This file is indexed.

/usr/share/singular/LIB/nfmodsyz.lib is in singular-data 1:4.1.0-p3+ds-2build1.

This file is owned by root:root, with mode 0o644.

The actual contents of the file can be viewed below.

  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
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
////////////////////////////////////////////////////////////////////////////////
version="Id";  // $Id: 6667bfb150105848c367d6aba1002b3a22dcfaa8 $
category="Commutative Algebra";
info="

LIBRARY:   nfmodsyz.lib    Syzygy modules of submodules of free modules over
                           algebraic number fields
AUTHORS:   D.K. Boku       boku@mathematik.uni-kl.de
@*         W. Decker       decker@mathematik.uni-kl.de
@*         C. Fieker       fieker@mathematik.uni-kl.de

OVERVIEW:
  A library for computing the syzygy module of a given submodule I in a polynomial ring
  over an algebraic number field Q(t), where t is an algebraic number, using modular methods.
  For the case Q(t)=Q, that is, where t is an element of Q, we compute, following
  [1], the syzygy module of I as follows: For a submodule I of A^m with A = Q[X], we first
  choose a sufficiently large set of primes P and compute the reduced Groebner basis of the
  syzygy module of I_p, for each p in P, in parallel. We then use the Chinese remainder
  algorithm and rational reconstruction to obtain the syzygy module of I over Q.
  For the case where t is not in Q, we compute, following [2], the syzygy module of I as
  follows:
  Let f be the minimal polynomial of t. For a submodule I in A^m with A = Q(t)[X], we map I
  to a submodule I' in A^m with A = (Q[t]/<f>)[X] via the map sending t to t + <f>. We first
  choose a prime p such that f has at least two factors in characteristic p. For each
  factor f_{i,p} of f_p:= (f mod p), we set I'_{i,p} := (I'_p mod f_{i,p}). We then
  compute the reduced Groebner bases G'_i of the syzygy modules of I'_{i,p} over
  F_p[t]/<f_{i,p}> and combine the G'_i to G_p (the syzygy module of I'_p) using chinese
  remaindering for polynomials. As described in [2], the procedure is repeated for many primes
  p, where we compute the G_p in parallel until the number of primes is sufficiently large to
  recover the correct generating set for the syzygy module G' of I' which is, considered over
  Q(t), also a generating set for the syzygy module of I.

REFERENCES:
  [1] E. A. Arnold: Modular algorithms for computing Groebner bases.
      J. Symb. Comp. 35, 403-419 (2003).
  [2] D. Boku, W. Decker, C. Fieker, and A. Steenpass. Groebner bases over algebraic
      number fields. In: Proceedings of the 2015 International Workshop on Parallel
      Symb. Comp. PASCO'15, pages 16-24 (2015).

PROCEDURES:
  nfmodSyz(I);          syzygy module of I over algebraic number field using modular
                        methods
";

LIB "nfmodstd.lib";

////////////////////////////////////////////////////////////////////////////////

static proc testPrime(int p, list args)
{
    /*
     * test whether a prime p divides the denominator(s)
     * and leading coefficients of generating set of ideal
     */
    int i,j,k;
    vector f;
    number num;
    module I = args[1];
    bigint d1,d2,d3;
    for(i = 1; i <= ncols(I); i++)
    {
        f = cleardenom(I[i]);
        if(f == 0)
        {
            return(0);
        }
        num = leadcoef(I[i])/leadcoef(f);
        d1 = bigint(numerator(num));
        d2 = bigint(denominator(num));
        if( (d1 mod p) == 0)
        {
            return(0);
        }
        if((d2 mod p) == 0)
        {
            return(0);
        }
        for(j = nrows(f); j > 0; j--)
        {
            for(k=1;k<=size(f[j]);k++)
            {
                d3 = bigint(leadcoef(f[j][k]));
                if((d3 mod p) == 0)
                {
                    return(0);
                }
            }
        }
    }
    return(1);
}

////////////////////////////////////////////////////////////////////////////////
/* return 1 if the number of factors are in the required bound , 0 else */

static proc minpolyTask(poly f,int p)
{
    /*
     * bound for irreducible factor(s) of (f mod p)
     * see testfact()
     */
    int nr,k,ur;
    ur=deg(f);
    list L=factmodp(f,p);
    if(degtest(L[2])==1)
    {
        // now each factor is squarefree
        if(ur<=3)
        {
            return(1);
        }
        else
        {
            nr = testfact(ur);
            k=ncols(L[1]);
            if(nr < k && k < (ur-nr)) // set a bound for k
            {
                return(1);
            }
        }
    }
    return(0);
}

////////////////////////////////////////////////////////////////////////////////
/* return 1 if both testPrime(p,J) and minpolyTask(f,p) is true, 0 else */

static proc PrimeTestTask_syz(int p, list L)
{
    /* L=list(I), I=J,f; J ideal , f minpoly */
    int sz,nr;
    module J = L[1];
    sz=ncols(J);
    def f=J[sz];
    poly g = f[1];
    if(!testPrime(p,list(J)) or !minpolyTask(g,p))
    {
        return(0);
    }
    return(1);
}

////////////////////////////////////////////////////////////////////////////////
/* compute factors of f mod p with multiplicity */

static proc factmodp(poly f, int p)
{
    def R=basering;
    list l=ringlist(R);
    l[1]=p;
    def S=ring(l);
    setring S;
    list L=factorize(imap(R,f),2);
    ideal J=L[1];
    intvec v=L[2];
    list scx=J,v;
    setring R;
    return(imap(S,scx));
    kill S;
}

////////////////////////////////////////////////////////////////////////////////
/* set a bound for number of factors w.r.t degree nr*/

static proc testfact(int nr)
{
    // nr must be greater than 3
    int i;
    if(nr>3 and nr<=5)
    {
        i=1;
    }
    if(nr>5 and nr<=10)
    {
        i=2;
    }
    if(nr>10 and nr<=15)
    {
        i=3;
    }
    if(nr>15 and nr<=20)
    {
        i=4;
    }
    if(nr>20 and nr<=25)
    {
        i=5;
    }
    if(nr>25 and nr<=30)
    {
        i=6;
    }
    if(nr>30)
    {
        i=10;
    }
    return(i);
}

///////////////////////////////////////////////////////////////////////////////
// return 1 if v[i]>1 , 0 else

static proc degtest(intvec v)
{
    for(int j=1;j<=nrows(v);j++)
    {
        if(v[j]>1)
        {
            return(0);
        }
    }
    return(1);
}

////////////////////////////////////////////////////////////////////////////////

static proc check_leadmonom_and_size(list L)
{
    /*
     * compare the size of ideals in the list and
     * check the corresponding leading monomials
     * size(L)>=2
     */
    def J=L[1];
    int i=size(L);
    int sc=ncols(J);
    int j,k;
    def g=leadmonom(J[1]);
    for(j=1;j<=i;j++)
    {
        if(ncols(L[j])!=sc)
        {
            return(0);
        }
    }
    for(k=2;k<=i;k++)
    {
        for(j=1;j<=sc;j++)
        {
            if(leadmonom(J[j])!=leadmonom(L[k][j]))
            {
                return(0);
            }
        }
    }
    return(1);
}

////////////////////////////////////////////////////////////////////////////////

static proc LiftPolyCRT_syz(def I)
{
    /*
     * compute syz for each factor and combine this result
     * to modulo minpoly via CRT for poly over char p>0
     */
    def sl;
    int u,in,j;
    list LL,Lk,T2;
    module J,II;
    vector f;
    u=ncols(I);
    J=I[1..u-1];
    f=I[u];
    poly ff = f[1];
    ideal K=factorize(ff,1);
    in=ncols(K);
    def Ls = basering;
    list l = ringlist(Ls);
    if(l[3][1][1]=="c")
    {
        l[1] = list(l[1]) + list(list(l[2][size(l[2])])) +
        list(list(l[3][size(l[3])]))+list(ideal(0));
        l[2] = delete(l[2],size(l[2]));
        l[3] = delete(l[3],size(l[3]));
    }
    else
    {
        l[1] = list(l[1]) + list(list(l[2][size(l[2])])) +
        list(list(l[3][size(l[3])-1]))+list(ideal(0));
        l[2] = delete(l[2],size(l[2]));
        l[3] = delete(l[3],size(l[3])-1);
    }

    def S1 = ring(l);
    setring S1;
    number Num= number(imap(Ls,ff));
    list l = ringlist(S1);
    l[1][4][1] = Num;
    S1 = ring(l);
    setring S1;
    ideal K = imap(Ls,K);
    def S2;
    module II;
    number Num;
    /* ++++++ if minpoly is irreducible then K will be the zero ideal +++ */
    if(size(K)==0)
    {
        module M = syz(imap(Ls,J));
        if(size(M)==0)
        {
            setring Ls;
            return(module([0]));
        }
        II = normalize(M);
    }
    else
    {
        for(j=1;j<=in;j++)
        {
            LL[j]=K[j];
            Num = number(K[j]);
            T2 = ringlist(S1);
            T2[1][4][1] = Num;
            S2 = ring(T2);
            setring S2;
            module M = syz(imap(Ls,J));
            if(size(M)==0)
            {
                setring Ls;
                return(module([0]));
                break;
            }
            setring S1;
            Lk[j] = imap(S2,M);
        }

        if(check_leadmonom_and_size(Lk))
        {
            // apply CRT for polynomials
            setring Ls;
            II =chinrempoly(imap(S1,Lk),imap(S1,LL));
            setring S1;
            II = normalize(imap(Ls,II));
        }
        else
        {
            setring S1;
            II=[0];
        }
     }
     setring Ls;
     return(imap(S1,II));
}

////////////////////////////////////////////////////////////////////////////////

static proc final_Test_syz(string command, alias list args, def result)
{
    /*
     * test if the set generating 'result' also generates the syzygy module
     * of args[1] in characteristic zero
     */
    def Ls = basering;
    def Ip = args[1];
    vector f;
    int u=ncols(Ip);
    module J=Ip[1..u-1];
    f=Ip[u];
    poly ff = f[1];
    list l = ringlist(Ls);

    if(l[3][1][1]=="c")
    {
        l[1] = list(l[1]) + list(list(l[2][size(l[2])])) +
        list(list(l[3][size(l[3])]))+list(ideal(0));
        l[2] = delete(l[2],size(l[2]));
        l[3] = delete(l[3],size(l[3]));
    }
    else
    {
        l[1] = list(l[1]) + list(list(l[2][size(l[2])])) +
        list(list(l[3][size(l[3])-1]))+list(ideal(0));
        l[2] = delete(l[2],size(l[2]));
        l[3] = delete(l[3],size(l[3])-1);
    }

    def S1 = ring(l);
    setring S1;
    number Num= number(imap(Ls,ff));
    list l = ringlist(S1);
    l[1][4][1] = Num;
    S1 = ring(l);
    setring S1;
    def result2 = imap(Ls,result);
    def M = imap(Ls,J);
    if(size(result2)==0)
    {
        return(1);
    }
    else
    {
        if(size(module(matrix(M)*matrix(result2)))!=0)
        {
            return(0);
        }
        return(1);
    }
}

////////////////////////////////////////////////////////////////////////////////

static proc final_test(string command, alias list args, def result)
{
    /*
     * test if the set  generating 'result' also generates the syzygy module
     * of args[1] in characteristic zero
     */
    module M=args[1];
    if(size(result)==0)
    {
        return(1);
    }
    else
    {
        if(size(module(matrix(M)*matrix(result)))!=0)
        {
            return(0);
        }
        return(1);
    }
}

////////////////////////////////////////////////////////////////////////////////
// ------------------------ test in characteristic p ------------
static proc pTest_syzmod(string command, list args, def result, int p)
{
     /*
      * This procedure performs the first test in positive characteristic to
      * verify whether the set generating 'result' also generates the syzygy
      * module of the submodule args[1]. Note that this test works only
      * over Z_p
      */
     def br = basering;
     if(size(result)==0)
     {
         return(1);
     }
     list lbr = ringlist(br);
     if (typeof(lbr[1]) == "int")
     {
         lbr[1] = p;
     }
     else
     {
         lbr[1][1] = p;
     }
     def rp = ring(lbr);
     setring(rp);
     module Jp = imap(br, args)[1];
     module Gp = imap(br, result);
     module Ip = syz(Jp);
     // test if Ip is contained in Gp
     attrib(Gp, "isSB", 1);
     for (int i = ncols(Ip); i > 0; i--)
     {
          if (reduce(Ip[i], Gp, 1) != 0)
          {
              setring(br);
              return(0);
          }
     }
     // test if Gp is contained in syz(Jp)
     if(size(module(matrix(Jp)*matrix(Gp)))!=0)
     {
         setring br;
         return(0);
     }
     setring br;
     return(1);
}

////////////////////////////////////////////////////////////////////////////////
 // ------------------------ test in characteristic p ------------
static proc pTest_syz(string command, list args, def result, int p)
{
     /*
      * This procedure performs the first test in positive characteristic to
      * verify whether the set generating 'result' also generates the syzygy
      * module of args[1]. Note that this test works only over Z_p(t) where
      * t is an algebraic number which is not in Z_p.
     */

     def br = basering;
     if(size(result)==0)
     {
         return(1);
     }
     list lbr = ringlist(br);
     if (typeof(lbr[1]) == "int")
     {
         lbr[1] = p;
     }
     else
     {
         lbr[1][1] = p;
     }
     def rp = ring(lbr);
     setring(rp);
     def Ip = imap(br, args)[1];

     int u,in,j,i;
     list LL,Lk,T2;
     module J,II;
     vector f;
     u=ncols(Ip);
     J=Ip[1..u-1];
     f=Ip[u];
     poly ff = f[1];
     ideal K=factorize(ff,1);
     in=ncols(K);
     def Ls = basering;
     list l = ringlist(Ls);
     if(l[3][1][1]=="c")
     {
        l[1] = list(l[1]) + list(list(l[2][size(l[2])])) +
        list(list(l[3][size(l[3])]))+list(ideal(0));
        l[2] = delete(l[2],size(l[2]));
        l[3] = delete(l[3],size(l[3]));
     }
     else
     {
        l[1] = list(l[1]) + list(list(l[2][size(l[2])])) +
        list(list(l[3][size(l[3])-1]))+list(ideal(0));
        l[2] = delete(l[2],size(l[2]));
        l[3] = delete(l[3],size(l[3])-1);
     }

     def S1 = ring(l);
     setring S1;
     number Num= number(imap(Ls,ff));
     list l = ringlist(S1);
     l[1][4][1] = Num;
     S1 = ring(l);
     setring S1;
     ideal K = imap(Ls,K);
     module Jp = imap(Ls,J);
     def S2;
     module Ip;
     number Num;
     /* ++++++ if the minpoly is irreducible then K = ideal(0) +++ */
     if(size(K)==0)
     {
          module M = syz(Jp);
          Ip = normalize(M);
     }
     else
     {
         for(j=1;j<=ncols(K);j++)
         {
              LL[j]=K[j];
              Num = number(K[j]);
              T2 = ringlist(S1);
              T2[1][4][1] = Num;
              S2 = ring(T2);
              setring S2;
              module M = syz(imap(Ls,J));
              setring S1;
              Lk[j]= imap(S2,M);
         }
         if(check_leadmonom_and_size(Lk))
         {
              // apply CRT for polynomials
              setring Ls;
              II =chinrempoly(imap(S1,Lk),imap(S1,LL));
              setring S1;
              Ip = normalize(imap(Ls,II));
         }
         else
         {
              setring S1;
              Ip=[0];
         }
     }
     setring S1;
     module Gp = imap(br, result);
     // test if Ip is contained in Gp
     attrib(Gp, "isSB", 1);
     for (i = ncols(Ip); i > 0; i--)
     {
          if (reduce(Ip[i], Gp, 1) != 0)
          {
              setring(br);
              return(0);
          }
     }
     // test if Gp is contained in syz(Jp)
     if(size(module(matrix(Jp)*matrix(Gp)))!=0)
     {
         setring br;
         return(0);
     }
     setring br;
     return(1);
}

 ////////////////////////////////////////////////////////////////////////////////

 static proc cleardenomIdeal(def I)
 {
     int t=ncols(I);
     if(size(I)==0)
     {
         return(I);
     }
     else
     {
         for(int i=1;i<=t;i++)
         {
             I[i]=cleardenom(I[i]);
         }
     }
     return(I);
 }

////////////////////////////////////////////////////////////////////////////////

 static proc modStdparallelized_syzSB(module I, list #)
 {
     /* save options */
     intvec opt = option(get);
     option(redSB);
     option(returnSB);
     /*------ if these options are set, the Singular command syz returns the
      reduced Groebner basis of I ---------------------------------------*/

     // apply modular command from modular.lib
     if(size(#)>0)
     {
         I = modular("syz", list(I), testPrime, Modstd::deleteUnluckyPrimes_std,
             pTest_syzmod, final_test, 536870909);
     }
     else
     {
         I = modular("Nfmodsyz::LiftPolyCRT_syz", list(I), PrimeTestTask_syz,
             Modstd::deleteUnluckyPrimes_std,pTest_syz, final_Test_syz,536870909);
     }
     attrib(I, "isSB", 1);
     option(set,opt);
     return(I);
 }

////////////////////////////////////////////////////////////////////////////////
/* main procedure */
proc nfmodSyz(def I)
"USAGE:  nfmodSyz(I); I ideal or module
RETURN:  syzygy module of I over an algebraic number field
SEE ALSO: syz
EXAMPLE: example nfmodSyz; shows an example
"
{
     if(typeof(I)!="ideal" and typeof(I)!="module")
     {
        ERROR("type of input must be either ideal or module");
     }
     else
     {
        module F = I;
        kill I;
        module I = F;
     }
     def Rbs=basering;
     poly f;
     int n=nvars(Rbs);
     if(size(I)==0)
     {
         return(module([0]));
     }
     if(npars(Rbs)==0)
     {
        module M = modStdparallelized_syzSB(I,1); //if algebraic number is in Q
        return(M);
     }

     def S;
     list rl=ringlist(Rbs);
     f=rl[1][4][1];

     if(rl[3][1][1]!="c")
     {
        rl[2] = rl[2] + rl[1][2];
        rl[3] = insert(rl[3], rl[1][3][1],1);
        rl[1] = rl[1][1];
     }
     else
     {
        rl[2] = rl[2] + rl[1][2];
        rl[3][size(rl[3])+1] = rl[1][3][1];
        rl[1] = rl[1][1];
     }

     S = ring(rl);
     setring S;
     poly f=imap(Rbs,f);
     def I=imap(Rbs,I);
     I = simplify(I,2); // eraze the zero generatos
     if(f==0)
     {
         ERROR("minpoly must be non-zero");
     }
     I=I,f;
     def J_I = modStdparallelized_syzSB(I);
     setring Rbs;
     def J=imap(S,J_I);
     J=simplify(J,2);
     return(J);
}
example
{ "EXAMPLE:"; echo = 2;
    ring r1 =(0,a),(x,y),(c,dp);
    minpoly = (a^3+2a+7);
    module M1 = [(a/2+1)*y, 3*x-a*y],
                [y-x,y2],
                [x2-xy, ax-y];
    nfmodSyz(M1);
    ring r2 = (0,a),(x,y,z),(dp,c);
    minpoly = (a3+a+1);
    module M2 = [x2z+x+(-a)*y,z2+(a+2)*x],
                [y2+(a)*z+(a),(a+3)*z3+(-a)*x2],
                [-xz+(a2+3)*yz,xy+(a2)*z];
    nfmodSyz(M2);
    ring r3=0,(x,y),dp; // ring without parameter
    module M3 = [x2 + y, xy], [-7y, 2x], [x2-y, 0];
    nfmodSyz(M3);
    ring r4=0,(x,y),(c,dp); // ring without parameter
    module M4 = [xy, x-y],
                [x2 + y, 5y],
                [- 7y, 2x],
                [x2-y, 0];
    nfmodSyz(M4);
}