/* g in Z[X] potentially defines a subfield of Q[X]/f. It is a subfield iff A
* (cf subfield) was a block system; then there
* exists h in Q[X] such that f | g o h. listdelta determines h s.t f | g o h
* in Fp[X] (cf chinese_retrieve_pol). Try to lift it; den is a
* multiplicative bound for denominator of lift. */
static GEN
embedding(GEN g, GEN DATA, primedata *S, GEN den, GEN listdelta)
{
GEN TR, w0_Q, w0, w1_Q, w1, wpow, h0, gp, T, q2, q, maxp, a, p = S->p;
long rt;
pari_sp av;
T = gel(DATA,1); rt = brent_kung_optpow(degpol(T), 2);
maxp= gel(DATA,7);
gp = derivpol(g); av = avma;
w0 = chinese_retrieve_pol(DATA, S, listdelta);
w0_Q = centermod(gmul(w0,den), p);
h0 = FpXQ_inv(FpX_FpXQ_compo(gp,w0, T,p), T,p); /* = 1/g'(w0) mod (T,p) */
wpow = NULL; q = sqri(p);
for(;;)
{/* Given g,w0,h0 in Z[x], s.t. h0.g'(w0) = 1 and g(w0) = 0 mod (T,p), find
* [w1,h1] satisfying the same conditions mod p^2, [w1,h1] = [w0,h0] (mod p)
* (cf. Dixon: J. Austral. Math. Soc., Series A, vol.49, 1990, p.445) */
if (DEBUGLEVEL>1)
fprintferr("lifting embedding mod p^k = %Z^%ld\n",p, Z_pval(q,p));
/* w1 := w0 - h0 g(w0) mod (T,q) */
if (wpow)
a = FpX_FpXQV_compo(g,wpow, T,q);
else
a = FpX_FpXQ_compo(g,w0, T,q); /* first time */
/* now, a = 0 (p) */
a = gmul(gneg(h0), gdivexact(a, p));
w1 = gadd(w0, gmul(p, FpX_rem(a, T,p)));
w1_Q = centermod(gmul(w1, remii(den,q)), q);
if (gequal(w1_Q, w0_Q) || cmpii(q,maxp) > 0)
{
GEN G = gcmp1(den)? g: RgX_rescale(g,den);
if (gcmp0(RgX_RgXQ_compo(G, w1_Q, T))) break;
}
if (cmpii(q, maxp) > 0)
{
if (DEBUGLEVEL) fprintferr("coeff too big for embedding\n");
return NULL;
}
gerepileall(av, 5, &w1,&h0,&w1_Q,&q,&p);
q2 = sqri(q);
wpow = FpXQ_powers(w1, rt, T, q2);
/* h0 := h0 * (2 - h0 g'(w1)) mod (T,q)
* = h0 + h0 * (1 - h0 g'(w1)) */
a = gmul(gneg(h0), FpX_FpXQV_compo(gp, FpXV_red(wpow,q),T,q));
a = ZX_Z_add(FpX_rem(a, T,q), gen_1); /* 1 - h0 g'(w1) = 0 (p) */
a = gmul(h0, gdivexact(a, p));
h0 = gadd(h0, gmul(p, FpX_rem(a, T,p)));
w0 = w1; w0_Q = w1_Q; p = q; q = q2;
}
TR = gel(DATA,5);
if (!gcmp0(TR)) w1_Q = translate_pol(w1_Q, TR);
return gdiv(w1_Q,den);
}
/* return C in Z[X]/(p,T), C[ D[1] ] = 1, C[ D[i] ] = 0 otherwise. H is the
* list of degree 1 polynomials X - D[i] (come directly from factorization) */
static GEN
interpol(GEN H, GEN T, GEN p)
{
long i, m = lg(H);
GEN X = pol_x[0],d,p1,p2,a;
p1=pol_1[0]; p2=gen_1; a = gneg(constant_term(gel(H,1))); /* = D[1] */
for (i=2; i<m; i++)
{
d = constant_term(gel(H,i)); /* -D[i] */
p1 = FpXQX_mul(p1,gadd(X,d), T,p);
p2 = Fq_mul(p2, gadd(a,d), T,p);
}
return FqX_Fq_mul(p1,Fq_inv(p2, T,p), T,p);
}
static GEN
nf_DDF_roots(GEN pol, GEN polred, GEN nfpol, GEN lt, GEN init_fa, long nbf,
long fl, nflift_t *L)
{
long Cltx_r[] = { evaltyp(t_POL)|_evallg(4), 0,0,0 };
long i, m;
GEN C2ltpol, C = L->topowden;
GEN Clt = mul_content(C, lt);
GEN C2lt = mul_content(C,Clt);
GEN z;
if (L->Tpk)
{
int cof = (degpol(pol) > nbf); /* non trivial cofactor ? */
z = FqX_split_roots(init_fa, L->Tp, L->p, cof? polred: NULL);
z = hensel_lift_fact(polred, z, L->Tpk, L->p, L->pk, L->k);
if (cof) setlg(z, lg(z)-1); /* remove cofactor */
z = roots_from_deg1(z);
}
else
z = rootpadicfast(polred, L->p, L->k);
Cltx_r[1] = evalsigne(1) | evalvarn(varn(pol));
gel(Cltx_r,3) = Clt? Clt: gen_1;
C2ltpol = C2lt? gmul(C2lt, pol): pol;
for (m=1,i=1; i<lg(z); i++)
{
GEN q, r = gel(z,i);
r = nf_bestlift_to_pol(lt? gmul(lt,r): r, NULL, L);
gel(Cltx_r,2) = gneg(r); /* check P(r) == 0 */
q = RgXQX_divrem(C2ltpol, Cltx_r, nfpol, ONLY_DIVIDES); /* integral */
if (q) {
C2ltpol = C2lt? gmul(Clt,q): q;
if (Clt) r = gdiv(r, Clt);
gel(z,m++) = r;
}
else if (fl == 2) return cgetg(1, t_VEC);
}
z[0] = evaltyp(t_VEC) | evallg(m);
return z;
}
/* return the factorization of the square-free polynomial x.
The coeffs of x are in Z_nf and its leading term is a rational integer.
deg(x) > 1, deg(nfpol) > 1
If fl = 1, return only the roots of x in nf
If fl = 2, as fl=1 if pol splits, [] otherwise */
static GEN
nfsqff(GEN nf, GEN pol, long fl)
{
long n, nbf, dpol = degpol(pol);
GEN pr, C0, polbase, init_fa = NULL;
GEN N2, rep, polmod, polred, lt, nfpol = gel(nf,1);
nfcmbf_t T;
nflift_t L;
pari_timer ti, ti_tot;
if (DEBUGLEVEL>2) { TIMERstart(&ti); TIMERstart(&ti_tot); }
n = degpol(nfpol);
polbase = unifpol(nf, pol, t_COL);
if (typ(polbase) != t_POL) pari_err(typeer, "nfsqff");
polmod = lift_intern( unifpol(nf, pol, t_POLMOD) );
if (dpol == 1) return mkvec(QXQX_normalize(polmod, nfpol));
/* heuristic */
if (dpol*3 < n)
{
GEN z, t;
long i;
if (DEBUGLEVEL>2) fprintferr("Using Trager's method\n");
z = (GEN)polfnf(polmod, nfpol)[1];
if (fl) {
long l = lg(z);
for (i = 1; i < l; i++)
{
t = gel(z,i); if (degpol(t) > 1) break;
gel(z,i) = gneg(gdiv(gel(t,2), gel(t,3)));
}
setlg(z, i);
if (fl == 2 && i != l) return cgetg(1,t_VEC);
}
return z;
}
nbf = nf_pick_prime(5, nf, polbase, fl, <, &init_fa, &pr, &L.Tp);
if (fl == 2 && nbf < dpol) return cgetg(1,t_VEC);
if (nbf <= 1)
{
if (!fl) return mkvec(QXQX_normalize(polmod, nfpol)); /* irreducible */
if (!nbf) return cgetg(1,t_VEC); /* no root */
}
if (DEBUGLEVEL>2) {
msgTIMER(&ti, "choice of a prime ideal");
fprintferr("Prime ideal chosen: %Z\n", pr);
}
pol = simplify_i(lift(polmod));
L.tozk = gel(nf,8);
L.topow= Q_remove_denom(gel(nf,7), &L.topowden);
T.ZC = L2_bound(nf, L.tozk, &(T.dn));
T.Br = nf_root_bounds(pol, nf); if (lt) T.Br = gmul(T.Br, lt);
if (fl) C0 = normlp(T.Br, 2, n);
else C0 = nf_factor_bound(nf, polbase); /* bound for T_2(Q_i), Q | P */
T.bound = mulrr(T.ZC, C0); /* bound for |Q_i|^2 in Z^n on chosen Z-basis */
N2 = mulsr(dpol*dpol, normlp(T.Br, 4, n)); /* bound for T_2(lt * S_2) */
T.BS_2 = mulrr(T.ZC, N2); /* bound for |S_2|^2 on chosen Z-basis */
if (DEBUGLEVEL>2) {
msgTIMER(&ti, "bound computation");
fprintferr(" 1) T_2 bound for %s: %Z\n", fl?"root":"factor", C0);
fprintferr(" 2) Conversion from T_2 --> | |^2 bound : %Z\n", T.ZC);
fprintferr(" 3) Final bound: %Z\n", T.bound);
}
L.p = gel(pr,1);
if (L.Tp && degpol(L.Tp) == 1) L.Tp = NULL;
bestlift_init(0, nf, pr, T.bound, &L);
if (DEBUGLEVEL>2) TIMERstart(&ti);
polred = ZqX_normalize(polbase, lt, &L); /* monic */
if (fl) {
GEN z = nf_DDF_roots(pol, polred, nfpol, lt, init_fa, nbf, fl, &L);
if (lg(z) == 1) return cgetg(1, t_VEC);
return z;
}
{
pari_sp av = avma;
if (L.Tp)
rep = FqX_split_all(init_fa, L.Tp, L.p);
else
{
long d;
rep = cgetg(dpol + 1, t_VEC); gel(rep,1) = FpX_red(polred,L.p);
d = FpX_split_Berlekamp((GEN*)(rep + 1), L.p);
setlg(rep, d + 1);
}
T.fact = gerepilecopy(av, sort_vecpol(rep, &cmp_pol));
}
if (DEBUGLEVEL>2) msgTIMER(&ti, "splitting mod %Z", pr);
T.pr = pr;
T.L = &L;
T.polbase = polbase;
T.pol = pol;
T.nf = nf;
T.hint = 1; /* useless */
rep = nf_combine_factors(&T, polred, L.p, L.k, dpol-1);
if (DEBUGLEVEL>2)
fprintferr("Total Time: %ld\n===========\n", TIMER(&ti_tot));
return rep;
}
static GEN
bnflog_i(GEN bnf, GEN ell)
{
long prec0, prec;
GEN nf, US, vdegS, S, T, M, CLp, CLt, Ftilde, vtG, ellk;
GEN D, Ap, cycAp, bnfS;
long i, j, lS, lvAp;
checkbnf(bnf);
nf = checknf(bnf);
S = idealprimedec(nf, ell);
bnfS = bnfsunit0(bnf, S, nf_GENMAT, LOWDEFAULTPREC); /* S-units */
US = leafcopy(gel(bnfS,1));
prec0 = maxss(30, vtilde_prec(nf, US, ell));
US = shallowconcat(bnf_get_fu(bnf), US);
settyp(US, t_COL);
T = padicfact(nf, S, prec0);
lS = lg(S); Ftilde = cgetg(lS, t_VECSMALL);
for (j = 1; j < lS; j++) Ftilde[j] = ftilde(nf, gel(S,j), gel(T,j));
CLp = CL_prime(bnf, ell, S);
cycAp = gel(CLp,1);
Ap = gel(CLp,2);
for(;;)
{
CLt = CL_tilde(nf, US, ell, T, Ftilde, &vtG, prec0);
if (CLt) break;
prec0 <<= 1;
T = padicfact(nf, S, prec0);
}
prec = ellexpo(cycAp, ell) + ellexpo(CLt,ell) + 1;
if (prec == 1) return mkvec3(cgetg(1,t_VEC), cgetg(1,t_VEC), cgetg(1,t_VEC));
vdegS = get_vdegS(Ftilde, ell, prec0);
ellk = powiu(ell, prec);
lvAp = lg(Ap);
if (lvAp > 1)
{
GEN Kcyc = bnf_get_cyc(bnf);
GEN C = zeromatcopy(lvAp-1, lS-1);
GEN Rell = gel(CLp,3), Uell = gel(CLp,4), ordS = gel(CLp,5);
for (i = 1; i < lvAp; i++)
{
GEN a, b, bi, A = gel(Ap,i), d = gel(cycAp,i);
bi = isprincipal(bnf, A);
a = vecmodii(ZC_Z_mul(bi,d), Kcyc);
/* a in subgroup generated by S = Rell; hence b integral */
b = hnf_invimage(Rell, a);
b = vecmodii(ZM_ZC_mul(Uell, ZC_neg(b)), ordS);
A = mkvec2(A, cgetg(1,t_MAT));
A = idealpowred(nf, A, d);
/* find a principal representative of A_i^cycA_i up to elements of S */
a = isprincipalfact(bnf,gel(A,1),S,b,nf_GENMAT|nf_FORCE);
if (!gequal0(gel(a,1))) pari_err_BUG("bnflog");
a = famat_mul_shallow(gel(A,2), gel(a,2)); /* principal part */
if (lg(a) == 1) continue;
for (j = 1; j < lS; j++)
gcoeff(C,i,j) = vtilde(nf, a, gel(T,j), gel(vdegS,j), ell, prec0);
}
C = gmod(gneg(C),ellk);
C = shallowtrans(C);
M = mkmat2(mkcol2(diagonal_shallow(cycAp), C), mkcol2(gen_0, vtG));
M = shallowmatconcat(M); /* relation matrix */
}
else
M = vtG;
M = ZM_hnfmodid(M, ellk);
D = matsnf0(M, 4);
if (lg(D) == 1 || !dvdii(gel(D,1), ellk))
pari_err_BUG("bnflog [missing Z_l component]");
D = vecslice(D,2,lg(D)-1);
return mkvec3(D, CLt, ellsylow(cycAp, ell));
}
