aboutsummaryrefslogtreecommitdiff
path: root/src/algebra/mfinfact.spad.pamphlet
blob: 5292b22244a668f8bcab877f08581c110bdedf62 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra mfinfact.spad}
\author{Patrizia Gianni}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package MFINFACT MultFiniteFactorize}
<<package MFINFACT MultFiniteFactorize>>=
)abbrev package MFINFACT MultFiniteFactorize
++ Author: P. Gianni
++ Date Created: Summer 1990
++ Date Last Updated: 19 March 1992
++ Basic Functions:
++ Related Constructors: PrimeField, FiniteField, Polynomial
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description: Package for factorization of multivariate polynomials
++ over finite fields.


MultFiniteFactorize(OV,E,F,PG) : C == T
 where
  F          :   FiniteFieldCategory
  OV         :   OrderedSet
  E          :   OrderedAbelianMonoidSup
  PG         :   PolynomialCategory(F,E,OV)
  SUP        ==> SparseUnivariatePolynomial
  R          ==> SUP F
  P          ==> SparseMultivariatePolynomial(R,OV)
  Z          ==> Integer
  FFPOLY     ==> FiniteFieldPolynomialPackage(F)
  MParFact   ==> Record(irr:P,pow:Z)
  MFinalFact ==> Record(contp:R,factors:List MParFact)
  SUParFact    ==> Record(irr:SUP P,pow:Z)
  SUPFinalFact ==> Record(contp:R,factors:List SUParFact)

                 --  contp   =  content,
                 --  factors =  List of irreducible factors with exponent

  C == with

    factor      : PG     ->  Factored PG
      ++ factor(p) produces the complete factorization of the multivariate
      ++ polynomial p over a finite field.
    factor      : SUP PG     ->  Factored SUP PG
      ++ factor(p) produces the complete factorization of the multivariate
      ++ polynomial p over a finite field. p is represented as a univariate
      ++ polynomial with multivariate coefficients over a finite field.

  T == add

    import LeadingCoefDetermination(OV,IndexedExponents OV,R,P)
    import MultivariateLifting(IndexedExponents OV,OV,R,P)
    import FactoringUtilities(IndexedExponents OV,OV,R,P)
    import FactoringUtilities(E,OV,F,PG)
    import GenExEuclid(R,SUP R)

    NNI       ==> NonNegativeInteger
    L         ==> List
    UPCF2     ==> UnivariatePolynomialCategoryFunctions2
    LeadFact  ==> Record(polfac:L P,correct:R,corrfact:L SUP R)
    ContPrim  ==> Record(cont:P,prim:P)
    ParFact   ==> Record(irr:SUP R,pow:Z)
    FinalFact ==> Record(contp:R,factors:L ParFact)
    NewOrd    ==> Record(npol:SUP P,nvar:L OV,newdeg:L NNI)
    Valuf     ==> Record(inval:L L R,unvfact:L SUP R,lu:R,complead:L R)

                   ----  Local Functions  ----
    ran       :                   Z              -> R
    mFactor   :                (P,Z)             -> MFinalFact
    supFactor :              (SUP P,Z)           -> SUPFinalFact
    mfconst   :        (SUP P,Z,L OV,L NNI)      -> L SUP P
    mfpol     :        (SUP P,Z,L OV,L NNI)      -> L SUP P
    varChoose :           (P,L OV,L NNI)         -> NewOrd
    simplify  :         (P,Z,L OV,L NNI)         -> MFinalFact
    intChoose :        (SUP P,L OV,R,L P,L L R)  -> Valuf
    pretest   :         (P,NNI,L OV,L R)         -> FinalFact
    checkzero :            (SUP P,SUP R)         -> Boolean
    pushdcoef :                  PG              -> P
    pushdown  :                (PG,OV)           -> P
    pushupconst :               (R,OV)           -> PG
    pushup    :                 (P,OV)           -> PG
    norm      :               L SUP R            -> Integer
    constantCase :        (P,L MParFact)         -> MFinalFact
    pM          :             L SUP R            -> R
    intfact     :     (SUP P,L OV,L NNI,MFinalFact,L L R) -> L SUP P

    basicVar:OV:=NIL$Lisp pretend OV  -- variable for the basic step


    convertPUP(lfg:MFinalFact): SUPFinalFact ==
      [lfg.contp,[[lff.irr ::SUP P,lff.pow]$SUParFact
                    for lff in lfg.factors]]$SUPFinalFact

    supFactor(um:SUP P,dx:Z) : SUPFinalFact ==
        degree(um)=0 => convertPUP(mFactor(ground um,dx))
        lvar:L OV:= "setUnion"/[variables cf for cf in coefficients um]
        lcont:SUP P
        lf:L SUP P

        flead : SUPFinalFact:=[0,empty()]$SUPFinalFact
        factorlist:L SUParFact :=empty()

        mdeg :=minimumDegree um     ---- is the Mindeg > 0? ----
        if mdeg>0 then
          f1:SUP P:=monomial(1,mdeg)
          um:=(um exquo f1)::SUP P
          factorlist:=cons([monomial(1,1),mdeg],factorlist)
          if degree um=0 then return
            lfg:=convertPUP mFactor(ground um, dx)
            [lfg.contp,append(factorlist,lfg.factors)]


        om:=map(pushup(#1,basicVar),um)$UPCF2(P,SUP P,PG,SUP PG)
        sqfacs:=squareFree(om)
        lcont:=map(pushdown(#1,basicVar),unit sqfacs)$UPCF2(PG,SUP PG,P,SUP P)

                                   ----   Factorize the content  ----
        if ground? lcont then
          flead:=convertPUP constantCase(ground lcont,empty())
        else
          flead:=supFactor(lcont,dx)

        factorlist:=flead.factors

                                 ----  Make the polynomial square-free  ----
        sqqfact:=[[map(pushdown(#1,basicVar),ff.factor),ff.exponent]
                      for ff in factors sqfacs]

                        ---  Factorize the primitive square-free terms ---
        for fact in sqqfact repeat
          ffactor:SUP P:=fact.irr
          ffexp:=fact.pow
          ffcont:=content ffactor
          coefs := coefficients ffactor
          ldeg:= ["max"/[degree(fc,xx) for fc in coefs] for xx in lvar]
          if ground?(leadingCoefficient ffactor) then
             lf:= mfconst(ffactor,dx,lvar,ldeg)
          else lf:=mfpol(ffactor,dx,lvar,ldeg)
          auxfl:=[[lfp,ffexp]$SUParFact  for lfp in lf]
          factorlist:=append(factorlist,auxfl)
        lcfacs := */[leadingCoefficient leadingCoefficient(f.irr)**((f.pow)::NNI)
                             for f in factorlist]
        [(leadingCoefficient leadingCoefficient(um) exquo lcfacs)::R,
                       factorlist]$SUPFinalFact

    factor(um:SUP PG):Factored SUP PG ==
        lv:List OV:=variables um
        ld:=degree(um,lv)
        dx:="min"/ld
        basicVar:=lv.position(dx,ld)
        cm:=map(pushdown(#1,basicVar),um)$UPCF2(PG,SUP PG,P,SUP P)
        flist := supFactor(cm,dx)
        pushupconst(flist.contp,basicVar)::SUP(PG) *
          (*/[primeFactor(map(pushup(#1,basicVar),u.irr)$UPCF2(P,SUP P,PG,SUP PG),
                 u.pow) for u in flist.factors])



    mFactor(m:P,dx:Z) : MFinalFact ==
      ground?(m) => constantCase(m,empty())
      lvar:L OV:= variables m
      lcont:P
      lf:L SUP P
      flead : MFinalFact:=[1,empty()]$MFinalFact
      factorlist:L MParFact :=empty()
                                  ---- is the Mindeg > 0? ----
      lmdeg :=minimumDegree(m,lvar)
      or/[n>0 for n in lmdeg] => simplify(m,dx,lvar,lmdeg)
                              ----  Make the polynomial square-free  ----
      om:=pushup(m,basicVar)
      sqfacs:=squareFree(om)
      lcont := pushdown(unit sqfacs,basicVar)

                                  ----  Factorize the content  ----
      if ground? lcont then
        flead:=constantCase(lcont,empty())
      else
        flead:=mFactor(lcont,dx)
      factorlist:=flead.factors
      sqqfact:List Record(factor:P,exponent:Integer)
      sqqfact:=[[pushdown(ff.factor,basicVar),ff.exponent]
                                              for ff in factors sqfacs]
                       ---  Factorize the primitive square-free terms ---
      for fact in sqqfact repeat
        ffactor:P:=fact.factor
        ffexp := fact.exponent
        ground? ffactor =>
          for lterm in constantCase(ffactor,empty()).factors repeat
            factorlist:=cons([lterm.irr,lterm.pow * ffexp], factorlist)
        lvar := variables ffactor
        x:OV:=lvar.1
        ldeg:=degree(ffactor,lvar)
             ---  Is the polynomial linear in one of the variables ? ---
        member?(1,ldeg) =>
          x:OV:=lvar.position(1,ldeg)
          lcont:= gcd coefficients(univariate(ffactor,x))
          ffactor:=(ffactor exquo lcont)::P
          factorlist:=cons([ffactor,ffexp]$MParFact,factorlist)
          for lcterm in mFactor(lcont,dx).factors repeat
           factorlist:=cons([lcterm.irr,lcterm.pow * ffexp], factorlist)

        varch:=varChoose(ffactor,lvar,ldeg)
        um:=varch.npol


        ldeg:=ldeg.rest
        lvar:=lvar.rest
        if varch.nvar.1 ~= x then
          lvar:= varch.nvar
          x := lvar.1
          lvar:=lvar.rest
          pc:= gcd coefficients um
          if pc~=1 then
            um:=(um exquo pc)::SUP P
            ffactor:=multivariate(um,x)
            for lcterm in mFactor(pc,dx).factors repeat
              factorlist:=cons([lcterm.irr,lcterm.pow*ffexp],factorlist)
          ldeg:= degree(ffactor,lvar)

        -- should be unitNormal if unified, but for now it is easier
        lcum:F:= leadingCoefficient leadingCoefficient
                leadingCoefficient um
        if lcum ~=1  then
          um:=((inv lcum)::R::P) * um
          flead.contp := (lcum::R) *flead.contp

        if ground?(leadingCoefficient um)
        then lf:= mfconst(um,dx,lvar,ldeg)
        else lf:=mfpol(um,dx,lvar,ldeg)
        auxfl:=[[multivariate(lfp,x),ffexp]$MParFact  for lfp in lf]
        factorlist:=append(factorlist,auxfl)
      flead.factors:= factorlist
      flead


    pM(lum:L SUP R) : R ==
      x := monomial(1,1)$R
      for i in 1..size()$F repeat
         p := x + (index(i::PositiveInteger)$F) ::R
         testModulus(p,lum) => return p
      for e in 2.. repeat
          p :=  (createIrreduciblePoly(e::PositiveInteger))$FFPOLY
          testModulus(p,lum) => return p
          while not((q := nextIrreduciblePoly(p)$FFPOLY) case "failed") repeat
             p := q::SUP F
             if testModulus(p, lum)$GenExEuclid(R, SUP R) then return p

      ----  push x in the coefficient domain for a term ----
    pushdcoef(t:PG):P ==
       map(coerce(#1)$R,t)$MPolyCatFunctions2(OV,E,
                                           IndexedExponents OV,F,R,PG,P)


              ----  internal function, for testing bad cases  ----
    intfact(um:SUP P,lvar: L OV,ldeg:L NNI,
            tleadpol:MFinalFact,ltry:L L R):  L SUP P ==
      polcase:Boolean:=(not empty? tleadpol.factors )
      vfchoo:Valuf:=
        polcase =>
          leadpol:L P:=[ff.irr for ff in tleadpol.factors]
          intChoose(um,lvar,tleadpol.contp,leadpol,ltry)
        intChoose(um,lvar,1,empty(),empty())
      unifact:List SUP R := vfchoo.unvfact
      nfact:NNI := #unifact
      nfact=1 => [um]
      ltry:L L R:= vfchoo.inval
      lval:L R:=first ltry
      dd:= vfchoo.lu
      lpol:List P:=empty()
      leadval:List R:=empty()
      if polcase then
        leadval := vfchoo.complead
        distf := distFact(vfchoo.lu,unifact,tleadpol,leadval,lvar,lval)
        distf case "failed" =>
             return intfact(um,lvar,ldeg,tleadpol,ltry)
        dist := distf :: LeadFact
          -- check the factorization of leading coefficient
        lpol:= dist.polfac
        dd := dist.correct
        unifact:=dist.corrfact
      if dd~=1 then
        unifact := [dd*unifact.i for i in 1..nfact]
        um := ((dd**(nfact-1)::NNI)::P)*um
      (ffin:= lifting(um,lvar,unifact,lval,lpol,ldeg,pM(unifact)))
           case "failed" => intfact(um,lvar,ldeg,tleadpol,ltry)
      factfin: L SUP P:=ffin :: L SUP P
      if dd~=1 then
        factfin:=[primitivePart ff  for ff in  factfin]
      factfin

-- the following functions are used to "push" x in the coefficient ring -
               ----  push back the variable  ----
    pushup(f:P,x:OV) :PG ==
       ground? f => pushupconst((retract f)@R,x)
       rr:PG:=0
       while f~=0 repeat
         lf:=leadingMonomial f
         cf:=pushupconst(leadingCoefficient f,x)
         lvf:=variables lf
         rr:=rr+monomial(cf,lvf, degree(lf,lvf))$PG
         f:=reductum f
       rr

        ----  push x in the coefficient domain for a polynomial ----
    pushdown(g:PG,x:OV) : P ==
       ground? g => ((retract g)@F)::R::P
       rf:P:=0$P
       ug:=univariate(g,x)
       while ug~=0 repeat
         cf:=monomial(1,degree ug)$R
         rf:=rf+cf*pushdcoef(leadingCoefficient ug)
         ug := reductum ug
       rf

      ----  push x back from the coefficient domain ----
    pushupconst(r:R,x:OV):PG ==
       ground? r => (retract r)@F ::PG
       rr:PG:=0
       while r~=0 repeat
         rr:=rr+monomial((leadingCoefficient r)::PG,x,degree r)$PG
         r:=reductum r
       rr

    -- This function has to be added to Eucliden domain
    ran(k1:Z) : R ==
      --if R case Integer then random()$R rem (2*k1)-k1
      --else
      +/[monomial(random()$F,i)$R for i in 0..k1]

    checkzero(u:SUP P,um:SUP R) : Boolean ==
      u=0 => um =0
      um = 0 => false
      degree u = degree um => checkzero(reductum u, reductum um)
      false

              ---  Choose the variable of least degree  ---
    varChoose(m:P,lvar:L OV,ldeg:L NNI) : NewOrd ==
      k:="min"/[d for d in ldeg]
      k=degree(m,first lvar) =>
                             [univariate(m,first lvar),lvar,ldeg]$NewOrd
      i:=position(k,ldeg)
      x:OV:=lvar.i
      ldeg:=cons(k,delete(ldeg,i))
      lvar:=cons(x,delete(lvar,i))
      [univariate(m,x),lvar,ldeg]$NewOrd


    norm(lum: L SUP R): Integer == "max"/[degree lup for lup in lum]

          ---  Choose the values to reduce to the univariate case  ---
    intChoose(um:SUP P,lvar:L OV,clc:R,plist:L P,ltry:L L R) : Valuf ==
      -- declarations
      degum:NNI := degree um
      nvar1:=#lvar
      range:NNI:=0
      unifact:L SUP R
      ctf1 : R := 1
      testp:Boolean :=             -- polynomial leading coefficient
        plist = empty() => false
        true
      leadcomp,leadcomp1 : L R
      leadcomp:=leadcomp1:=empty()
      nfatt:NNI := degum+1
      lffc:R:=1
      lffc1:=lffc
      newunifact : L SUP R:=empty()
      leadtest:=true --- the lc test with polCase has to be performed
      int:L R:=empty()

   --  New sets of values are chosen until we find twice the
   --  same number of "univariate" factors:the set smaller in modulo is
   --  is chosen.
      while true repeat
       lval := [ ran(range) for i in 1..nvar1]
       member?(lval,ltry) => range:=1+range
       ltry := cons(lval,ltry)
       leadcomp1:=[retract eval(pol,lvar,lval) for pol in plist]
       testp and or/[unit? epl for epl in leadcomp1] => range:=range+1
       newm:SUP R:=completeEval(um,lvar,lval)
       degum ~= degree newm or minimumDegree newm ~=0 => range:=range+1
       lffc1:=content newm
       newm:=(newm exquo lffc1)::SUP R
       testp and leadtest and not polCase(lffc1*clc,#plist,leadcomp1)
                           => range:=range+1
       Dnewm := differentiate newm
       D2newm := map(differentiate, newm)
       degree(gcd [newm,Dnewm,D2newm])~=0 => range:=range+1
      -- if R has Integer then luniv:=henselFact(newm,false)$
      -- else
       lcnm:F:=1
        -- should be unitNormal if unified, but for now it is easier
       if (lcnm:=leadingCoefficient leadingCoefficient newm)~=1 then
         newm:=((inv lcnm)::R)*newm
       dx:="max"/[degree uc  for uc in coefficients newm]
       luniv:=generalTwoFactor(newm)$TwoFactorize(F)
       lunivf:= factors luniv
       nf:= #lunivf

       nf=0 or nf>nfatt => "next values"      ---  pretest failed ---

                        --- the univariate polynomial is irreducible ---
       if nf=1 then leave (unifact:=[newm])

       lffc1:=lcnm * retract(unit luniv)@R * lffc1

   --  the new integer give the same number of factors
       nfatt = nf =>
       -- if this is the first univariate factorization with polCase=true
       -- or if the last factorization has smaller norm and satisfies
       -- polCase
         if leadtest or
           ((norm unifact > norm [ff.factor for ff in lunivf]) and
             (not testp or polCase(lffc1*clc,#plist,leadcomp1))) then
                unifact:=[uf.factor for uf in lunivf]
                int:=lval
                lffc:=lffc1
                if testp then leadcomp:=leadcomp1
         leave "foundit"

   --  the first univariate factorization, inizialize
       nfatt > degum =>
         unifact:=[uf.factor for uf in lunivf]
         lffc:=lffc1
         if testp then leadcomp:=leadcomp1
         int:=lval
         leadtest := false
         nfatt := nf

       nfatt>nf =>  -- for the previous values there were more factors
         if testp then leadtest := not polCase(lffc*clc,#plist,leadcomp)
         else leadtest:= false
         -- if polCase=true we can consider the univariate decomposition
         if not leadtest then
           unifact:=[uf.factor for uf in lunivf]
           lffc:=lffc1
           if testp then leadcomp:=leadcomp1
           int:=lval
         nfatt := nf
      [cons(int,ltry),unifact,lffc,leadcomp]$Valuf


    constantCase(m:P,factorlist:List MParFact) : MFinalFact ==
    --if R case Integer then [const m,factorlist]$MFinalFact
    --else
      lunm:=distdfact((retract m)@R,false)$DistinctDegreeFactorize(F,R)
      [(lunm.cont)::R, append(factorlist,
           [[(pp.irr)::P,pp.pow] for pp in lunm.factors])]$MFinalFact

                ----  The polynomial has mindeg>0   ----

    simplify(m:P,dm:Z,lvar:L OV,lmdeg:L NNI):MFinalFact ==
      factorlist:L MParFact:=empty()
      pol1:P:= 1$P
      for x in lvar repeat
        i := lmdeg.(position(x,lvar))
        i=0 => "next value"
        pol1:=pol1*monomial(1$P,x,i)
        factorlist:=cons([x::P,i]$MParFact,factorlist)
      m := (m exquo pol1)::P
      ground? m => constantCase(m,factorlist)
      flead:=mFactor(m,dm)
      flead.factors:=append(factorlist,flead.factors)
      flead

                ----  m square-free,primitive,lc constant  ----
    mfconst(um:SUP P,dm:Z,lvar:L OV,ldeg:L NNI):L SUP P ==
      nsign:Boolean
      factfin:L SUP P:=empty()
      empty? lvar =>
          um1:SUP R:=map(ground,
              um)$UPCF2(P,SUP P,R,SUP R)
          lum:= generalTwoFactor(um1)$TwoFactorize(F)
          [map(coerce,lumf.factor)$UPCF2(R,SUP R,P,SUP P)
                for lumf in factors lum]
      intfact(um,lvar,ldeg,[0,empty()]$MFinalFact,empty())

              --- m is square-free,primitive,lc is a polynomial  ---
    mfpol(um:SUP P,dm:Z,lvar:L OV,ldeg:L NNI):L SUP P ==
      dist : LeadFact
      tleadpol:=mFactor(leadingCoefficient um,dm)
      intfact(um,lvar,ldeg,tleadpol,empty())

    factor(m:PG):Factored PG ==
       lv:=variables m
       lv=empty() => makeFR(m,empty() )
    -- reduce to multivariate over SUP
       ld:=[degree(m,x) for x in lv]
       dx:="min"/ld
       basicVar:=lv(position(dx,ld))
       cm:=pushdown(m,basicVar)
       flist := mFactor(cm,dx)
       pushupconst(flist.contp,basicVar) *
          (*/[primeFactor(pushup(u.irr,basicVar),u.pow)
                                                 for u in flist.factors])

@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
--    - Redistributions of source code must retain the above copyright
--      notice, this list of conditions and the following disclaimer.
--
--    - Redistributions in binary form must reproduce the above copyright
--      notice, this list of conditions and the following disclaimer in
--      the documentation and/or other materials provided with the
--      distribution.
--
--    - Neither the name of The Numerical ALgorithms Group Ltd. nor the
--      names of its contributors may be used to endorse or promote products
--      derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
@
<<*>>=
<<license>>

<<package MFINFACT MultFiniteFactorize>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}