Code coverage tests

This page documents the degree to which the PARI/GP source code is tested by our public test suite, distributed with the source distribution in directory src/test/. This is measured by the gcov utility; we then process gcov output using the lcov frond-end.

We test a few variants depending on Configure flags on the pari.math.u-bordeaux.fr machine (x86_64 architecture), and agregate them in the final report:

The target is to exceed 90% coverage for all mathematical modules (given that branches depending on DEBUGLEVEL or DEBUGMEM are not covered). This script is run to produce the results below.

LCOV - code coverage report
Current view: top level - basemath - QX_factor.c (source / functions) Hit Total Coverage
Test: PARI/GP v2.14.0 lcov report (development 27775-aca467eab2) Lines: 772 801 96.4 %
Date: 2022-07-03 07:33:15 Functions: 46 47 97.9 %
Legend: Lines: hit not hit

          Line data    Source code
       1             : /* Copyright (C) 2000  The PARI group.
       2             : 
       3             : This file is part of the PARI/GP package.
       4             : 
       5             : PARI/GP is free software; you can redistribute it and/or modify it under the
       6             : terms of the GNU General Public License as published by the Free Software
       7             : Foundation; either version 2 of the License, or (at your option) any later
       8             : version. It is distributed in the hope that it will be useful, but WITHOUT
       9             : ANY WARRANTY WHATSOEVER.
      10             : 
      11             : Check the License for details. You should have received a copy of it, along
      12             : with the package; see the file 'COPYING'. If not, write to the Free Software
      13             : Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA. */
      14             : #include "pari.h"
      15             : #include "paripriv.h"
      16             : 
      17             : #define DEBUGLEVEL DEBUGLEVEL_factor
      18             : 
      19             : /* x,y two ZX, y non constant. Return q = x/y if y divides x in Z[X] and NULL
      20             :  * otherwise. If not NULL, B is a t_INT upper bound for ||q||_oo. */
      21             : static GEN
      22     6516730 : ZX_divides_i(GEN x, GEN y, GEN B)
      23             : {
      24             :   long dx, dy, dz, i, j;
      25             :   pari_sp av;
      26             :   GEN z,p1,y_lead;
      27             : 
      28     6516730 :   dy=degpol(y);
      29     6516730 :   dx=degpol(x);
      30     6516730 :   dz=dx-dy; if (dz<0) return NULL;
      31     6515666 :   z=cgetg(dz+3,t_POL); z[1] = x[1];
      32     6515664 :   x += 2; y += 2; z += 2;
      33     6515664 :   y_lead = gel(y,dy);
      34     6515664 :   if (equali1(y_lead)) y_lead = NULL;
      35             : 
      36     6515664 :   p1 = gel(x,dx);
      37     6515664 :   if (y_lead) {
      38             :     GEN r;
      39       16058 :     p1 = dvmdii(p1,y_lead, &r);
      40       16058 :     if (r != gen_0) return NULL;
      41             :   }
      42     6499606 :   else p1 = icopy(p1);
      43     6511504 :   gel(z,dz) = p1;
      44     8060613 :   for (i=dx-1; i>=dy; i--)
      45             :   {
      46     1554246 :     av = avma; p1 = gel(x,i);
      47     5423828 :     for (j=i-dy+1; j<=i && j<=dz; j++)
      48     3869646 :       p1 = subii(p1, mulii(gel(z,j),gel(y,i-j)));
      49     1554182 :     if (y_lead) {
      50             :       GEN r;
      51       33887 :       p1 = dvmdii(p1,y_lead, &r);
      52       33887 :       if (r != gen_0) return NULL;
      53             :     }
      54     1552774 :     if (B && abscmpii(p1, B) > 0) return NULL;
      55     1549053 :     p1 = gerepileuptoint(av, p1);
      56     1549109 :     gel(z,i-dy) = p1;
      57             :   }
      58     6506367 :   av = avma;
      59    16920193 :   for (; i >= 0; i--)
      60             :   {
      61    10451884 :     p1 = gel(x,i);
      62             :     /* we always enter this loop at least once */
      63    23463105 :     for (j=0; j<=i && j<=dz; j++)
      64    13011238 :       p1 = subii(p1, mulii(gel(z,j),gel(y,i-j)));
      65    10451867 :     if (signe(p1)) return NULL;
      66    10413828 :     set_avma(av);
      67             :   }
      68     6468309 :   return z - 2;
      69             : }
      70             : static GEN
      71     6490420 : ZX_divides(GEN x, GEN y) { return ZX_divides_i(x,y,NULL); }
      72             : 
      73             : #if 0
      74             : /* cf Beauzamy et al: upper bound for
      75             :  *      lc(x) * [2^(5/8) / pi^(3/8)] e^(1/4n) 2^(n/2) sqrt([x]_2)/ n^(3/8)
      76             :  * where [x]_2 = sqrt(\sum_i=0^n x[i]^2 / binomial(n,i)). One factor has
      77             :  * all coeffs less than then bound */
      78             : static GEN
      79             : two_factor_bound(GEN x)
      80             : {
      81             :   long i, j, n = lg(x) - 3;
      82             :   pari_sp av = avma;
      83             :   GEN *invbin, c, r = cgetr(3), z;
      84             : 
      85             :   x += 2; invbin = (GEN*)new_chunk(n+1);
      86             :   z = real_1(LOWDEFAULTPREC); /* invbin[i] = 1 / binomial(n, i) */
      87             :   for (i=0,j=n; j >= i; i++,j--)
      88             :   {
      89             :     invbin[i] = invbin[j] = z;
      90             :     z = divru(mulru(z, i+1), n-i);
      91             :   }
      92             :   z = invbin[0]; /* = 1 */
      93             :   for (i=0; i<=n; i++)
      94             :   {
      95             :     c = gel(x,i); if (!signe(c)) continue;
      96             :     affir(c, r);
      97             :     z = addrr(z, mulrr(sqrr(r), invbin[i]));
      98             :   }
      99             :   z = shiftr(sqrtr(z), n);
     100             :   z = divrr(z, dbltor(pow((double)n, 0.75)));
     101             :   z = roundr_safe(sqrtr(z));
     102             :   z = mulii(z, absi_shallow(gel(x,n)));
     103             :   return gerepileuptoint(av, shifti(z, 1));
     104             : }
     105             : #endif
     106             : 
     107             : /* A | S ==> |a_i| <= binom(d-1, i-1) || S ||_2 + binom(d-1, i) lc(S) */
     108             : static GEN
     109       57282 : Mignotte_bound(GEN S)
     110             : {
     111       57282 :   long i, d = degpol(S);
     112       57282 :   GEN C, N2, t, binlS, lS = leading_coeff(S), bin = vecbinomial(d-1);
     113             : 
     114       57282 :   N2 = sqrtr(RgX_fpnorml2(S,DEFAULTPREC));
     115       57282 :   binlS = is_pm1(lS)? bin: ZC_Z_mul(bin, lS);
     116             : 
     117             :   /* i = 0 */
     118       57282 :   C = gel(binlS,1);
     119             :   /* i = d */
     120       57282 :   t = N2; if (gcmp(C, t) < 0) C = t;
     121      511395 :   for (i = 1; i < d; i++)
     122             :   {
     123      454116 :     t = addri(mulir(gel(bin,i), N2), gel(binlS,i+1));
     124      454111 :     if (mpcmp(C, t) < 0) C = t;
     125             :   }
     126       57279 :   return C;
     127             : }
     128             : /* A | S ==> |a_i|^2 <= 3^{3/2 + d} / (4 \pi d) [P]_2^2,
     129             :  * where [P]_2 is Bombieri's 2-norm */
     130             : static GEN
     131       57282 : Beauzamy_bound(GEN S)
     132             : {
     133       57282 :   const long prec = DEFAULTPREC;
     134       57282 :   long i, d = degpol(S);
     135             :   GEN bin, lS, s, C;
     136       57282 :   bin = vecbinomial(d);
     137             : 
     138       57282 :   s = real_0(prec);
     139      625963 :   for (i=0; i<=d; i++)
     140             :   {
     141      568681 :     GEN c = gel(S,i+2);
     142      568681 :     if (gequal0(c)) continue;
     143             :     /* s += P_i^2 / binomial(d,i) */
     144      479331 :     s = addrr(s, divri(itor(sqri(c), prec), gel(bin,i+1)));
     145             :   }
     146             :   /* s = [S]_2^2 */
     147       57282 :   C = powruhalf(utor(3,prec), 3 + 2*d); /* 3^{3/2 + d} */
     148       57282 :   C = divrr(mulrr(C, s), mulur(4*d, mppi(prec)));
     149       57282 :   lS = absi_shallow(leading_coeff(S));
     150       57282 :   return mulir(lS, sqrtr(C));
     151             : }
     152             : 
     153             : static GEN
     154       57282 : factor_bound(GEN S)
     155             : {
     156       57282 :   pari_sp av = avma;
     157       57282 :   GEN a = Mignotte_bound(S);
     158       57282 :   GEN b = Beauzamy_bound(S);
     159       57282 :   if (DEBUGLEVEL>2)
     160             :   {
     161           0 :     err_printf("Mignotte bound: %Ps\n",a);
     162           0 :     err_printf("Beauzamy bound: %Ps\n",b);
     163             :   }
     164       57282 :   return gerepileupto(av, ceil_safe(gmin_shallow(a, b)));
     165             : }
     166             : 
     167             : /* Naive recombination of modular factors: combine up to maxK modular
     168             :  * factors, degree <= klim
     169             :  *
     170             :  * target = polynomial we want to factor
     171             :  * famod = array of modular factors.  Product should be congruent to
     172             :  * target/lc(target) modulo p^a
     173             :  * For true factors: S1,S2 <= p^b, with b <= a and p^(b-a) < 2^31 */
     174             : static GEN
     175       47007 : cmbf(GEN pol, GEN famod, GEN bound, GEN p, long a, long b,
     176             :      long klim, long *pmaxK, int *done)
     177             : {
     178       47007 :   long K = 1, cnt = 1, i,j,k, curdeg, lfamod = lg(famod)-1;
     179             :   ulong spa_b, spa_bs2, Sbound;
     180       47007 :   GEN lc, lcpol, pa = powiu(p,a), pas2 = shifti(pa,-1);
     181       47007 :   GEN trace1   = cgetg(lfamod+1, t_VECSMALL);
     182       47007 :   GEN trace2   = cgetg(lfamod+1, t_VECSMALL);
     183       47007 :   GEN ind      = cgetg(lfamod+1, t_VECSMALL);
     184       47007 :   GEN deg      = cgetg(lfamod+1, t_VECSMALL);
     185       47007 :   GEN degsofar = cgetg(lfamod+1, t_VECSMALL);
     186       47007 :   GEN listmod  = cgetg(lfamod+1, t_VEC);
     187       47007 :   GEN fa       = cgetg(lfamod+1, t_VEC);
     188             : 
     189       47007 :   *pmaxK = cmbf_maxK(lfamod);
     190       47007 :   lc = absi_shallow(leading_coeff(pol));
     191       47007 :   if (equali1(lc)) lc = NULL;
     192       47007 :   lcpol = lc? ZX_Z_mul(pol, lc): pol;
     193             : 
     194             :   {
     195       47007 :     GEN pa_b,pa_bs2,pb, lc2 = lc? sqri(lc): NULL;
     196             : 
     197       47007 :     pa_b = powiu(p, a-b); /* < 2^31 */
     198       47007 :     pa_bs2 = shifti(pa_b,-1);
     199       47007 :     pb= powiu(p, b);
     200      167534 :     for (i=1; i <= lfamod; i++)
     201             :     {
     202      120527 :       GEN T1,T2, P = gel(famod,i);
     203      120527 :       long d = degpol(P);
     204             : 
     205      120527 :       deg[i] = d; P += 2;
     206      120527 :       T1 = gel(P,d-1);/* = - S_1 */
     207      120527 :       T2 = sqri(T1);
     208      120527 :       if (d > 1) T2 = subii(T2, shifti(gel(P,d-2),1));
     209      120527 :       T2 = modii(T2, pa); /* = S_2 Newton sum */
     210      120527 :       if (lc)
     211             :       {
     212        3059 :         T1 = Fp_mul(lc, T1, pa);
     213        3059 :         T2 = Fp_mul(lc2,T2, pa);
     214             :       }
     215      120527 :       uel(trace1,i) = itou(diviiround(T1, pb));
     216      120527 :       uel(trace2,i) = itou(diviiround(T2, pb));
     217             :     }
     218       47007 :     spa_b   = uel(pa_b,2); /* < 2^31 */
     219       47007 :     spa_bs2 = uel(pa_bs2,2); /* < 2^31 */
     220             :   }
     221       47007 :   degsofar[0] = 0; /* sentinel */
     222             : 
     223             :   /* ind runs through strictly increasing sequences of length K,
     224             :    * 1 <= ind[i] <= lfamod */
     225       87438 : nextK:
     226       87438 :   if (K > *pmaxK || 2*K > lfamod) goto END;
     227       52564 :   if (DEBUGLEVEL > 3)
     228           0 :     err_printf("\n### K = %d, %Ps combinations\n", K,binomial(utoipos(lfamod), K));
     229       52564 :   setlg(ind, K+1); ind[1] = 1;
     230       52564 :   Sbound = (ulong) ((K+1)>>1);
     231       52564 :   i = 1; curdeg = deg[ind[1]];
     232             :   for(;;)
     233             :   { /* try all combinations of K factors */
     234      618752 :     for (j = i; j < K; j++)
     235             :     {
     236       85684 :       degsofar[j] = curdeg;
     237       85684 :       ind[j+1] = ind[j]+1; curdeg += deg[ind[j+1]];
     238             :     }
     239      533068 :     if (curdeg <= klim) /* trial divide */
     240             :     {
     241             :       GEN y, q, list;
     242             :       pari_sp av;
     243             :       ulong t;
     244             : 
     245             :       /* d - 1 test */
     246     1348709 :       for (t=uel(trace1,ind[1]),i=2; i<=K; i++)
     247      815641 :         t = Fl_add(t, uel(trace1,ind[i]), spa_b);
     248      533068 :       if (t > spa_bs2) t = spa_b - t;
     249      533068 :       if (t > Sbound)
     250             :       {
     251      427208 :         if (DEBUGLEVEL>6) err_printf(".");
     252      427208 :         goto NEXT;
     253             :       }
     254             :       /* d - 2 test */
     255      231092 :       for (t=uel(trace2,ind[1]),i=2; i<=K; i++)
     256      125232 :         t = Fl_add(t, uel(trace2,ind[i]), spa_b);
     257      105860 :       if (t > spa_bs2) t = spa_b - t;
     258      105860 :       if (t > Sbound)
     259             :       {
     260       57228 :         if (DEBUGLEVEL>6) err_printf("|");
     261       57228 :         goto NEXT;
     262             :       }
     263             : 
     264       48632 :       av = avma;
     265             :       /* check trailing coeff */
     266       48632 :       y = lc;
     267      145402 :       for (i=1; i<=K; i++)
     268             :       {
     269       96770 :         GEN q = constant_coeff(gel(famod,ind[i]));
     270       96770 :         if (y) q = mulii(y, q);
     271       96770 :         y = centermodii(q, pa, pas2);
     272             :       }
     273       48632 :       if (!signe(y) || !dvdii(constant_coeff(lcpol), y))
     274             :       {
     275       22482 :         if (DEBUGLEVEL>3) err_printf("T");
     276       22482 :         set_avma(av); goto NEXT;
     277             :       }
     278       26150 :       y = lc; /* full computation */
     279       58905 :       for (i=1; i<=K; i++)
     280             :       {
     281       32755 :         GEN q = gel(famod,ind[i]);
     282       32755 :         if (y) q = gmul(y, q);
     283       32755 :         y = centermod_i(q, pa, pas2);
     284             :       }
     285             : 
     286             :       /* y is the candidate factor */
     287       26150 :       if (! (q = ZX_divides_i(lcpol,y,bound)) )
     288             :       {
     289        3749 :         if (DEBUGLEVEL>3) err_printf("*");
     290        3749 :         set_avma(av); goto NEXT;
     291             :       }
     292             :       /* found a factor */
     293       22401 :       list = cgetg(K+1, t_VEC);
     294       22401 :       gel(listmod,cnt) = list;
     295       45509 :       for (i=1; i<=K; i++) list[i] = famod[ind[i]];
     296             : 
     297       22401 :       y = Q_primpart(y);
     298       22401 :       gel(fa,cnt++) = y;
     299             :       /* fix up pol */
     300       22401 :       pol = q;
     301       22401 :       if (lc) pol = Q_div_to_int(pol, leading_coeff(y));
     302       84338 :       for (i=j=k=1; i <= lfamod; i++)
     303             :       { /* remove used factors */
     304       61937 :         if (j <= K && i == ind[j]) j++;
     305             :         else
     306             :         {
     307       38829 :           gel(famod,k) = gel(famod,i);
     308       38829 :           uel(trace1,k) = uel(trace1,i);
     309       38829 :           uel(trace2,k) = uel(trace2,i);
     310       38829 :           deg[k] = deg[i]; k++;
     311             :         }
     312             :       }
     313       22401 :       lfamod -= K;
     314       22401 :       *pmaxK = cmbf_maxK(lfamod);
     315       22401 :       if (lfamod < 2*K) goto END;
     316       10268 :       i = 1; curdeg = deg[ind[1]];
     317       10268 :       bound = factor_bound(pol);
     318       10268 :       if (lc) lc = absi_shallow(leading_coeff(pol));
     319       10268 :       lcpol = lc? ZX_Z_mul(pol, lc): pol;
     320       10268 :       if (DEBUGLEVEL>3)
     321           0 :         err_printf("\nfound factor %Ps\nremaining modular factor(s): %ld\n",
     322             :                    y, lfamod);
     323       10268 :       continue;
     324             :     }
     325             : 
     326           0 : NEXT:
     327      510667 :     for (i = K+1;;)
     328             :     {
     329      636075 :       if (--i == 0) { K++; goto nextK; }
     330      595644 :       if (++ind[i] <= lfamod - K + i)
     331             :       {
     332      470236 :         curdeg = degsofar[i-1] + deg[ind[i]];
     333      470236 :         if (curdeg <= klim) break;
     334             :       }
     335             :     }
     336             :   }
     337       47007 : END:
     338       47007 :   *done = 1;
     339       47007 :   if (degpol(pol) > 0)
     340             :   { /* leftover factor */
     341       47007 :     if (signe(leading_coeff(pol)) < 0) pol = ZX_neg(pol);
     342       47007 :     if (lfamod >= 2*K) *done = 0;
     343             : 
     344       47007 :     setlg(famod, lfamod+1);
     345       47007 :     gel(listmod,cnt) = leafcopy(famod);
     346       47007 :     gel(fa,cnt++) = pol;
     347             :   }
     348       47007 :   if (DEBUGLEVEL>6) err_printf("\n");
     349       47007 :   setlg(listmod, cnt);
     350       47007 :   setlg(fa, cnt); return mkvec2(fa, listmod);
     351             : }
     352             : 
     353             : /* recombination of modular factors: van Hoeij's algorithm */
     354             : 
     355             : /* Q in Z[X], return Q(2^n) */
     356             : static GEN
     357      161548 : shifteval(GEN Q, long n)
     358             : {
     359      161548 :   pari_sp av = avma;
     360      161548 :   long i, l = lg(Q);
     361             :   GEN s;
     362             : 
     363      161548 :   if (!signe(Q)) return gen_0;
     364      161548 :   s = gel(Q,l-1);
     365      910521 :   for (i = l-2; i > 1; i--)
     366             :   {
     367      748973 :     s = addii(gel(Q,i), shifti(s, n));
     368      748973 :     if (gc_needed(av,1)) s = gerepileuptoint(av, s);
     369             :   }
     370      161548 :   return s;
     371             : }
     372             : 
     373             : /* return integer y such that all |a| <= y if P(a) = 0 */
     374             : static GEN
     375       97185 : root_bound(GEN P0)
     376             : {
     377       97185 :   GEN Q = leafcopy(P0), lP = absi_shallow(leading_coeff(Q)), x,y,z;
     378       97185 :   long k, d = degpol(Q);
     379             : 
     380             :   /* P0 = lP x^d + Q, deg Q < d */
     381       97185 :   Q = normalizepol_lg(Q, d+2);
     382      645939 :   for (k=lg(Q)-1; k>1; k--) gel(Q,k) = absi_shallow(gel(Q,k));
     383       97185 :   k = (long)(fujiwara_bound(P0));
     384      161878 :   for (  ; k >= 0; k--)
     385             :   {
     386      161549 :     pari_sp av = avma;
     387             :     /* y = 2^k; Q(y) >= lP y^d ? */
     388      161549 :     if (cmpii(shifteval(Q,k), shifti(lP, d*k)) >= 0) break;
     389       64693 :     set_avma(av);
     390             :   }
     391       97185 :   if (k < 0) k = 0;
     392       97185 :   y = int2n(k+1);
     393       97185 :   if (d > 2000) return y; /* likely to be expensive, don't bother */
     394       97185 :   x = int2n(k);
     395       97185 :   for(k=0; ; k++)
     396             :   {
     397      532159 :     z = shifti(addii(x,y), -1);
     398      532159 :     if (equalii(x,z) || k > 5) break;
     399      434974 :     if (cmpii(ZX_Z_eval(Q,z), mulii(lP, powiu(z, d))) < 0)
     400      232639 :       y = z;
     401             :     else
     402      202335 :       x = z;
     403             :   }
     404       97185 :   return y;
     405             : }
     406             : 
     407             : GEN
     408         350 : chk_factors_get(GEN lt, GEN famod, GEN c, GEN T, GEN N)
     409             : {
     410         350 :   long i = 1, j, l = lg(famod);
     411         350 :   GEN V = cgetg(l, t_VEC);
     412        8414 :   for (j = 1; j < l; j++)
     413        8064 :     if (signe(gel(c,j))) gel(V,i++) = gel(famod,j);
     414         350 :   if (lt && i > 1) gel(V,1) = RgX_Rg_mul(gel(V,1), lt);
     415         350 :   setlg(V, i);
     416         350 :   return T? FpXQXV_prod(V, T, N): FpXV_prod(V,N);
     417             : }
     418             : 
     419             : static GEN
     420         140 : chk_factors(GEN P, GEN M_L, GEN bound, GEN famod, GEN pa)
     421             : {
     422             :   long i, r;
     423         140 :   GEN pol = P, list, piv, y, ltpol, lt, paov2;
     424             : 
     425         140 :   piv = ZM_hnf_knapsack(M_L);
     426         140 :   if (!piv) return NULL;
     427          70 :   if (DEBUGLEVEL>7) err_printf("ZM_hnf_knapsack output:\n%Ps\n",piv);
     428             : 
     429          70 :   r  = lg(piv)-1;
     430          70 :   list = cgetg(r+1, t_VEC);
     431          70 :   lt = absi_shallow(leading_coeff(pol));
     432          70 :   if (equali1(lt)) lt = NULL;
     433          70 :   ltpol = lt? ZX_Z_mul(pol, lt): pol;
     434          70 :   paov2 = shifti(pa,-1);
     435          70 :   for (i = 1;;)
     436             :   {
     437         161 :     if (DEBUGLEVEL) err_printf("LLL_cmbf: checking factor %ld\n",i);
     438         161 :     y = chk_factors_get(lt, famod, gel(piv,i), NULL, pa);
     439         161 :     y = FpX_center_i(y, pa, paov2);
     440         161 :     if (! (pol = ZX_divides_i(ltpol,y,bound)) ) return NULL;
     441         133 :     if (lt) y = Q_primpart(y);
     442         133 :     gel(list,i) = y;
     443         133 :     if (++i >= r) break;
     444             : 
     445          91 :     if (lt)
     446             :     {
     447          35 :       pol = ZX_Z_divexact(pol, leading_coeff(y));
     448          35 :       lt = absi_shallow(leading_coeff(pol));
     449          35 :       ltpol = ZX_Z_mul(pol, lt);
     450             :     }
     451             :     else
     452          56 :       ltpol = pol;
     453             :   }
     454          42 :   y = Q_primpart(pol);
     455          42 :   gel(list,i) = y; return list;
     456             : }
     457             : 
     458             : GEN
     459        1547 : LLL_check_progress(GEN Bnorm, long n0, GEN m, int final, long *ti_LLL)
     460             : {
     461             :   GEN norm, u;
     462             :   long i, R;
     463             :   pari_timer T;
     464             : 
     465        1547 :   if (DEBUGLEVEL>2) timer_start(&T);
     466        1547 :   u = ZM_lll_norms(m, final? 0.999: 0.75, LLL_INPLACE, &norm);
     467        1547 :   if (DEBUGLEVEL>2) *ti_LLL += timer_delay(&T);
     468       10542 :   for (R=lg(m)-1; R > 0; R--)
     469       10542 :     if (cmprr(gel(norm,R), Bnorm) < 0) break;
     470       16156 :   for (i=1; i<=R; i++) setlg(u[i], n0+1);
     471        1547 :   if (R <= 1)
     472             :   {
     473         105 :     if (!R) pari_err_BUG("LLL_cmbf [no factor]");
     474         105 :     return NULL; /* irreducible */
     475             :   }
     476        1442 :   setlg(u, R+1); return u;
     477             : }
     478             : 
     479             : static ulong
     480          14 : next2pow(ulong a)
     481             : {
     482          14 :   ulong b = 1;
     483         112 :   while (b < a) b <<= 1;
     484          14 :   return b;
     485             : }
     486             : 
     487             : /* Recombination phase of Berlekamp-Zassenhaus algorithm using a variant of
     488             :  * van Hoeij's knapsack
     489             :  *
     490             :  * P = squarefree in Z[X].
     491             :  * famod = array of (lifted) modular factors mod p^a
     492             :  * bound = Mignotte bound for the size of divisors of P (for the sup norm)
     493             :  * previously recombined all set of factors with less than rec elts */
     494             : static GEN
     495         126 : LLL_cmbf(GEN P, GEN famod, GEN p, GEN pa, GEN bound, long a, long rec)
     496             : {
     497         126 :   const long N0 = 1; /* # of traces added at each step */
     498         126 :   double BitPerFactor = 0.4; /* nb bits in p^(a-b) / modular factor */
     499         126 :   long i,j,tmax,n0,C, dP = degpol(P);
     500         126 :   double logp = log((double)itos(p)), LOGp2 = M_LN2/logp;
     501         126 :   double b0 = log((double)dP*2) / logp, logBr;
     502             :   GEN lP, Br, Bnorm, Tra, T2, TT, CM_L, m, list, ZERO;
     503             :   pari_sp av, av2;
     504         126 :   long ti_LLL = 0, ti_CF  = 0;
     505             : 
     506         126 :   lP = absi_shallow(leading_coeff(P));
     507         126 :   if (equali1(lP)) lP = NULL;
     508         126 :   Br = root_bound(P);
     509         126 :   if (lP) Br = mulii(lP, Br);
     510         126 :   logBr = gtodouble(glog(Br, DEFAULTPREC)) / logp;
     511             : 
     512         126 :   n0 = lg(famod) - 1;
     513         126 :   C = (long)ceil( sqrt(N0 * n0 / 4.) ); /* > 1 */
     514         126 :   Bnorm = dbltor(n0 * (C*C + N0*n0/4.) * 1.00001);
     515         126 :   ZERO = zeromat(n0, N0);
     516             : 
     517         126 :   av = avma;
     518         126 :   TT = cgetg(n0+1, t_VEC);
     519         126 :   Tra  = cgetg(n0+1, t_MAT);
     520        2401 :   for (i=1; i<=n0; i++)
     521             :   {
     522        2275 :     TT[i]  = 0;
     523        2275 :     gel(Tra,i) = cgetg(N0+1, t_COL);
     524             :   }
     525         126 :   CM_L = scalarmat_s(C, n0);
     526             :   /* tmax = current number of traces used (and computed so far) */
     527         126 :   for (tmax = 0;; tmax += N0)
     528         469 :   {
     529         595 :     long b, bmin, bgood, delta, tnew = tmax + N0, r = lg(CM_L)-1;
     530             :     GEN M_L, q, CM_Lp, oldCM_L;
     531         595 :     int first = 1;
     532             :     pari_timer ti2, TI;
     533             : 
     534         595 :     bmin = (long)ceil(b0 + tnew*logBr);
     535         595 :     if (DEBUGLEVEL>2)
     536           0 :       err_printf("\nLLL_cmbf: %ld potential factors (tmax = %ld, bmin = %ld)\n",
     537             :                  r, tmax, bmin);
     538             : 
     539             :     /* compute Newton sums (possibly relifting first) */
     540         595 :     if (a <= bmin)
     541             :     {
     542          14 :       a = (long)ceil(bmin + 3*N0*logBr) + 1; /* enough for 3 more rounds */
     543          14 :       a = (long)next2pow((ulong)a);
     544             : 
     545          14 :       pa = powiu(p,a);
     546          14 :       famod = ZpX_liftfact(P, famod, pa, p, a);
     547         266 :       for (i=1; i<=n0; i++) TT[i] = 0;
     548             :     }
     549       11557 :     for (i=1; i<=n0; i++)
     550             :     {
     551       10962 :       GEN p1 = gel(Tra,i);
     552       10962 :       GEN p2 = polsym_gen(gel(famod,i), gel(TT,i), tnew, NULL, pa);
     553       10962 :       gel(TT,i) = p2;
     554       10962 :       p2 += 1+tmax; /* ignore traces number 0...tmax */
     555       21924 :       for (j=1; j<=N0; j++) gel(p1,j) = gel(p2,j);
     556       10962 :       if (lP)
     557             :       { /* make Newton sums integral */
     558        1848 :         GEN lPpow = powiu(lP, tmax);
     559        3696 :         for (j=1; j<=N0; j++)
     560             :         {
     561        1848 :           lPpow = mulii(lPpow,lP);
     562        1848 :           gel(p1,j) = mulii(gel(p1,j), lPpow);
     563             :         }
     564             :       }
     565             :     }
     566             : 
     567             :     /* compute truncation parameter */
     568         595 :     if (DEBUGLEVEL>2) { timer_start(&ti2); timer_start(&TI); }
     569         595 :     oldCM_L = CM_L;
     570         595 :     av2 = avma;
     571         595 :     delta = b = 0; /* -Wall */
     572        1071 : AGAIN:
     573        1071 :     M_L = Q_div_to_int(CM_L, utoipos(C));
     574        1071 :     T2 = centermod( ZM_mul(Tra, M_L), pa );
     575        1071 :     if (first)
     576             :     { /* initialize lattice, using few p-adic digits for traces */
     577         595 :       double t = gexpo(T2) - maxdd(32.0, BitPerFactor*r);
     578         595 :       bgood = (long) (t * LOGp2);
     579         595 :       b = maxss(bmin, bgood);
     580         595 :       delta = a - b;
     581             :     }
     582             :     else
     583             :     { /* add more p-adic digits and continue reduction */
     584         476 :       long b0 = (long)(gexpo(T2) * LOGp2);
     585         476 :       if (b0 < b) b = b0;
     586         476 :       b = maxss(b-delta, bmin);
     587         476 :       if (b - delta/2 < bmin) b = bmin; /* near there. Go all the way */
     588             :     }
     589             : 
     590        1071 :     q = powiu(p, b);
     591        1071 :     m = vconcat( CM_L, gdivround(T2, q) );
     592        1071 :     if (first)
     593             :     {
     594         595 :       GEN P1 = scalarmat(powiu(p, a-b), N0);
     595         595 :       first = 0;
     596         595 :       m = shallowconcat( m, vconcat(ZERO, P1) );
     597             :       /*     [ C M_L        0     ]
     598             :        * m = [                    ]   square matrix
     599             :        *     [  T2'  p^(a-b) I_N0 ]   T2' = Tra * M_L  truncated
     600             :        */
     601             :     }
     602             : 
     603        1071 :     CM_L = LLL_check_progress(Bnorm, n0, m, b == bmin, /*dbg:*/ &ti_LLL);
     604        1071 :     if (DEBUGLEVEL>2)
     605           0 :       err_printf("LLL_cmbf: (a,b) =%4ld,%4ld; r =%3ld -->%3ld, time = %ld\n",
     606           0 :                  a,b, lg(m)-1, CM_L? lg(CM_L)-1: 1, timer_delay(&TI));
     607        1071 :     if (!CM_L) { list = mkvec(P); break; }
     608         987 :     if (b > bmin)
     609             :     {
     610         476 :       CM_L = gerepilecopy(av2, CM_L);
     611         476 :       goto AGAIN;
     612             :     }
     613         511 :     if (DEBUGLEVEL>2) timer_printf(&ti2, "for this block of traces");
     614             : 
     615         511 :     i = lg(CM_L) - 1;
     616         511 :     if (i == r && ZM_equal(CM_L, oldCM_L))
     617             :     {
     618          63 :       CM_L = oldCM_L;
     619          63 :       set_avma(av2); continue;
     620             :     }
     621             : 
     622         448 :     CM_Lp = FpM_image(CM_L, utoipos(27449)); /* inexpensive test */
     623         448 :     if (lg(CM_Lp) != lg(CM_L))
     624             :     {
     625           7 :       if (DEBUGLEVEL>2) err_printf("LLL_cmbf: rank decrease\n");
     626           7 :       CM_L = ZM_hnf(CM_L);
     627             :     }
     628             : 
     629         448 :     if (i <= r && i*rec < n0)
     630             :     {
     631             :       pari_timer ti;
     632         140 :       if (DEBUGLEVEL>2) timer_start(&ti);
     633         140 :       list = chk_factors(P, Q_div_to_int(CM_L,utoipos(C)), bound, famod, pa);
     634         140 :       if (DEBUGLEVEL>2) ti_CF += timer_delay(&ti);
     635         140 :       if (list) break;
     636          98 :       if (DEBUGLEVEL>2) err_printf("LLL_cmbf: chk_factors failed");
     637             :     }
     638         406 :     CM_L = gerepilecopy(av2, CM_L);
     639         406 :     if (gc_needed(av,1))
     640             :     {
     641           0 :       if(DEBUGMEM>1) pari_warn(warnmem,"LLL_cmbf");
     642           0 :       gerepileall(av, 5, &CM_L, &TT, &Tra, &famod, &pa);
     643             :     }
     644             :   }
     645         126 :   if (DEBUGLEVEL>2)
     646           0 :     err_printf("* Time LLL: %ld\n* Time Check Factor: %ld\n",ti_LLL,ti_CF);
     647         126 :   return list;
     648             : }
     649             : 
     650             : /* Find a,b minimal such that A < q^a, B < q^b, 1 << q^(a-b) < 2^31 */
     651             : static int
     652       47007 : cmbf_precs(GEN q, GEN A, GEN B, long *pta, long *ptb, GEN *qa, GEN *qb)
     653             : {
     654       47007 :   long a,b,amin,d = (long)(31 * M_LN2/gtodouble(glog(q,DEFAULTPREC)) - 1e-5);
     655       47007 :   int fl = 0;
     656             : 
     657       47007 :   b = logintall(B, q, qb) + 1;
     658       47007 :   *qb = mulii(*qb, q);
     659       47007 :   amin = b + d;
     660       47007 :   if (gcmp(powiu(q, amin), A) <= 0)
     661             :   {
     662       13902 :     a = logintall(A, q, qa) + 1;
     663       13902 :     *qa = mulii(*qa, q);
     664       13902 :     b = a - d; *qb = powiu(q, b);
     665             :   }
     666             :   else
     667             :   { /* not enough room */
     668       33105 :     a = amin;  *qa = powiu(q, a);
     669       33105 :     fl = 1;
     670             :   }
     671       47007 :   if (DEBUGLEVEL > 3) {
     672           0 :     err_printf("S_2   bound: %Ps^%ld\n", q,b);
     673           0 :     err_printf("coeff bound: %Ps^%ld\n", q,a);
     674             :   }
     675       47007 :   *pta = a;
     676       47007 :   *ptb = b; return fl;
     677             : }
     678             : 
     679             : /* use van Hoeij's knapsack algorithm */
     680             : static GEN
     681       47007 : combine_factors(GEN target, GEN famod, GEN p, long klim)
     682             : {
     683             :   GEN la, B, A, res, L, pa, pb, listmod;
     684       47007 :   long a,b, l, maxK, n = degpol(target);
     685             :   int done;
     686             :   pari_timer T;
     687             : 
     688       47007 :   A = factor_bound(target);
     689             : 
     690       47007 :   la = absi_shallow(leading_coeff(target));
     691       47007 :   B = mului(n, sqri(mulii(la, root_bound(target)))); /* = bound for S_2 */
     692             : 
     693       47007 :   (void)cmbf_precs(p, A, B, &a, &b, &pa, &pb);
     694             : 
     695       47007 :   if (DEBUGLEVEL>2) timer_start(&T);
     696       47007 :   famod = ZpX_liftfact(target, famod, pa, p, a);
     697       47007 :   if (DEBUGLEVEL>2) timer_printf(&T, "Hensel lift (mod %Ps^%ld)", p,a);
     698       47007 :   L = cmbf(target, famod, A, p, a, b, klim, &maxK, &done);
     699       47007 :   if (DEBUGLEVEL>2) timer_printf(&T, "Naive recombination");
     700             : 
     701       47007 :   res     = gel(L,1);
     702       47007 :   listmod = gel(L,2); l = lg(listmod)-1;
     703       47007 :   famod = gel(listmod,l);
     704       47007 :   if (maxK > 0 && lg(famod)-1 > 2*maxK)
     705             :   {
     706         126 :     if (l!=1) A = factor_bound(gel(res,l));
     707         126 :     if (DEBUGLEVEL > 4) err_printf("last factor still to be checked\n");
     708         126 :     L = LLL_cmbf(gel(res,l), famod, p, pa, A, a, maxK);
     709         126 :     if (DEBUGLEVEL>2) timer_printf(&T,"Knapsack");
     710             :     /* remove last elt, possibly unfactored. Add all new ones. */
     711         126 :     setlg(res, l); res = shallowconcat(res, L);
     712             :   }
     713       47007 :   return res;
     714             : }
     715             : 
     716             : /* Assume 'a' a squarefree ZX; return 0 if no root (fl=1) / irreducible (fl=0).
     717             :  * Otherwise return prime p such that a mod p has fewest roots / factors */
     718             : static ulong
     719      838886 : pick_prime(GEN a, long fl, pari_timer *T)
     720             : {
     721      838886 :   pari_sp av = avma, av1;
     722      838886 :   const long MAXNP = 7, da = degpol(a);
     723      838886 :   long nmax = da+1, np;
     724      838886 :   ulong chosenp = 0;
     725      838886 :   GEN lead = gel(a,da+2);
     726             :   forprime_t S;
     727      838886 :   if (equali1(lead)) lead = NULL;
     728      838886 :   u_forprime_init(&S, 2, ULONG_MAX);
     729      838885 :   av1 = avma;
     730     4596164 :   for (np = 0; np < MAXNP; set_avma(av1))
     731             :   {
     732     4499100 :     ulong p = u_forprime_next(&S);
     733             :     long nfacp;
     734             :     GEN z;
     735             : 
     736     4499098 :     if (!p) pari_err_OVERFLOW("DDF [out of small primes]");
     737     4499098 :     if (lead && !umodiu(lead,p)) continue;
     738     4443544 :     z = ZX_to_Flx(a, p);
     739     4443541 :     if (!Flx_is_squarefree(z, p)) continue;
     740             : 
     741     2808920 :     if (fl)
     742             :     {
     743     2248034 :       nfacp = Flx_nbroots(z, p);
     744     2248034 :       if (!nfacp) { chosenp = 0; break; } /* no root */
     745             :     }
     746             :     else
     747             :     {
     748      560886 :       nfacp = Flx_nbfact(z, p);
     749      560892 :       if (nfacp == 1) { chosenp = 0; break; } /* irreducible */
     750             :     }
     751     2067098 :     if (DEBUGLEVEL>4)
     752           0 :       err_printf("...tried prime %3lu (%-3ld %s). Time = %ld\n",
     753             :                   p, nfacp, fl? "roots": "factors", timer_delay(T));
     754     2067104 :     if (nfacp < nmax)
     755             :     {
     756      666212 :       nmax = nfacp; chosenp = p;
     757      666212 :       if (da > 100 && nmax < 5) break; /* large degree, few factors. Enough */
     758             :     }
     759     2067097 :     np++;
     760             :   }
     761      838897 :   return gc_ulong(av, chosenp);
     762             : }
     763             : 
     764             : /* Assume A a squarefree ZX; return the vector of its rational roots */
     765             : static GEN
     766      643586 : DDF_roots(GEN A)
     767             : {
     768             :   GEN p, lc, lcpol, z, pe, pes2, bound;
     769             :   long i, m, e, lz;
     770             :   ulong pp;
     771             :   pari_sp av;
     772             :   pari_timer T;
     773             : 
     774      643586 :   if (DEBUGLEVEL>2) timer_start(&T);
     775      643586 :   pp = pick_prime(A, 1, &T);
     776      643586 :   if (!pp) return cgetg(1,t_VEC); /* no root */
     777       50052 :   p = utoipos(pp);
     778       50052 :   lc = leading_coeff(A);
     779       50052 :   if (is_pm1(lc))
     780       46586 :   { lc = NULL; lcpol = A; }
     781             :   else
     782        3466 :   { lc = absi_shallow(lc); lcpol = ZX_Z_mul(A, lc); }
     783       50052 :   bound = root_bound(A); if (lc) bound = mulii(lc, bound);
     784       50052 :   e = logintall(addiu(shifti(bound, 1), 1), p, &pe) + 1;
     785       50052 :   pe = mulii(pe, p);
     786       50052 :   pes2 = shifti(pe, -1);
     787       50052 :   if (DEBUGLEVEL>2) timer_printf(&T, "Root bound");
     788       50052 :   av = avma;
     789       50052 :   z = ZpX_roots(A, p, e); lz = lg(z);
     790       50052 :   z = deg1_from_roots(z, varn(A));
     791       50052 :   if (DEBUGLEVEL>2) timer_printf(&T, "Hensel lift (mod %lu^%ld)", pp,e);
     792      107923 :   for (m=1, i=1; i < lz; i++)
     793             :   {
     794       57871 :     GEN q, r, y = gel(z,i);
     795       57871 :     if (lc) y = ZX_Z_mul(y, lc);
     796       57871 :     y = centermod_i(y, pe, pes2);
     797       57871 :     if (! (q = ZX_divides(lcpol, y)) ) continue;
     798             : 
     799       18575 :     lcpol = q;
     800       18575 :     r = negi( constant_coeff(y) );
     801       18575 :     if (lc) {
     802        2754 :       r = gdiv(r,lc);
     803        2754 :       lcpol = Q_primpart(lcpol);
     804        2754 :       lc = absi_shallow( leading_coeff(lcpol) );
     805        2754 :       if (is_pm1(lc)) lc = NULL; else lcpol = ZX_Z_mul(lcpol, lc);
     806             :     }
     807       18575 :     gel(z,m++) = r;
     808       18575 :     if (gc_needed(av,2))
     809             :     {
     810           0 :       if (DEBUGMEM>1) pari_warn(warnmem,"DDF_roots, m = %ld", m);
     811           0 :       gerepileall(av, lc? 3:2, &z, &lcpol, &lc);
     812             :     }
     813             :   }
     814       50052 :   if (DEBUGLEVEL>2) timer_printf(&T, "Recombination");
     815       50052 :   z[0] = evaltyp(t_VEC) | evallg(m); return z;
     816             : }
     817             : 
     818             : /* Assume a squarefree ZX, deg(a) > 0, return rational factors.
     819             :  * In fact, a(0) != 0 but we don't use this */
     820             : static GEN
     821      195301 : DDF(GEN a)
     822             : {
     823             :   GEN ap, prime, famod, z;
     824      195301 :   long ti = 0;
     825      195301 :   ulong p = 0;
     826      195301 :   pari_sp av = avma;
     827             :   pari_timer T, T2;
     828             : 
     829      195301 :   if (DEBUGLEVEL>2) { timer_start(&T); timer_start(&T2); }
     830      195301 :   p = pick_prime(a, 0, &T2);
     831      195301 :   if (!p) return mkvec(a);
     832       47007 :   prime = utoipos(p);
     833       47007 :   ap = Flx_normalize(ZX_to_Flx(a, p), p);
     834       47007 :   famod = gel(Flx_factor(ap, p), 1);
     835       47007 :   if (DEBUGLEVEL>2)
     836             :   {
     837           0 :     if (DEBUGLEVEL>4) timer_printf(&T2, "splitting mod p = %lu", p);
     838           0 :     ti = timer_delay(&T);
     839           0 :     err_printf("Time setup: %ld\n", ti);
     840             :   }
     841       47007 :   z = combine_factors(a, FlxV_to_ZXV(famod), prime, degpol(a)-1);
     842       47007 :   if (DEBUGLEVEL>2)
     843           0 :     err_printf("Total Time: %ld\n===========\n", ti + timer_delay(&T));
     844       47007 :   return gerepilecopy(av, z);
     845             : }
     846             : 
     847             : /* Distinct Degree Factorization (deflating first)
     848             :  * Assume x squarefree, degree(x) > 0, x(0) != 0 */
     849             : GEN
     850      147165 : ZX_DDF(GEN x)
     851             : {
     852             :   GEN L;
     853             :   long m;
     854      147165 :   x = ZX_deflate_max(x, &m);
     855      147167 :   L = DDF(x);
     856      147168 :   if (m > 1)
     857             :   {
     858       46103 :     GEN e, v, fa = factoru(m);
     859             :     long i,j,k, l;
     860             : 
     861       46103 :     e = gel(fa,2); k = 0;
     862       46103 :     fa= gel(fa,1); l = lg(fa);
     863       92529 :     for (i=1; i<l; i++) k += e[i];
     864       46103 :     v = cgetg(k+1, t_VECSMALL); k = 1;
     865       92529 :     for (i=1; i<l; i++)
     866       94189 :       for (j=1; j<=e[i]; j++) v[k++] = fa[i];
     867       93866 :     for (k--; k; k--)
     868             :     {
     869       47763 :       GEN L2 = cgetg(1,t_VEC);
     870       95897 :       for (i=1; i < lg(L); i++)
     871       48134 :               L2 = shallowconcat(L2, DDF(RgX_inflate(gel(L,i), v[k])));
     872       47763 :       L = L2;
     873             :     }
     874             :   }
     875      147168 :   return L;
     876             : }
     877             : 
     878             : /* SquareFree Factorization in Z[X] (char 0 is enough, if ZX_gcd -> RgX_gcd)
     879             :  * f = prod Q[i]^E[i], E[1] < E[2] < ..., and Q[i] squarefree and coprime.
     880             :  * Return Q, set *pE = E. For efficiency, caller should have used ZX_valrem
     881             :  * so that f(0) != 0 */
     882             : GEN
     883      109742 : ZX_squff(GEN f, GEN *pE)
     884             : {
     885             :   GEN T, V, P, E;
     886      109742 :   long i, k, n = 1 + degpol(f);
     887             : 
     888      109742 :   if (signe(leading_coeff(f)) < 0) f = ZX_neg(f);
     889      109742 :   E = cgetg(n, t_VECSMALL);
     890      109742 :   P = cgetg(n, t_COL);
     891      109742 :   T = ZX_gcd_all(f, ZX_deriv(f), &V);
     892      109743 :   for (k = i = 1;; k++)
     893        3620 :   {
     894      113363 :     GEN W = ZX_gcd_all(T,V, &T); /* V and W are squarefree */
     895      113363 :     long dW = degpol(W), dV = degpol(V);
     896             :     /* f = prod_i T_i^{e_i}
     897             :      * W = prod_{i: e_i > k} T_i,
     898             :      * V = prod_{i: e_i >= k} T_i,
     899             :      * T = prod_{i: e_i > k} T_i^{e_i - k} */
     900      113363 :     if (!dW)
     901             :     {
     902      109743 :       if (dV) { gel(P,i) = Q_primpart(V); E[i] = k; i++; }
     903      109743 :       break;
     904             :     }
     905        3620 :     if (dW == dV)
     906             :     {
     907             :       GEN U;
     908        1659 :       while ( (U = ZX_divides(T, V)) ) { k++; T = U; }
     909             :     }
     910             :     else
     911             :     {
     912        2367 :       gel(P,i) = Q_primpart(RgX_div(V,W));
     913        2367 :       E[i] = k; i++; V = W;
     914             :     }
     915             :   }
     916      109743 :   setlg(P,i);
     917      109743 :   setlg(E,i); *pE = E; return P;
     918             : }
     919             : 
     920             : static GEN
     921       47510 : fact_from_DDF(GEN Q, GEN E, long n)
     922             : {
     923       47510 :   GEN v,w, y = cgetg(3, t_MAT);
     924       47510 :   long i,j,k, l = lg(Q);
     925             : 
     926       47510 :   v = cgetg(n+1, t_COL); gel(y,1) = v;
     927       47510 :   w = cgetg(n+1, t_COL); gel(y,2) = w;
     928       96805 :   for (k = i = 1; i < l; i++)
     929             :   {
     930       49295 :     GEN L = gel(Q,i), e = utoipos(E[i]);
     931       49295 :     long J = lg(L);
     932      120683 :     for (j = 1; j < J; j++,k++)
     933             :     {
     934       71388 :       gel(v,k) = ZX_copy(gel(L,j));
     935       71388 :       gel(w,k) = e;
     936             :     }
     937             :   }
     938       47510 :   return y;
     939             : }
     940             : 
     941             : /* Factor T in Z[x] */
     942             : static GEN
     943       47516 : ZX_factor_i(GEN T)
     944             : {
     945             :   GEN Q, E, y;
     946             :   long n, i, l, v;
     947             : 
     948       47516 :   if (!signe(T)) return prime_fact(T);
     949       47509 :   v = ZX_valrem(T, &T);
     950       47509 :   Q = ZX_squff(T, &E); l = lg(Q);
     951       95265 :   for (i = 1, n = 0; i < l; i++)
     952             :   {
     953       47755 :     gel(Q,i) = ZX_DDF(gel(Q,i));
     954       47755 :     n += lg(gel(Q,i)) - 1;
     955             :   }
     956       47510 :   if (v)
     957             :   {
     958        1540 :     Q = vec_append(Q, mkvec(pol_x(varn(T))));
     959        1540 :     E = vecsmall_append(E, v); n++;
     960             :   }
     961       47510 :   y = fact_from_DDF(Q, E, n);
     962       47510 :   return sort_factor_pol(y, cmpii);
     963             : }
     964             : GEN
     965       46788 : ZX_factor(GEN x)
     966             : {
     967       46788 :   pari_sp av = avma;
     968       46788 :   return gerepileupto(av, ZX_factor_i(x));
     969             : }
     970             : GEN
     971         728 : QX_factor(GEN x)
     972             : {
     973         728 :   pari_sp av = avma;
     974         728 :   return gerepileupto(av, ZX_factor_i(Q_primpart(x)));
     975             : }
     976             : 
     977             : long
     978       99729 : ZX_is_irred(GEN x)
     979             : {
     980       99729 :   pari_sp av = avma;
     981       99729 :   long l = lg(x);
     982             :   GEN y;
     983       99729 :   if (l <= 3) return 0; /* degree < 1 */
     984       99729 :   if (l == 4) return 1; /* degree 1 */
     985       96534 :   if (ZX_val(x)) return 0;
     986       96338 :   if (!ZX_is_squarefree(x)) return 0;
     987       96268 :   y = ZX_DDF(x); set_avma(av);
     988       96271 :   return (lg(y) == 2);
     989             : }
     990             : 
     991             : GEN
     992      643586 : nfrootsQ(GEN x)
     993             : {
     994      643586 :   pari_sp av = avma;
     995             :   GEN z;
     996             :   long val;
     997             : 
     998      643586 :   if (typ(x)!=t_POL) pari_err_TYPE("nfrootsQ",x);
     999      643586 :   if (!signe(x)) pari_err_ROOTS0("nfrootsQ");
    1000      643586 :   x = Q_primpart(x);
    1001      643586 :   RgX_check_ZX(x,"nfrootsQ");
    1002      643586 :   val = ZX_valrem(x, &x);
    1003      643586 :   z = DDF_roots( ZX_radical(x) );
    1004      643586 :   if (val) z = vec_append(z, gen_0);
    1005      643586 :   return gerepileupto(av, sort(z));
    1006             : }
    1007             : 
    1008             : /************************************************************************
    1009             :  *                   GCD OVER Z[X] / Q[X]                               *
    1010             :  ************************************************************************/
    1011             : int
    1012      203126 : ZX_is_squarefree(GEN x)
    1013             : {
    1014      203126 :   pari_sp av = avma;
    1015             :   GEN d;
    1016             :   long m;
    1017      203126 :   if (lg(x) == 2) return 0;
    1018      203126 :   m = ZX_deflate_order(x);
    1019      203129 :   if (m > 1)
    1020             :   {
    1021       85442 :     if (!signe(gel(x,2))) return 0;
    1022       85204 :     x = RgX_deflate(x, m);
    1023             :   }
    1024      202890 :   d = ZX_gcd(x,ZX_deriv(x));
    1025      202904 :   return gc_bool(av, lg(d) == 3);
    1026             : }
    1027             : 
    1028             : static int
    1029      102115 : ZX_gcd_filter(GEN *pt_A, GEN *pt_P)
    1030             : {
    1031      102115 :   GEN A = *pt_A, P = *pt_P;
    1032      102115 :   long i, j, l = lg(A), n = 1, d = degpol(gel(A,1));
    1033             :   GEN B, Q;
    1034      205236 :   for (i=2; i<l; i++)
    1035             :   {
    1036      103121 :     long di = degpol(gel(A,i));
    1037      103121 :     if (di==d) n++;
    1038          36 :     else if (d > di)
    1039          36 :     { n=1; d = di; }
    1040             :   }
    1041      102115 :   if (n == l-1)
    1042      102079 :     return 0;
    1043          36 :   B = cgetg(n+1, t_VEC);
    1044          36 :   Q = cgetg(n+1, typ(P));
    1045         156 :   for (i=1, j=1; i<l; i++)
    1046             :   {
    1047         120 :     if (degpol(gel(A,i))==d)
    1048             :     {
    1049          84 :       gel(B,j) = gel(A,i);
    1050          84 :       Q[j] = P[i];
    1051          84 :       j++;
    1052             :     }
    1053             :   }
    1054          36 :   *pt_A = B; *pt_P = Q; return 1;
    1055             : }
    1056             : 
    1057             : static GEN
    1058     3316518 : ZX_gcd_Flx(GEN a, GEN b, ulong g, ulong p)
    1059             : {
    1060     3316518 :   GEN H = Flx_gcd(a, b, p);
    1061     3316518 :   if (!g)
    1062     3300533 :     return Flx_normalize(H, p);
    1063             :   else
    1064             :   {
    1065       15985 :     ulong t = Fl_mul(g, Fl_inv(Flx_lead(H), p), p);
    1066       15985 :     return Flx_Fl_mul(H, t, p);
    1067             :   }
    1068             : }
    1069             : 
    1070             : static GEN
    1071     3310612 : ZX_gcd_slice(GEN A, GEN B, GEN g, GEN P, GEN *mod)
    1072             : {
    1073     3310612 :   pari_sp av = avma;
    1074     3310612 :   long i, n = lg(P)-1;
    1075             :   GEN H, T;
    1076     3310612 :   if (n == 1)
    1077             :   {
    1078     3305684 :     ulong p = uel(P,1), gp = g ? umodiu(g, p): 0;
    1079     3305684 :     GEN a = ZX_to_Flx(A, p), b = ZX_to_Flx(B, p);
    1080     3305682 :     GEN Hp = ZX_gcd_Flx(a, b, gp, p);
    1081     3305684 :     H = gerepileupto(av, Flx_to_ZX(Hp));
    1082     3305685 :     *mod = utoi(p);
    1083     3305685 :     return H;
    1084             :   }
    1085        4928 :   T = ZV_producttree(P);
    1086        4928 :   A = ZX_nv_mod_tree(A, P, T);
    1087        4928 :   B = ZX_nv_mod_tree(B, P, T);
    1088        4928 :   g = g ?  Z_ZV_mod_tree(g, P, T): NULL;
    1089        4928 :   H = cgetg(n+1, t_VEC);
    1090       15764 :   for(i=1; i <= n; i++)
    1091             :   {
    1092       10836 :     ulong p = P[i];
    1093       10836 :     GEN a = gel(A,i), b = gel(B,i);
    1094       10836 :     gel(H,i) = ZX_gcd_Flx(a, b, g? g[i]: 0, p);
    1095             :   }
    1096        4928 :   if (ZX_gcd_filter(&H, &P))
    1097          12 :     T = ZV_producttree(P);
    1098        4928 :   H = nxV_chinese_center_tree(H, P, T, ZV_chinesetree(P, T));
    1099        4928 :   *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
    1100             : }
    1101             : 
    1102             : GEN
    1103     3310613 : ZX_gcd_worker(GEN P, GEN A, GEN B, GEN g)
    1104             : {
    1105     3310613 :   GEN V = cgetg(3, t_VEC);
    1106     3310612 :   gel(V,1) = ZX_gcd_slice(A, B, equali1(g)? NULL: g , P, &gel(V,2));
    1107     3310613 :   return V;
    1108             : }
    1109             : 
    1110             : static GEN
    1111       97187 : ZX_gcd_chinese(GEN A, GEN P, GEN *mod)
    1112             : {
    1113       97187 :   ZX_gcd_filter(&A, &P);
    1114       97187 :   return nxV_chinese_center(A, P, mod);
    1115             : }
    1116             : 
    1117             : GEN
    1118    10756424 : ZX_gcd_all(GEN A, GEN B, GEN *Anew)
    1119             : {
    1120    10756424 :   pari_sp av = avma;
    1121    10756424 :   long k, valH, valA, valB, vA = varn(A), dA = degpol(A), dB = degpol(B);
    1122    10756442 :   GEN worker, c, cA, cB, g, Ag, Bg, H = NULL, mod = gen_1, R;
    1123             :   GEN Ap, Bp, Hp;
    1124             :   forprime_t S;
    1125             :   ulong pp;
    1126    10756442 :   if (dA < 0) { if (Anew) *Anew = pol_0(vA); return ZX_copy(B); }
    1127    10756421 :   if (dB < 0) { if (Anew) *Anew = pol_1(vA); return ZX_copy(A); }
    1128    10755308 :   A = Q_primitive_part(A, &cA);
    1129    10755302 :   B = Q_primitive_part(B, &cB);
    1130    10755306 :   valA = ZX_valrem(A, &A); dA -= valA;
    1131    10755312 :   valB = ZX_valrem(B, &B); dB -= valB;
    1132    10755316 :   valH = minss(valA, valB);
    1133    10755314 :   valA -= valH; /* valuation(Anew) */
    1134    10755314 :   c = (cA && cB)? gcdii(cA, cB): NULL; /* content(gcd) */
    1135    10755318 :   if (!dA || !dB)
    1136             :   {
    1137     6205920 :     if (Anew) *Anew = RgX_shift_shallow(A, valA);
    1138     6205920 :     return monomial(c? c: gen_1, valH, vA);
    1139             :   }
    1140     4549398 :   g = gcdii(leading_coeff(A), leading_coeff(B)); /* multiple of lead(gcd) */
    1141     4549367 :   if (is_pm1(g)) {
    1142     4440267 :     g = NULL;
    1143     4440267 :     Ag = A;
    1144     4440267 :     Bg = B;
    1145             :   } else {
    1146      109104 :     Ag = ZX_Z_mul(A,g);
    1147      109104 :     Bg = ZX_Z_mul(B,g);
    1148             :   }
    1149     4549371 :   init_modular_big(&S);
    1150             :   do {
    1151     4549410 :     pp = u_forprime_next(&S);
    1152     4549410 :     Ap = ZX_to_Flx(Ag, pp);
    1153     4549414 :     Bp = ZX_to_Flx(Bg, pp);
    1154     4549407 :   } while (degpol(Ap) != dA || degpol(Bp) != dB);
    1155     4549389 :   if (degpol(Flx_gcd(Ap, Bp, pp)) == 0)
    1156             :   {
    1157     1335997 :     if (Anew) *Anew = RgX_shift_shallow(A, valA);
    1158     1335997 :     return monomial(c? c: gen_1, valH, vA);
    1159             :   }
    1160     3213398 :   worker = snm_closure(is_entry("_ZX_gcd_worker"), mkvec3(A, B, g? g: gen_1));
    1161     3213398 :   av = avma;
    1162     3213398 :   for (k = 1; ;k *= 2)
    1163             :   {
    1164     3308932 :     gen_inccrt_i("ZX_gcd", worker, g, (k+1)>>1, 0, &S, &H, &mod, ZX_gcd_chinese, NULL);
    1165     3308934 :     gerepileall(av, 2, &H, &mod);
    1166     3308932 :     Hp = ZX_to_Flx(H, pp);
    1167     3308934 :     if (lgpol(Flx_rem(Ap, Hp, pp)) || lgpol(Flx_rem(Bp, Hp, pp))) continue;
    1168     3217464 :     if (!ZX_divides(Bg, H)) continue;
    1169     3213428 :     R = ZX_divides(Ag, H);
    1170     3213426 :     if (R) break;
    1171             :   }
    1172             :   /* lead(H) = g */
    1173     3213396 :   if (g) H = Q_primpart(H);
    1174     3213396 :   if (c) H = ZX_Z_mul(H,c);
    1175     3213396 :   if (DEBUGLEVEL>5) err_printf("done\n");
    1176     3213397 :   if (Anew) *Anew = RgX_shift_shallow(R, valA);
    1177     3213397 :   return valH? RgX_shift_shallow(H, valH): H;
    1178             : }
    1179             : 
    1180             : #if 0
    1181             : /* ceil( || p ||_oo / lc(p) ) */
    1182             : static GEN
    1183             : maxnorm(GEN p)
    1184             : {
    1185             :   long i, n = degpol(p), av = avma;
    1186             :   GEN x, m = gen_0;
    1187             : 
    1188             :   p += 2;
    1189             :   for (i=0; i<n; i++)
    1190             :   {
    1191             :     x = gel(p,i);
    1192             :     if (abscmpii(x,m) > 0) m = x;
    1193             :   }
    1194             :   m = divii(m, gel(p,n));
    1195             :   return gerepileuptoint(av, addiu(absi_shallow(m),1));
    1196             : }
    1197             : #endif
    1198             : 
    1199             : GEN
    1200     9724065 : ZX_gcd(GEN A, GEN B)
    1201             : {
    1202     9724065 :   pari_sp av = avma;
    1203     9724065 :   return gerepilecopy(av, ZX_gcd_all(A,B,NULL));
    1204             : }
    1205             : 
    1206             : GEN
    1207      804838 : ZX_radical(GEN A) { GEN B; (void)ZX_gcd_all(A,ZX_deriv(A),&B); return B; }
    1208             : 
    1209             : static GEN
    1210       12935 : _gcd(GEN a, GEN b)
    1211             : {
    1212       12935 :   if (!a) a = gen_1;
    1213       12935 :   if (!b) b = gen_1;
    1214       12935 :   return Q_gcd(a,b);
    1215             : }
    1216             : /* A0 and B0 in Q[X] */
    1217             : GEN
    1218       12936 : QX_gcd(GEN A0, GEN B0)
    1219             : {
    1220             :   GEN a, b, D;
    1221       12936 :   pari_sp av = avma, av2;
    1222             : 
    1223       12936 :   D = ZX_gcd(Q_primitive_part(A0, &a), Q_primitive_part(B0, &b));
    1224       12935 :   av2 = avma; a = _gcd(a,b);
    1225       12934 :   if (isint1(a)) set_avma(av2); else D = ZX_Q_mul(D, a);
    1226       12934 :   return gerepileupto(av, D);
    1227             : }
    1228             : 
    1229             : /*****************************************************************************
    1230             :  * Variants of the Bradford-Davenport algorithm: look for cyclotomic         *
    1231             :  * factors, and decide whether a ZX is cyclotomic or a product of cyclotomic *
    1232             :  *****************************************************************************/
    1233             : /* f of degree 1, return a cyclotomic factor (Phi_1 or Phi_2) or NULL */
    1234             : static GEN
    1235           0 : BD_deg1(GEN f)
    1236             : {
    1237           0 :   GEN a = gel(f,3), b = gel(f,2); /* f = ax + b */
    1238           0 :   if (!absequalii(a,b)) return NULL;
    1239           0 :   return polcyclo((signe(a) == signe(b))? 2: 1, varn(f));
    1240             : }
    1241             : 
    1242             : /* f a squarefree ZX; not divisible by any Phi_n, n even */
    1243             : static GEN
    1244         406 : BD_odd(GEN f)
    1245             : {
    1246         406 :   while(degpol(f) > 1)
    1247             :   {
    1248         406 :     GEN f1 = ZX_graeffe(f); /* contain all cyclotomic divisors of f */
    1249         406 :     if (ZX_equal(f1, f)) return f; /* product of cyclotomics */
    1250           0 :     f = ZX_gcd(f, f1);
    1251             :   }
    1252           0 :   if (degpol(f) == 1) return BD_deg1(f);
    1253           0 :   return NULL; /* no cyclotomic divisor */
    1254             : }
    1255             : 
    1256             : static GEN
    1257        2310 : myconcat(GEN v, GEN x)
    1258             : {
    1259        2310 :   if (typ(x) != t_VEC) x = mkvec(x);
    1260        2310 :   if (!v) return x;
    1261        1470 :   return shallowconcat(v, x);
    1262             : }
    1263             : 
    1264             : /* Bradford-Davenport algorithm.
    1265             :  * f a squarefree ZX of degree > 0, return NULL or a vector of coprime
    1266             :  * cyclotomic factors of f [ possibly reducible ] */
    1267             : static GEN
    1268        2359 : BD(GEN f)
    1269             : {
    1270        2359 :   GEN G = NULL, Gs = NULL, Gp = NULL, Gi = NULL;
    1271             :   GEN fs2, fp, f2, f1, fe, fo, fe1, fo1;
    1272        2359 :   RgX_even_odd(f, &fe, &fo);
    1273        2359 :   fe1 = ZX_eval1(fe);
    1274        2359 :   fo1 = ZX_eval1(fo);
    1275        2359 :   if (absequalii(fe1, fo1)) /* f(1) = 0 or f(-1) = 0 */
    1276             :   {
    1277        1519 :     long i, v = varn(f);
    1278        1519 :     if (!signe(fe1))
    1279         371 :       G = mkvec2(polcyclo(1, v), polcyclo(2, v)); /* both 0 */
    1280        1148 :     else if (signe(fe1) == signe(fo1))
    1281         693 :       G = mkvec(polcyclo(2, v)); /*f(-1) = 0*/
    1282             :     else
    1283         455 :       G = mkvec(polcyclo(1, v)); /*f(1) = 0*/
    1284        3409 :     for (i = lg(G)-1; i; i--) f = RgX_div(f, gel(G,i));
    1285             :   }
    1286             :   /* f no longer divisible by Phi_1 or Phi_2 */
    1287        2359 :   if (degpol(f) <= 1) return G;
    1288        2058 :   f1 = ZX_graeffe(f); /* has at most square factors */
    1289        2058 :   if (ZX_equal(f1, f)) return myconcat(G,f); /* f = product of Phi_n, n odd */
    1290             : 
    1291        1183 :   fs2 = ZX_gcd_all(f1, ZX_deriv(f1), &f2); /* fs2 squarefree */
    1292        1183 :   if (degpol(fs2))
    1293             :   { /* fs contains all Phi_n | f, 4 | n; and only those */
    1294             :     /* In that case, Graeffe(Phi_n) = Phi_{n/2}^2, and Phi_n = Phi_{n/2}(x^2) */
    1295        1029 :     GEN fs = RgX_inflate(fs2, 2);
    1296        1029 :     (void)ZX_gcd_all(f, fs, &f); /* remove those Phi_n | f, 4 | n */
    1297        1029 :     Gs = BD(fs2);
    1298        1029 :     if (Gs)
    1299             :     {
    1300             :       long i;
    1301        2555 :       for (i = lg(Gs)-1; i; i--) gel(Gs,i) = RgX_inflate(gel(Gs,i), 2);
    1302             :       /* prod Gs[i] is the product of all Phi_n | f, 4 | n */
    1303        1029 :       G = myconcat(G, Gs);
    1304             :     }
    1305             :     /* f2 = f1 / fs2 */
    1306        1029 :     f1 = RgX_div(f2, fs2); /* f1 / fs2^2 */
    1307             :   }
    1308        1183 :   fp = ZX_gcd(f, f1); /* contains all Phi_n | f, n > 1 odd; and only those */
    1309        1183 :   if (degpol(fp))
    1310             :   {
    1311         196 :     Gp = BD_odd(fp);
    1312             :     /* Gp is the product of all Phi_n | f, n odd */
    1313         196 :     if (Gp) G = myconcat(G, Gp);
    1314         196 :     f = RgX_div(f, fp);
    1315             :   }
    1316        1183 :   if (degpol(f))
    1317             :   { /* contains all Phi_n originally dividing f, n = 2 mod 4, n > 2;
    1318             :      * and only those
    1319             :      * In that case, Graeffe(Phi_n) = Phi_{n/2}, and Phi_n = Phi_{n/2}(-x) */
    1320         210 :     Gi = BD_odd(ZX_z_unscale(f, -1));
    1321         210 :     if (Gi)
    1322             :     { /* N.B. Phi_2 does not divide f */
    1323         210 :       Gi = ZX_z_unscale(Gi, -1);
    1324             :       /* Gi is the product of all Phi_n | f, n = 2 mod 4 */
    1325         210 :       G = myconcat(G, Gi);
    1326             :     }
    1327             :   }
    1328        1183 :   return G;
    1329             : }
    1330             : 
    1331             : /* Let f be a nonzero QX, return the (squarefree) product of cyclotomic
    1332             :  * divisors of f */
    1333             : GEN
    1334         315 : polcyclofactors(GEN f)
    1335             : {
    1336         315 :   pari_sp av = avma;
    1337         315 :   if (typ(f) != t_POL || !signe(f)) pari_err_TYPE("polcyclofactors",f);
    1338         315 :   (void)RgX_valrem(f, &f);
    1339         315 :   f = Q_primpart(f);
    1340         315 :   RgX_check_ZX(f,"polcyclofactors");
    1341         315 :   if (degpol(f))
    1342             :   {
    1343         315 :     f = BD(ZX_radical(f));
    1344         315 :     if (f) return gerepilecopy(av, f);
    1345             :   }
    1346           0 :   set_avma(av); return cgetg(1,t_VEC);
    1347             : }
    1348             : 
    1349             : /* return t*x mod T(x), T a monic ZX. Assume deg(t) < deg(T) */
    1350             : static GEN
    1351       50105 : ZXQ_mul_by_X(GEN t, GEN T)
    1352             : {
    1353             :   GEN lt;
    1354       50105 :   t = RgX_shift_shallow(t, 1);
    1355       50105 :   if (degpol(t) < degpol(T)) return t;
    1356        7363 :   lt = leading_coeff(t);
    1357        7363 :   if (is_pm1(lt)) return signe(lt) > 0 ? ZX_sub(t, T): ZX_add(t, T);
    1358         217 :   return ZX_sub(t, ZX_Z_mul(T, leading_coeff(t)));
    1359             : }
    1360             : /* f a product of Phi_n, all n odd; deg f > 1. Is it irreducible ? */
    1361             : static long
    1362        2078 : BD_odd_iscyclo(GEN f)
    1363             : {
    1364             :   pari_sp av;
    1365             :   long d, e, n, bound;
    1366             :   GEN t;
    1367        2078 :   f = ZX_deflate_max(f, &e);
    1368        2078 :   av = avma;
    1369             :   /* The original f is cyclotomic (= Phi_{ne}) iff the present one is Phi_n,
    1370             :    * where all prime dividing e also divide n. If current f is Phi_n,
    1371             :    * then n is odd and squarefree */
    1372        2078 :   d = degpol(f); /* = phi(n) */
    1373             :   /* Let e > 0, g multiplicative such that
    1374             :        g(p) = p / (p-1)^(1+e) < 1 iff p < (p-1)^(1+e)
    1375             :      For all squarefree odd n, we have g(n) < C, hence n < C phi(n)^(1+e), where
    1376             :        C = \prod_{p odd | p > (p-1)^(1+e)} g(p)
    1377             :      For e = 1/10,   we obtain p = 3, 5 and C < 1.523
    1378             :      For e = 1/100,  we obtain p = 3, 5, ..., 29 and C < 2.573
    1379             :      In fact, for n <= 10^7 odd & squarefree, we have n < 2.92 * phi(n)
    1380             :      By the above, n<10^7 covers all d <= (10^7/2.573)^(1/(1+1/100)) < 3344391.
    1381             :   */
    1382        2078 :   if (d <= 3344391)
    1383        2078 :     bound = (long)(2.92 * d);
    1384             :   else
    1385           0 :     bound = (long)(2.573 * pow(d,1.01));
    1386             :   /* IF f = Phi_n, n squarefree odd, then n <= bound */
    1387        2078 :   t = pol_xn(d-1, varn(f));
    1388       50140 :   for (n = d; n <= bound; n++)
    1389             :   {
    1390       50105 :     t = ZXQ_mul_by_X(t, f);
    1391             :     /* t = (X mod f(X))^d */
    1392       50106 :     if (degpol(t) == 0) break;
    1393       48062 :     if (gc_needed(av,1))
    1394             :     {
    1395         461 :       if(DEBUGMEM>1) pari_warn(warnmem,"BD_odd_iscyclo");
    1396         461 :       t = gerepilecopy(av, t);
    1397             :     }
    1398             :   }
    1399        2079 :   if (n > bound || eulerphiu(n) != (ulong)d) return 0;
    1400             : 
    1401        2016 :   if (e > 1) return (u_ppo(e, n) == 1)? e * n : 0;
    1402        1848 :   return n;
    1403             : }
    1404             : 
    1405             : /* Checks if f, monic squarefree ZX with |constant coeff| = 1, is a cyclotomic
    1406             :  * polynomial. Returns n if f = Phi_n, and 0 otherwise */
    1407             : static long
    1408       10332 : BD_iscyclo(GEN f)
    1409             : {
    1410       10332 :   pari_sp av = avma;
    1411             :   GEN f2, fn, f1;
    1412             : 
    1413       10332 :   if (degpol(f) == 1) return isint1(gel(f,2))? 2: 1;
    1414       10003 :   f1 = ZX_graeffe(f);
    1415             :   /* f = product of Phi_n, n odd */
    1416       10002 :   if (ZX_equal(f, f1)) return gc_long(av, BD_odd_iscyclo(f));
    1417             : 
    1418        8980 :   fn = ZX_z_unscale(f, -1); /* f(-x) */
    1419             :   /* f = product of Phi_n, n = 2 mod 4 */
    1420        8980 :   if (ZX_equal(f1, fn)) return gc_long(av, 2*BD_odd_iscyclo(fn));
    1421             : 
    1422        7924 :   if (issquareall(f1, &f2))
    1423             :   {
    1424        1344 :     GEN lt = leading_coeff(f2);
    1425             :     long c;
    1426        1344 :     if (signe(lt) < 0) f2 = ZX_neg(f2);
    1427        1344 :     c = BD_iscyclo(f2);
    1428        1344 :     return odd(c)? 0: 2*c;
    1429             :   }
    1430        6580 :   return gc_long(av, 0);
    1431             : }
    1432             : long
    1433       98723 : poliscyclo(GEN f)
    1434             : {
    1435             :   long d;
    1436       98723 :   if (typ(f) != t_POL) pari_err_TYPE("poliscyclo", f);
    1437       98716 :   d = degpol(f);
    1438       98716 :   if (d <= 0 || !RgX_is_ZX(f)) return 0;
    1439       98711 :   if (!equali1(gel(f,d+2)) || !is_pm1(gel(f,2))) return 0;
    1440        9051 :   if (d == 1) return signe(gel(f,2)) > 0? 2: 1;
    1441        8988 :   return ZX_is_squarefree(f)? BD_iscyclo(f): 0;
    1442             : }
    1443             : 
    1444             : long
    1445        1029 : poliscycloprod(GEN f)
    1446             : {
    1447        1029 :   pari_sp av = avma;
    1448        1029 :   long i, d = degpol(f);
    1449        1029 :   if (typ(f) != t_POL) pari_err_TYPE("poliscycloprod",f);
    1450        1029 :   if (!RgX_is_ZX(f)) return 0;
    1451        1029 :   if (!ZX_is_monic(f) || !is_pm1(constant_coeff(f))) return 0;
    1452        1029 :   if (d < 2) return (d == 1);
    1453        1022 :   if ( degpol(ZX_gcd_all(f, ZX_deriv(f), &f)) )
    1454             :   {
    1455          14 :     d = degpol(f);
    1456          14 :     if (d == 1) return 1;
    1457             :   }
    1458        1015 :   f = BD(f); if (!f) return 0;
    1459        3619 :   for (i = lg(f)-1; i; i--) d -= degpol(gel(f,i));
    1460        1015 :   return gc_long(av, d == 0);
    1461             : }

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