 2022: This has never happened, i.e., "must now sub for this 2" has never been reported, }
    if recheck then
      for i = 3, n_ do
        if Deg(^archeck, vlist[i]) > 0 and rx[i] = 0 then
          rx[i] := Rand;
          ry[i] := Rand;
          recheck := 1;
          {arg := arg#(Var(vlist[i]) = Var(vlist[i]) + ry[i]*Var(fir) + rx[i]*Var(sec))  new Oct 2022: }
          k := vlist[i];
          deg := Deg(^arg,k);
          arg := Sigma<j=0,deg> [ (Var(k) + ry[i]*Var(fir) + rx[i]*Var(sec))^j * Coef(^arg,Var(k),j) ]
        fi
      od
    fi
  fi;

  { Create fmk from arg }
  Mkgood_arg;
  Array e_[lastvar];
  { Feb 2024:  return success or failure. }
  Return(Cont_(0)).;


;;========= Insert
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;; changes global fkn, and array fkd.  [fkd] is array of distinct degrees of factors of fk.  last actual entry is at fkn. Nov 2022
Func Insert(d; i,j) =
  for i = 1, fkn do
    if d = fkd[i] then
      Return(0)
    fi;
    if d < fkd[i] then
      fkn :+ ;
      for j = fkn, i+1, -1 do
        fkd[j] := fkd[j-1]
      od;
      fkd[i] := d;
      Return(0)
    fi
  od;
  { d must be larger than all previous }
  fkn :+ ;
  fkd[fkn] := d.;

;; Nov 2022. Is d in the sorted array fkd?
Func Not_there(d, i) =
  for i = 1, fkn do
    if d = fkd[i] then
      Return(0)
    fi;
    if d < fkd[i] then
      Return(1)
    fi
  od;
  Return(1).;

;; We plug in one unsubbed var.  We hope this is enough!  changes Oct 2022.  used by Cont_.
;; Dec 2022:  apparently never(?) happens, never called.
;; Apr 2023:  happened
;; Jun 2024:  added "if Verbose" to some messages here.
Func Recomp_fn(i, k, j, deg) =
  if recheck then
    if Verbose then
      !!('recheck ', Var(sec))
    fi;
    archeck := Homzw(arg);
  fi;
  { else can use previous archeck }
  for i = 3, n_ do
    if rx[i] = 0 then
      if Verbose then
        !!('****** Recomp. now sub for this ', vlist[i], Var(vlist[i]))
      fi;
      rx[i] := Rand;
      ry[i] := Rand;
      k := vlist[i];
      deg := Deg(^arg,k);
      arg := Sigma<j=0,deg> [ (Var(k) + ry[i]*Var(fir) + rx[i]*Var(sec))^j * Coef(^arg,Var(k),j) ]
    fi;
    &>
  od;
  archeck := Homzw(arg);
  recheck := 0;
  for i = 3, n_ do
    if Deg(^archeck, vlist[i]) > 0 and rx[i] = 0 then
      if Verbose then
        !!('****** Recomp, must now sub for this ', vlist[i], Var(vlist[i]))
      fi;
      rx[i] := Rand;
      ry[i] := Rand;
      recheck := 1;
      k := vlist[i];
      deg := Deg(^arg,k);
      arg := Sigma<j=0,deg> [ (Var(k) + ry[i]*Var(fir) + rx[i]*Var(sec))^j * Coef(^arg,Var(k),j) ]
    fi
  od;

  Mkgood_arg;

  fk := fmk#(Var(1) = [e_]);

  @[aaf];
  Totdeg(fk, [aaf]);
  { Use fk!  It's smaller! Oct 2022    fn := WDeg(fmk, [var], aaf[1]); }
  fn := WDeg(fk, [var], aaf[1]);
  

  degf := Deg(^fn, fir);
  if Verbose then
    !!'Recomp, fn '
  fi.;


;;====== Ca; now Cont_
;;=======================================================;
; modified Feb 2024 to return 1 and 0.  1 is a failure code. i is not used for much.
Func Cont_(i) =
  { move def of [e_] to First.  Oct 2022 }
  [e_] := [Rand];
  e_[fir] := Var(fir);
  e_[sec] := Var(sec);
  { fk is two-var version.  This suffices for a lot of the work here in Cont_. }
  fk := fmk#(Var(1) = [e_]);
  
  ;; fmkk := fmk;

  { Dec 2022: remove, never used: fkc := fk; }
  Totdeg(fk, [aaf]);
  { Use fk!  It's smaller! Oct 2022    fn := WDeg(fmk, [var], aaf[1]); }
  { fk is a two-var poly.  fn is its highest degree homogeneous piece. }
  fn := WDeg(fk, [var], aaf[1]);
  degf := Deg(^fn, fir);
  if degf <> Deg(^fn, sec) then
    { This has never been observed.  Apr 2023: yes it has. Feb 2024: }
    if Verbose then
       !!'*** Warning. fn has contents.';
       { Feb 2024. desperate:  Mar 2024: Recomp_fn used to be here, but that makes no sense. moved it }
       i:=1
    fi;
    Recomp_fn;
    if num_skip > 0 then
      if Verbose then
        !!'    Must recompute fn.'
      fi;
      Recomp_fn
    else
      Return(1)
    fi
  fi;
  { Array [a] is created: }
  Factor(fn, [a]);
  dd := Rows[a];
  { Create fkd = array of factor degrees. We assume the factors of fk have the same degrees as the factors of fmk. }
  { The array is of unique degrees and is sorted. }
  Array fkd[Deg(^fk,fir)];
  { fkn = number of distinct factors of fk. Nov 2022.  [fkd] is array of distinct degrees in factors of fk.  }
  { fk is the two-var version of fmk. }
  fkn := 0;
  
  { Mar 2024 }
origfk := fk;

  while True do
    Factor(fk,[a]);
    Insert(Deg(a[1],fir));
    SDivide(a[1],fk);
    @[a];
    if Deg(fk,fir) = 0 then
      &>
    fi
  od;
  Return(0).;
  

;;========== Next_
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;; fmk is needed now only to test the answer, faf.
;; uses fkn, dd, array [fkd].
Func Next_(i,j,ij,sum,tot,half,limit,prod) =
  Array g[degf+1], h[degf+1], hfk[degf+1];
  reuseg := 0;
  reuseh := 0;
  [hfk] := mm;
  hfzerg := 0;
  prevt := mm;
  { Put larger polys first.  NO, BAD IDEA
  for i = 1, dd \ 2 do
    j := a[i];
    a[i] := a[dd+1-i];
    a[dd+1-i] := j;
    j := a[i,2];
    a[i,2] := a[dd+1-i,2];
    a[dd+1-i,2] := j
  od;}
  { count number of polys with degree 1 and average degree. }
  { *********** remove? ***********}
  for i = 1, dd do
    sum := sum + Deg(a[i], fir);
    if Deg(a[i], fir) then
      tot :+
    fi
  od;
  { sum := average exponent }
  sum := _(sum/dd + 1);
  { Move up some of medium-sized exponent polys, move down the polys with exponent 1. }
  { Apr 2023: fixed bugs in this loop: }
  if dd > 5 and dd > tot then
    for i = 1, tot do
      ij := a[1];
      j := 2;
      while Deg(a[j], fir) < sum do
        a[j-1] := a[j];
        j :+ ;
        if j > dd then
          &>
        fi
      od;
      a[j-1] := ij
    od
  fi;
  sum := 0;
  tot := 0;
  Array sub[Rows[a]];
  half := _(Rows[a]\2 + 1);
  limit := Sigma<i = 1, half> [ Bin(Rows[a],i) ];
  incr := 0;
  for ij = 1, limit do
    Nextsub(half);
    sum := 0;
    for j=1,Deg[sub] do
      if sub[j] then
        sum := sum + Deg(a[j],fir)
      fi
    od;

    if Not_there(sum) then
      tot :+;
      &]
    fi;
 
    { Dec 2022: change dd to prod here: }
    prod := 1;
    for j = 1, Deg[sub] do
      if sub[j] then
        prod := prod*a[j]^a[j,2]
      fi
    od;
    faf := Fact(prod);
    if faf <> 0 then
      { success. the 0 here means nothing: }
      Return(0)
    fi
  od;
  faf := 0.;

;;========  Cancel
Func Cancel =
  { Change name to e2_  Jun 2024. remove maxn_ from list below. }
  @([e2_],[g], [h], [hfk]);
  @([var],[vv],[e_],[vlist]);
  @([rx],[ry],[fkd],[sub],[aaf],[a]) ;
  @([zer],[co],[vl],[size], fmk);
  @(pprevk,pprevr,pprevs,pprevt,prev_cg,prev_ch,prevk,prevr,prevs,prevt);
  @(recheck,reuseg,reuseh,u4u,v4v,archeck,bb_,b0_,aa_,degf,fmc);
  @(hfzerg,hcount,gcount,num_skip,gval,hval,fkn,fir,sec,incr);
  @(r_,s_,mm,faf,co,dd,gr_,hs_,fk,fn,fail).;
  
Func Cancel2 =
  @(pprevk,pprevr,pprevs,pprevt,prev_cg,prev_ch,prevk,prevr,prevs,prevt);
  @(recheck,reuseg,reuseh,u4u,v4v,archeck,bb_,b0_,aa_,degf,fmc,maxn_);
  @(hfzerg,hcount,gcount,num_skip,gval,hval,fkn,fir,sec,incr);
  @(r_,s_,mm,faf,co,dd,gr_,hs_,fk,fn,fail, failurr).;


;;=============================================================
;; Generate all the derangements of integers 1-n. Apr 2023
;; There are a couple methods here.  The easiest is just to use SDR.
;; The others are great if you want to list all derangements.
Func Isin(i, msk, s; j) =
  { Is i inside the array msk up through s? s <= Deg[msk], s could be 0. }
  for j = 1, s do
    if i = msk[j] then
      Return(1)
    fi
  od;
  0.;

Func Derang(n, k, sk, deg_sk; i,j) =
  if k > n then
    cnt :+
  fi;
  Array msk[1, deg_sk+1];
  for i = 1, deg_sk do
    msk[i] := sk[i]
  od;
  for i = 1, n do
    if i = k then
      &]
    fi;
    if Isin(i, [sk], deg_sk) then
      &]
    fi;
    msk[deg_sk+1] := i;
    Derang(n, k+1, [msk], deg_sk+1)
  od;
  @[msk].;

;; Generate all the derangements of integers 1-n.
Func Derang2(n, k, sk, deg_sk; i,j) =
  if k > n then
    cnt :+ ;
    if cnt < 101 then
      [ans[cnt,~]] := [sk]
    fi
  fi;
  if cnt > 20*mult then
    Return(0)
  fi;
  Array msk[1, deg_sk+1];
  for i = 1, deg_sk do
    msk[i] := sk[i]
  od;
  for i = 1, n do
    if i = k or i|3=0 then
      &]
    fi;
    if Isin(i, [sk], deg_sk) then
      &]
    fi;
    msk[deg_sk+1] := i;
    Derang2(n, k+1, [msk], deg_sk+1)
  od;
  for i = 1, n do
    if i = k or i|3=1 then
      &]
    fi;
    if Isin(i, [sk], deg_sk) then
      &]
    fi;
    msk[deg_sk+1] := i;
    Derang2(n, k+1, [msk], deg_sk+1)
  od;
  for i = 1, n do
    if i = k or i|3=2 then
      &]
    fi;
    if Isin(i, [sk], deg_sk) then
      &]
    fi;
    msk[deg_sk+1] := i;
    Derang2(n, k+1, [msk], deg_sk+1)
  od;
  @[msk].;

Func Dang(n; i,j) =
  Array de[1,1];
  for i = 2, n do
    de[1] := i;
    Derang(n, 2, [de], 1);
  od;
  @[de];
  cnt.;

Func Dang2(n, first; i,j,cnt) =
  Array ans[100,n], de[1,1];
  for i = 2, n do
    if i|3=0 then
      &]
    fi;
    de[1] := i;
    Derang2(n, 2, [de], 1);
  od;
  for i = 2, n do
    if i|3=1 then
      &]
    fi;
    de[1] := i;
    Derang2(n, 2, [de], 1);
  od;
  for i = 2, n do
    if i|3=2 then
      &]
    fi;
    de[1] := i;
    Derang2(n, 2, [de], 1);
  od;
  @[de];
  cnt.;

Func Dang2(n, first) =
  de[1] := first;
  Derang2(n, 2, [de], 1).;

Func DR(n, cnt, mult) =
  Array ans[100,n], de[1,1];
  mult := 0;
  Dang2(n, 2);
  mult := 1;
  Dang2(n, 5);
  mult := 2;
  Dang2(n, 7);
  mult := 3;
  Dang2(n, 4);
  mult := 4;
  Dang2(n, 3);
  mult := 5;
  Dang2(n, 6);
  @[de].;

;; find one random derangement.  n >= 3.  Assume Array ans[1,n] has been created.
;; Feb 2024: We find here derangements of the set 1,2,3...,n, which are stored in array [ans].
;; (So the output of this function is the array [ans] giving a derangement.)
;; However, the program needs derangements of the set of levels of poly vars that occur in the original
;; poly we are trying to factor. [Bug noticed Feb 2024.] That is not the same set.  Those integers are stored in [ovlist].
;; The procedures that call this have to do the conversion.
Func SDR(n,i,j,k) =
  for j = 1, n-1 do
    while True do
      k := Rand|n + 1;
      if k <> j and not Isin(k, [ans], j-1) then
        &>
      fi
    od;
    ans[j] := k
  od;
  { If n has been taken we are OK. }
  if Isin(n, [ans], n-1) then
    for i = 1, n-1 do
      if not Isin(i, [ans], n-1) then
        ans[n] := i
      fi
    od
  else
    { There's an easy fix. }
    ans[n] := ans[n-1];
    ans[n-1] := n
  fi;
  Return(1).;

;; END OF DERANGEMENT STUFF ;;;;;;;;;;;;;;;;;;;;;
;========================================================================;

;; Fin_ returns either 0 (failure) or 2 (sucess), meaning a nontrivial poly, the answer, fct.
;; Don't return fct, that's an unnecessary duplication.
Func Fin_(i, den, arp2, tt) =
  { Change name to e2_  Jun 2024 }
  Array e2_[lastvar];
  { faf is the factor of the poly formed by the substitutions. }
  if faf = 0 then
    fct := 1;
    { Apr 2023 }
    failurr :+;
    { This has been observed: }
    if failurr >= Deg[co] then
      { Give up. }
      fct := 0;
      Return(0);
    fi;
    Cancel;
    { Recursive call. Note the second argument 1 }
    Return(FCT(myar, 1))
  fi;
  { Nov 2022. potential bug.  6 lines down we assume n_ = lastvar }
  { Apr 2023:  n_ = lastvar is NONSENSE! But n_ below is correct. }
  [e2_] := 0;
  den := Var(sec) - b0_*Var(fir) - aa_;
  e2_[fir] := Var(fir) - bb_*Var(sec) + b0_*bb_*Var(fir);
  e2_[sec] := Var(sec) - b0_*Var(fir);
  for i = 3, n_ do
    e2_[vlist[i]] := Var(vlist[i]) - ry[i]*Var(fir) - rx[i]*Var(sec)
  od;
  [e2_] := [e2_]/den;
  { Nov 2022, fairly time consuming: }
  { Nov 2022.  bug fix attempt. was using [e2_] }
  fct := Numer(faf#(Var(1) = [e2_]));
  Cancel;
  if n_ > 4 then
    tt := PDivides(fct, myar)
  else
    tt := Divides(fct, myar)
  fi;
  if tt then
    @failurr;
    Return(2)
  else
   failurr :+;
   FCT(myar,1);
  fi.;
  
;==================================================================================================
;; Functions for factoring over Z.
  
; find next prime in the polymodding case.  p must be in the aux_prime list (if it exists of course).
Function Nexpmod(p, repeat; i,j) = `
   Integer;
   i := 1;
   while aux_prime[i] <> p do
     i :+
   od;
   i := i + 1 + Rand | 3 ;
   if repeat then
     i := i + 1 + Rand | 3
   fi;
   if i > Deg[aux_prime] then
     !!'ERROR. reached end of aux_prime array.'
   fi;
   Return(aux_prime[i]).;

; find next prime.  p must be odd.  If repeat, go further.
Function Nexp(p, repeat; i) = `
    Integer;`
    if Flevel > 0 then
      Return(Nexpmod(p, repeat))
    fi;
    p := p + 2;`
    while not Isprime(p) do`
      p := p + 2`
    od;
    if repeat then
      for i = 1, 1 + Rand | 10 do
        p := p + 2;
        while not Isprime(p) do`
          p := p + 2`
        od
      od
    fi;
    Return(p).;
    
;; N_ is used to generate all subsets of an array of factors, say [f]. [t_] is an array of indexes.  Initially [t_] is all 0s.  Each call to N_ changes
;; 0s and 1s.  recursive.  The subset of factors is indicated by the placement of 1s.  lim_ = 1 + size of factors. n_ = number of 1s in [t_];
;; it's a value-result parameter, so n is passed back.
;; Invoke with N_([f], limit, 1).  Yes, x_ is passed 1 always.  Call N 2^size - 2 times to generate all interesting subsets.
Func N_(t_, lim_, x_, ^n_) =  `
  if x_ = lim_ then `
     Return(0) `
  fi; `
  if t_[x_] = 0 then `
     n_ :+ ; `
     t_[x_] := 1 `
  else `
     t_[x_] := 0; `
     n_ :- ; `
     N_([t_], lim_, x_ + 1, n_) `
  fi.;
  
;; Search the array [facts] for a combination that works (i.e. lifts to Z).  modmularmode. fct is output.
;; fct is either a number or a reasonable candidate for a factor.
Func Mult_facts(facts, j, log2ar, lcof, ^fct; i, k, best, sum, fir, sec) =
  best := j;
  { 'best' is not pnemonic. }
  Array indx[1, best];
  fct := 1;
  { initialize array indx.  fir = first spot in indx with a 1, sec = second spot. k = number of 1s in indx. }
  [indx] := 0;
  
  { There are 2^best subsets.  We don't want the empty set or the full set. }
  { There are Bin(best,2) subsets of two. Those are generated by G. When i becomes Bin(best,2)+1, reinitialize. }
  { The first subset has been set above with fir = 1, sec = 2. }
  for i = 1, 2^best - 2 do
    N_([indx], best+1, 1, k);
    { k = number of 1s in indx] }
    if k > 1 and k < best\2+1 then
      { Reject a subset that has only one poly with the field variable. }
      if Flevel > 0 then
        sum := 0;
        for j = 1, best do
          if indx[j] then
            if Deg(facts[j], Flevel) > 0 then
              sum :+
            fi
          fi
        od;
        if sum then
          { cycle }
          &]
        fi
      fi;
      { Compute possible fct. }
      fct := 1;
      for j = 1, best do
        if indx[j] then
          fct := fct*facts[j]
        fi
      od;
      fct := fct/Flcoef(fct);
      fct := lcof*fct;
      if Log2(fct) <= log2ar then
        { Good. Promising looking factor. }
        @[indx];
        Return(0)
      fi
    fi
  od;
  fct := 1;
  @[indx].;
    
;; a and b must be numbers
Func Max__(a_, b_) =
  if a_ > b_ then
    a_
  else
    b_
  fi.;
  
;;====================================================================================================================

;; Factor polynomials over Z.  This works pretty well.  It will not handle cases with "large" numerical coefficients.
;; If all coefficients of ar are < half the primes here (46000 - 60000) it should work.  Furthermore, as
;; long as ar has a factor with all coefficients < half the primes here (46000 - 60000) it has a good chance of working.
;; Difficult cases, like z^4 + y^4, return -1, can't decide.  If your poly has larger coefficients you can try using larger
;; primes, up to 2^31-19.
;;   Invoke with FCTZ(your poly).
;;   Note use of cop.  If coeffs are too large, ar will not be brought back correctly from modular mode.
;; Returns 1 if arg is irreducible, 0 if arg is a constant or field element, -1 if can't decide, polynomial answer
;; placed in fct.  A return of -1  means it's probably irreducible.
;;   The primes can be be parameters to the function.
;;   Ideally it will return a nontrivial answer and put it in fct.  If one of
;; the coeffs of ar is divisible by one of these primes, it won't work.  If lcof is "large" it might not work.
;; Obviously the 16 iterations can be changed.
;;   FCTZ works by dropping in and out of modular mode.  It brings out only the vars ar, fct, lcof.  All other vars
;; created in mod mode should have been deleted in mod mode.  Be careful about modifying the code in Func Cancel.
;;   Modified December 20, 2023.  Added randomization of pr1 and pr2, recursion at end, 'again' to control recursion,
;; and irr to skip the test for irreducible.
;;   Revised in January 2024 for one-var polys, where previously it failed pretty often on factorable polys, reporting -1.
;;   It is assumed that no poly var named ezqp exists.  It could still fail (i.e. report -1).
;;   Jan 2024: remove irr. added function Max, Sqfree test, and better selection of primes when polymodding.  Needs Fermat v. 7.4.
;;   (now needs 7.5) If polymodding, you must have a field, and we assume that the array aux_prime has lots of valid entries.
;;   Note that in the last method, usually not used, a poly var called ezqp is created. Hopefully you don't already have such a name.
;;   Revised Feb 2024, as the previous version often failed when polymodding.  Now it usually works.
;;   Unlike SPF, the answer is returned.  Returning 1 means irreducible.
  
Func FCTZ(ar, pr1, pr2, again; i, best, k, j, fct1, ctt, cop, lcof, cop2, log2ar, levar) =
  if Modmode then
    !!'Error, must be in rational mode.';
    Return(-1)
  fi;
  if Polymod and Flevel = 0 then
    !!'Error, if polymodding, must have a field.';
    Return(-1)
  fi;
  { Bug fix, Apr 6. next two ifs. Should also be fixed for polymodding(?)  }
  if ar = 0 then
    !!'Error, input was 0.';
    Return(-1)
  fi;
  if Numb(ar) then
    Return(|ar|)
  fi;
  ctt := Content(^ar);
  if not Numb(ctt) then
    Return(ctt)
  fi;
  if not ctt then
    ar := ar/ctt
  fi;
  { Added January 29, 2024. Bug fix }
  if Flevel > 0 then
    Sqfree(ar, [q]);
    if Rows[q] > 1 or q[1,2] > 1 then
      k := q[1];
      @[q];
      Return(k)
    fi;
    @[q]
  fi;
  if Flevel > 0 and again = 0 then
    { try factoring without the field. Note the 0. }
    fct := Irred(ar, 0);
    if not Numb(fct) then
      Return(fct)
    fi
  fi;
  if pr1 = 0 then
    if Flevel > 0 then
      pr1 := aux_prime[Deg[aux_prime] \ 2 - Rand|4]
    else
      pr1 := 46445;
      pr1 := Nexp(pr1, 1)
    fi
  else
    pr1 := Nexp(Max__(pr1, pr2))
  fi;
  if pr2 = 0 then
    if Flevel > 0 then
      pr2 := aux_prime[Deg[aux_prime] \ 2 + Rand|4]
    else
      pr2 := 55947;
      pr2 := Nexp(pr2, 1)
    fi
  else
    pr2 := Nexp(Max__(pr1, pr2))
  fi;
  lcof := |Nlcoef(^ar)|;
  log2ar := Log2(ar);
  cop := ar;

  { fct may or may not exist as a global. set it or create it: }
  fct := 1;
  { Set maxn_ here.  It's used by FCT in mod p. }
  maxn_ := Numvars;
  
  { Look for a factor of ar mod pr which might be a real factor over Z. Check leading coefficients }
  &(p = pr1: ar, fct, lcof, log2ar, maxn_);
  { store the factors mod p that we find. }
  levar := Level(ar);
  Array facts[Deg(ar, levar)];
  j := 0;
  while True do
    j :+ ;
    { Factor mod p }
    SPF(ar);
    { Recall how SPF returns an answer: fct = 1 means irreducible. }
    { On first pass, we are done if fct = 1.  Feb 2024: }
    if fct then
      if j then
        &(p = _i: ar, fct, lcof, log2ar, maxn_);
        Return(1)
      else
        fct := ar
      fi
    fi;
    if fct = 0 then
      { extremely rare }
      &>
    fi;
    { try to find 'best' or most reasonable multiple of fct }
    fct := fct/Nlcoef(fct);
    best := Log2(fct);
    k := 1;
    for i = 1, lcof do
      if Log2(i*fct) < best then
        k := i;
        best := Log2(i*fct)
      fi
    od;
    fct := k*fct;
    facts[j] := fct;
    { See if lcof is divisible (over Z) by leading coeff of fct }
    if lcof|k = 0 and Log2(fct) <= log2ar then
      &>
    else
      SDivide(fct, ar);
      if Numb(ar) then
        &>
      fi
    fi
  od;
  
  { If ar is now a number, no reasonable candidate was found for fct.  Try to get a candidate by multiplying the j factors we found. }
  { They are in [facts]. }
  if Numb(ar) and j > 1 then
    Mult_facts([facts], j, log2ar, lcof, fct);
  fi;
  @[facts];
  &(p = _i: ar, fct, lcof, log2ar, maxn_);
  ctt := Content(^fct);
  if not ctt then
    SDivide(ctt, fct)
  fi;
  { Note that if polymodding the builtin Divides is not good enough: }
  if not Numb(fct) then
    if Flevel > 0 then
      fct := GCD(cop, fct);
      if not Numb(fct) then
        Cancel2;
        Return(fct)
      fi
    else
      if Divides(fct, cop) then
        Return(fct)
      fi
    fi
  fi;

  { End of phase 1 }
  { ***** We didn't return so it didn't work. Do it again. }
  fct1 := fct;
  ar := cop;
  log2ar := Log2(ar);
  &(p = pr2: ar, fct, lcof, log2ar, maxn_);
  { store the factors mod p that we find. }
  levar := Level(ar);
  Array facts[Deg(ar, Level(ar))];
  j := 0;
  while True do
    j :+ ;
    SPF(ar);
    { on first pass, we are done if fct = 1.  Feb 2024: }
    if fct then
      if j then
        &(p = _i: ar, fct, lcof, log2ar, maxn_);
        Cancel2;
        Return(1)
      else
        fct := ar
      fi
    fi;
    if fct = 0 then
      { extremely rare }
      &>
    fi;
    { try to find 'best' or most reasonable multiple of fct }
    fct := fct/Nlcoef(fct);
    best := Log2(fct);
    k := 1;
    for i = 1, lcof do
      if Log2(i*fct) < best then
        k := i;
        best := Log2(i*fct)
      fi
    od;
    fct := k*fct;
    facts[j] := fct;
    { See if lcof is divisible (over Z) by leading coeff of fct }
    if lcof|k = 0 and Log2(fct) <= log2ar then
      &>
    else
      SDivide(fct, ar);
      if Numb(ar) then
        &>
      fi
    fi
  od;
  
  { If ar is now a number, no reasonable candidate was found for fct.  Try to get a candidate by multiplying the j factors we found. }
  { They are in [facts]. }
  if Numb(ar) and j > 1 then
    Mult_facts([facts], j, log2ar, levar, lcof, fct);
  fi;
  @[facts];
  &(p = _i: ar, fct, lcof, log2ar, maxn_);
  ctt := Content(^fct);
  if not ctt then
    SDivide(ctt, fct)
  fi;
  if not Numb(fct) then
    if Divides(fct, cop) then
      Return(fct)
    fi
  fi;
  
  { We didn't return, so answer at this point is -1. Let's try again }
  if again < 4 and again >= 0 then
    fct1 := FCTZ(cop, pr1, pr2, again+1);
    Return(fct1)
  fi;
  { Another idea. For one-var polys, try adding a second var. If it can factor, this usually works! }
  { I'm not sure how complex the substitution has to be. }
  { Trying different primes seems to be key, more important than large k and ctt. }
  { Varsp(cop) = number of variables in cop above field_depth, = Vars(cop) if no polymodding. }
  if again = 4 then
    if Varsp(cop) then
      lcof := Level(cop);
      { We need to have an "unused" variable to add to cop, forming cop2 with two vars. }
      { if lcof > 1, that's easy, use Var(lcof-1). Else add new var at top.  If not polymodding, we could add it at bottom. }
      if lcof > 1 then
        ezqp := Var(lcof-1)
      else
        &(J = ezqp);
        lcof :+
      fi;
      [karr] := [[ 1, 1, -1, -1, 2 ]];
      [carr] := [[ 0, 1,  2,  3, 1 ]];
      for i = 1, 5 do
        k := karr[i];
        ctt := carr[i];
        cop2 := cop#(Var(lcof) = Var(lcof) + k*ezqp + ctt);
        fct1 := FCTZ(cop2, pr1, pr2, 5);
        fct := fct1#(Var(lcof) = Var(lcof) - k*ezqp - ctt);
        if fct <> -1 then
          if Level(ezqp) then
            &(J = -ezqp);
            lcof :-
          fi;
          Return(fct)
        fi;
        pr1 := Nexp(pr1, 1);
        pr2 := Nexp(pr2, 1)
      od;
      @([karr], [carr]);
      if Level(ezqp) then
        &(J = -ezqp);
        lcof :-
      fi;
      Return(fct)
    else
      Return(-1)
    fi
  else
    Return(-1)
  fi.;


&x;