// assert: poly is in Z[x]
Term t;
for ( int i=poly.degree(x); i>=poly.ldegree(x); --i ) {
- int coeff = ex_to<numeric>(poly.coeff(x,i)).to_int();
- if ( coeff ) {
+ cl_I coeff = the<cl_I>(ex_to<numeric>(poly.coeff(x,i)).to_cl_N());
+ if ( !zerop(coeff) ) {
t.c = R->canonhom(coeff);
if ( !zerop(t.c) ) {
t.exp = i;
maxcoeff += pow(abs(a.coeff(x, i)),2);
}
cl_I normmc = ceiling1(the<cl_R>(cln::sqrt(ex_to<numeric>(maxcoeff).to_cl_N())));
- unsigned int maxdegree = (u1_.degree() > w1_.degree()) ? u1_.degree() : w1_.degree();
- unsigned int B = cl_I_to_uint(normmc * expt_pos(cl_I(2), maxdegree));
+ cl_I maxdegree = (u1_.degree() > w1_.degree()) ? u1_.degree() : w1_.degree();
+ cl_I B = normmc * expt_pos(cl_I(2), maxdegree);
// step 1
ex alpha = a.lcoeff(x);
if ( gamma == 0 ) {
gamma = alpha;
}
- unsigned int gamma_ui = ex_to<numeric>(abs(gamma)).to_int();
+ numeric gamma_ui = ex_to<numeric>(abs(gamma));
a = a * gamma;
UniPoly nu1 = u1_;
nu1.unit_normal();
ex u = replace_lc(u1.to_ex(x), x, gamma);
ex w = replace_lc(w1.to_ex(x), x, alpha);
ex e = expand(a - u * w);
- unsigned int modulus = p;
+ numeric modulus = p;
+ const numeric maxmodulus = 2*numeric(B)*gamma_ui;
// step 4
- while ( !e.is_zero() && modulus < 2*B*gamma_ui ) {
+ while ( !e.is_zero() && modulus < maxmodulus ) {
ex c = e / modulus;
phi = expand(s.to_ex(x)*c);
UniPoly sigmatilde(R, phi, x);
vector<ex> anew = a;
for ( size_t i=0; i<r; ++i ) {
- a[i] = a[i].subs(xnu == alphanu);
+ anew[i] = anew[i].subs(xnu == alphanu);
}
ex cnew = c.subs(xnu == alphanu);
vector<EvalPoint> Inew = I;
Inew.pop_back();
- vector<ex> sigma = multivar_diophant(anew, x, cnew, Inew, d, p, k);
+ sigma = multivar_diophant(anew, x, cnew, Inew, d, p, k);
+ DCOUTVAR(sigma);
ex buf = c;
for ( size_t i=0; i<r; ++i ) {
ex e = buf;
e = make_modular(e, R);
+ e = e.expand();
+ DCOUTVAR(e);
+ DCOUTVAR(d);
ex monomial = 1;
for ( size_t m=1; m<=d; ++m ) {
- while ( !e.is_zero() ) {
+ DCOUTVAR(m);
+ while ( !e.is_zero() && e.has(xnu) ) {
monomial *= (xnu - alphanu);
monomial = expand(monomial);
+ DCOUTVAR(xnu);
+ DCOUTVAR(alphanu);
ex cm = e.diff(ex_to<symbol>(xnu), m).subs(xnu==alphanu) / factorial(m);
+ DCOUTVAR(cm);
if ( !cm.is_zero() ) {
vector<ex> delta_s = multivar_diophant(anew, x, cm, Inew, d, p, k);
+ DCOUTVAR(delta_s);
ex buf = e;
for ( size_t j=0; j<delta_s.size(); ++j ) {
delta_s[j] *= monomial;
sigma[j] += delta_s[j];
buf -= delta_s[j] * b[j];
}
- e = buf;
+ e = buf.expand();
e = make_modular(e, R);
}
}
}
}
else {
+ DCOUT(uniterm left);
UniPolyVec amod;
for ( size_t i=0; i<a.size(); ++i ) {
UniPoly up(R, a[i], x);
return sigma;
}
+#ifdef DEBUGFACTOR
+ostream& operator<<(ostream& o, const vector<EvalPoint>& v)
+{
+ for ( size_t i=0; i<v.size(); ++i ) {
+ o << "(" << v[i].x << "==" << v[i].evalpoint << ") ";
+ }
+ return o;
+}
+#endif // def DEBUGFACTOR
+
+
ex hensel_multivar(const ex& a, const ex& x, const vector<EvalPoint>& I, unsigned int p, const cl_I& l, const UniPolyVec& u, const vector<ex>& lcU)
{
DCOUT(hensel_multivar);
DCOUTVAR(a);
DCOUTVAR(x);
+ DCOUTVAR(I);
DCOUTVAR(p);
DCOUTVAR(l);
DCOUTVAR(u);
+ DCOUTVAR(lcU);
const size_t nu = I.size() + 1;
const cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),l));
DCOUTVAR(j);
vector<ex> U1 = U;
ex monomial = 1;
+ DCOUTVAR(U);
for ( size_t m=0; m<n; ++m) {
if ( lcU[m] != 1 ) {
ex coef = lcU[m];
U[m] = U[m] - U[m].lcoeff(x) * pow(x,deg) + coef * pow(x,deg);
}
}
+ DCOUTVAR(U);
ex Uprod = 1;
for ( size_t i=0; i<n; ++i ) {
Uprod *= U[i];
ex e = expand(A[j-1] - Uprod);
DCOUTVAR(e);
+ vector<EvalPoint> newI;
+ for ( size_t i=1; i<=j-2; ++i ) {
+ newI.push_back(I[i-1]);
+ }
+ DCOUTVAR(newI);
+
ex xj = I[j-2].x;
int alphaj = I[j-2].evalpoint;
size_t deg = A[j-1].degree(xj);
ex c = dif.subs(xj==alphaj) / factorial(k);
DCOUTVAR(c);
if ( !c.is_zero() ) {
- vector<EvalPoint> newI = I;
- newI.pop_back();
vector<ex> deltaU = multivar_diophant(U1, x, c, newI, maxdeg, p, cl_I_to_uint(l));
for ( size_t i=0; i<n; ++i ) {
DCOUTVAR(i);
deltaU[i] *= monomial;
U[i] += deltaU[i];
U[i] = make_modular(U[i], R);
+ U[i] = U[i].expand();
+ DCOUTVAR(U[i]);
}
ex Uprod = 1;
for ( size_t i=0; i<n; ++i ) {
Uprod *= U[i];
}
+ DCOUTVAR(Uprod.expand());
+ DCOUTVAR(A[j-1]);
e = expand(A[j-1] - Uprod);
e = make_modular(e, R);
DCOUTVAR(e);
for ( size_t i=0; i<U.size(); ++i ) {
res.append(U[i]);
}
+ DCOUTVAR(res);
+ DCOUT(END hensel_multivar);
return res;
}
else {
+ lst res;
+ DCOUTVAR(res);
+ DCOUT(END hensel_multivar);
return lst();
}
}
-static ex factor_multivariate(const ex& poly, const ex& x)
+static ex put_factors_into_lst(const ex& e)
{
- // TODO
- return 666;
+ DCOUT(put_factors_into_lst);
+ DCOUTVAR(e);
+
+ lst result;
+
+ if ( is_a<numeric>(e) ) {
+ result.append(e);
+ DCOUT(END put_factors_into_lst);
+ DCOUTVAR(result);
+ return result;
+ }
+ if ( is_a<power>(e) ) {
+ result.append(1);
+ result.append(e.op(0));
+ result.append(e.op(1));
+ DCOUT(END put_factors_into_lst);
+ DCOUTVAR(result);
+ return result;
+ }
+ if ( is_a<symbol>(e) || is_a<add>(e) ) {
+ result.append(1);
+ result.append(e);
+ result.append(1);
+ DCOUT(END put_factors_into_lst);
+ DCOUTVAR(result);
+ return result;
+ }
+ if ( is_a<mul>(e) ) {
+ ex nfac = 1;
+ for ( size_t i=0; i<e.nops(); ++i ) {
+ ex op = e.op(i);
+ if ( is_a<numeric>(op) ) {
+ nfac = op;
+ }
+ if ( is_a<power>(op) ) {
+ result.append(op.op(0));
+ result.append(op.op(1));
+ }
+ if ( is_a<symbol>(op) || is_a<add>(op) ) {
+ result.append(op);
+ result.append(1);
+ }
+ }
+ result.prepend(nfac);
+ DCOUT(END put_factors_into_lst);
+ DCOUTVAR(result);
+ return result;
+ }
+ throw runtime_error("put_factors_into_lst: bad term.");
+}
+
+#ifdef DEBUGFACTOR
+ostream& operator<<(ostream& o, const vector<numeric>& v)
+{
+ for ( size_t i=0; i<v.size(); ++i ) {
+ o << v[i] << " ";
+ }
+ return o;
+}
+#endif // def DEBUGFACTOR
+
+static bool checkdivisors(const lst& f, vector<numeric>& d)
+{
+ DCOUT(checkdivisors);
+ const int k = f.nops()-2;
+ DCOUTVAR(k);
+ DCOUTVAR(d.size());
+ numeric q, r;
+ d[0] = ex_to<numeric>(f.op(0) * f.op(f.nops()-1));
+ if ( d[0] == 1 && k == 1 && abs(f.op(1)) != 1 ) {
+ DCOUT(false);
+ DCOUT(END checkdivisors);
+ return false;
+ }
+ DCOUTVAR(d[0]);
+ for ( int i=1; i<=k; ++i ) {
+ DCOUTVAR(i);
+ DCOUTVAR(abs(f.op(i)));
+ q = ex_to<numeric>(abs(f.op(i)));
+ DCOUTVAR(q);
+ for ( int j=i-1; j>=0; --j ) {
+ r = d[j];
+ DCOUTVAR(r);
+ do {
+ r = gcd(r, q);
+ DCOUTVAR(r);
+ q = q/r;
+ DCOUTVAR(q);
+ } while ( r != 1 );
+ if ( q == 1 ) {
+ DCOUT(true);
+ DCOUT(END checkdivisors);
+ return true;
+ }
+ }
+ d[i] = q;
+ }
+ DCOUT(false);
+ DCOUT(END checkdivisors);
+ return false;
+}
+
+static bool generate_set(const ex& u, const ex& vn, const exset& syms, const ex& f, const numeric& modulus, vector<numeric>& a, vector<numeric>& d)
+{
+ // computation of d is actually not necessary
+ DCOUT(generate_set);
+ DCOUTVAR(u);
+ DCOUTVAR(vn);
+ DCOUTVAR(f);
+ DCOUTVAR(modulus);
+ const ex& x = *syms.begin();
+ bool trying = true;
+ do {
+ ex u0 = u;
+ ex vna = vn;
+ ex vnatry;
+ exset::const_iterator s = syms.begin();
+ ++s;
+ for ( size_t i=0; i<a.size(); ++i ) {
+ DCOUTVAR(*s);
+ do {
+ a[i] = mod(numeric(rand()), 2*modulus) - modulus;
+ vnatry = vna.subs(*s == a[i]);
+ } while ( vnatry == 0 );
+ vna = vnatry;
+ u0 = u0.subs(*s == a[i]);
+ ++s;
+ }
+ DCOUTVAR(a);
+ DCOUTVAR(u0);
+ if ( gcd(u0,u0.diff(ex_to<symbol>(x))) != 1 ) {
+ continue;
+ }
+ if ( is_a<numeric>(vn) ) {
+ trying = false;
+ }
+ else {
+ DCOUT(do substitution);
+ lst fnum;
+ lst::const_iterator i = ex_to<lst>(f).begin();
+ fnum.append(*i++);
+ bool problem = false;
+ while ( i!=ex_to<lst>(f).end() ) {
+ ex fs = *i;
+ if ( !is_a<numeric>(fs) ) {
+ s = syms.begin();
+ ++s;
+ for ( size_t j=0; j<a.size(); ++j ) {
+ fs = fs.subs(*s == a[j]);
+ ++s;
+ }
+ if ( abs(fs) == 1 ) {
+ problem = true;
+ break;
+ }
+ }
+ fnum.append(fs);
+ ++i; ++i;
+ }
+ if ( problem ) {
+ return true;
+ }
+ ex con = u0.content(x);
+ fnum.append(con);
+ DCOUTVAR(fnum);
+ trying = checkdivisors(fnum, d);
+ }
+ } while ( trying );
+ DCOUT(END generate_set);
+ return false;
+}
+
+#ifdef DEBUGFACTOR
+ex factor(const ex&);
+#endif
+
+static ex factor_multivariate(const ex& poly, const exset& syms)
+{
+ DCOUT(factor_multivariate);
+ DCOUTVAR(poly);
+
+ exset::const_iterator s;
+ const ex& x = *syms.begin();
+ DCOUTVAR(x);
+
+ /* make polynomial primitive */
+ ex p = poly.expand().collect(x);
+ DCOUTVAR(p);
+ ex cont = p.lcoeff(x);
+ for ( numeric i=p.degree(x)-1; i>=p.ldegree(x); --i ) {
+ cont = gcd(cont, p.coeff(x,ex_to<numeric>(i).to_int()));
+ if ( cont == 1 ) break;
+ }
+ DCOUTVAR(cont);
+ ex pp = expand(normal(p / cont));
+ DCOUTVAR(pp);
+ if ( !is_a<numeric>(cont) ) {
+#ifdef DEBUGFACTOR
+ return ::factor(cont) * ::factor(pp);
+#else
+ return factor(cont) * factor(pp);
+#endif
+ }
+
+ /* factor leading coefficient */
+ pp = pp.collect(x);
+ ex vn = pp.lcoeff(x);
+ pp = pp.expand();
+ ex vnlst;
+ if ( is_a<numeric>(vn) ) {
+ vnlst = lst(vn);
+ }
+ else {
+#ifdef DEBUGFACTOR
+ ex vnfactors = ::factor(vn);
+#else
+ ex vnfactors = factor(vn);
+#endif
+ vnlst = put_factors_into_lst(vnfactors);
+ }
+ DCOUTVAR(vnlst);
+
+ const numeric maxtrials = 3;
+ numeric modulus = (vnlst.nops()-1 > 3) ? vnlst.nops()-1 : 3;
+ DCOUTVAR(modulus);
+ numeric minimalr = -1;
+ vector<numeric> a(syms.size()-1, 0);
+ vector<numeric> d((vnlst.nops()-1)/2+1, 0);
+
+ while ( true ) {
+ numeric trialcount = 0;
+ ex u, delta;
+ unsigned int prime;
+ size_t factor_count;
+ ex ufac;
+ ex ufaclst;
+ while ( trialcount < maxtrials ) {
+ bool problem = generate_set(pp, vn, syms, vnlst, modulus, a, d);
+ DCOUTVAR(problem);
+ if ( problem ) {
+ ++modulus;
+ continue;
+ }
+ DCOUTVAR(a);
+ DCOUTVAR(d);
+ u = pp;
+ s = syms.begin();
+ ++s;
+ for ( size_t i=0; i<a.size(); ++i ) {
+ u = u.subs(*s == a[i]);
+ ++s;
+ }
+ delta = u.content(x);
+ DCOUTVAR(u);
+
+ // determine proper prime
+ prime = 3;
+ DCOUTVAR(prime);
+ cl_modint_ring R = find_modint_ring(prime);
+ DCOUTVAR(u.lcoeff(x));
+ while ( true ) {
+ if ( irem(ex_to<numeric>(u.lcoeff(x)), prime) != 0 ) {
+ UniPoly modpoly(R, u, x);
+ UniFactorVec sqrfree_ufv;
+ squarefree(modpoly, sqrfree_ufv);
+ DCOUTVAR(sqrfree_ufv);
+ if ( sqrfree_ufv.factors.size() == 1 && sqrfree_ufv.factors.front().exp == 1 ) break;
+ }
+ prime = next_prime(prime);
+ DCOUTVAR(prime);
+ R = find_modint_ring(prime);
+ }
+
+#ifdef DEBUGFACTOR
+ ufac = ::factor(u);
+#else
+ ufac = factor(u);
+#endif
+ DCOUTVAR(ufac);
+ ufaclst = put_factors_into_lst(ufac);
+ DCOUTVAR(ufaclst);
+ factor_count = (ufaclst.nops()-1)/2;
+ DCOUTVAR(factor_count);
+
+ if ( factor_count <= 1 ) {
+ DCOUTVAR(poly);
+ DCOUT(END factor_multivariate);
+ return poly;
+ }
+
+ if ( minimalr < 0 ) {
+ minimalr = factor_count;
+ }
+ else if ( minimalr == factor_count ) {
+ ++trialcount;
+ ++modulus;
+ }
+ else if ( minimalr > factor_count ) {
+ minimalr = factor_count;
+ trialcount = 0;
+ }
+ DCOUTVAR(trialcount);
+ DCOUTVAR(minimalr);
+ if ( minimalr <= 1 ) {
+ DCOUTVAR(poly);
+ DCOUT(END factor_multivariate);
+ return poly;
+ }
+ }
+
+ vector<numeric> ftilde((vnlst.nops()-1)/2+1);
+ ftilde[0] = ex_to<numeric>(vnlst.op(0));
+ for ( size_t i=1; i<ftilde.size(); ++i ) {
+ ex ft = vnlst.op((i-1)*2+1);
+ s = syms.begin();
+ ++s;
+ for ( size_t j=0; j<a.size(); ++j ) {
+ ft = ft.subs(*s == a[j]);
+ ++s;
+ }
+ ftilde[i] = ex_to<numeric>(ft);
+ }
+ DCOUTVAR(ftilde);
+
+ vector<bool> used_flag((vnlst.nops()-1)/2+1, false);
+ vector<ex> D(factor_count, 1);
+ for ( size_t i=0; i<=factor_count; ++i ) {
+ DCOUTVAR(i);
+ numeric prefac;
+ if ( i == 0 ) {
+ prefac = ex_to<numeric>(ufaclst.op(0));
+ ftilde[0] = ftilde[0] / prefac;
+ vnlst.let_op(0) = vnlst.op(0) / prefac;
+ continue;
+ }
+ else {
+ prefac = ex_to<numeric>(ufaclst.op(2*(i-1)+1).lcoeff(x));
+ }
+ DCOUTVAR(prefac);
+ for ( size_t j=(vnlst.nops()-1)/2+1; j>0; --j ) {
+ DCOUTVAR(j);
+ DCOUTVAR(prefac);
+ DCOUTVAR(ftilde[j-1]);
+ if ( abs(ftilde[j-1]) == 1 ) {
+ used_flag[j-1] = true;
+ continue;
+ }
+ numeric g = gcd(prefac, ftilde[j-1]);
+ DCOUTVAR(g);
+ if ( g != 1 ) {
+ DCOUT(has_common_prime);
+ prefac = prefac / g;
+ numeric count = abs(iquo(g, ftilde[j-1]));
+ DCOUTVAR(count);
+ used_flag[j-1] = true;
+ if ( i > 0 ) {
+ if ( j == 1 ) {
+ D[i-1] = D[i-1] * pow(vnlst.op(0), count);
+ }
+ else {
+ D[i-1] = D[i-1] * pow(vnlst.op(2*(j-2)+1), count);
+ }
+ }
+ else {
+ ftilde[j-1] = ftilde[j-1] / prefac;
+ DCOUT(BREAK);
+ DCOUTVAR(ftilde[j-1]);
+ break;
+ }
+ ++j;
+ }
+ }
+ }
+ DCOUTVAR(D);
+
+ bool some_factor_unused = false;
+ for ( size_t i=0; i<used_flag.size(); ++i ) {
+ if ( !used_flag[i] ) {
+ some_factor_unused = true;
+ break;
+ }
+ }
+ if ( some_factor_unused ) {
+ DCOUT(some factor unused!);
+ continue;
+ }
+
+ vector<ex> C(factor_count);
+ DCOUTVAR(C);
+ DCOUTVAR(delta);
+ if ( delta == 1 ) {
+ for ( size_t i=0; i<D.size(); ++i ) {
+ ex Dtilde = D[i];
+ s = syms.begin();
+ ++s;
+ for ( size_t j=0; j<a.size(); ++j ) {
+ Dtilde = Dtilde.subs(*s == a[j]);
+ ++s;
+ }
+ DCOUTVAR(Dtilde);
+ C[i] = D[i] * (ufaclst.op(2*i+1).lcoeff(x) / Dtilde);
+ }
+ }
+ else {
+ for ( size_t i=0; i<D.size(); ++i ) {
+ ex Dtilde = D[i];
+ s = syms.begin();
+ ++s;
+ for ( size_t j=0; j<a.size(); ++j ) {
+ Dtilde = Dtilde.subs(*s == a[j]);
+ ++s;
+ }
+ ex ui;
+ if ( i == 0 ) {
+ ui = ufaclst.op(0);
+ }
+ else {
+ ui = ufaclst.op(2*(i-1)+1);
+ }
+ while ( true ) {
+ ex d = gcd(ui.lcoeff(x), Dtilde);
+ C[i] = D[i] * ( ui.lcoeff(x) / d );
+ ui = ui * ( Dtilde[i] / d );
+ delta = delta / ( Dtilde[i] / d );
+ if ( delta == 1 ) break;
+ ui = delta * ui;
+ C[i] = delta * C[i];
+ pp = pp * pow(delta, D.size()-1);
+ }
+ }
+ }
+ DCOUTVAR(C);
+
+ EvalPoint ep;
+ vector<EvalPoint> epv;
+ s = syms.begin();
+ ++s;
+ for ( size_t i=0; i<a.size(); ++i ) {
+ ep.x = *s++;
+ ep.evalpoint = a[i].to_int();
+ epv.push_back(ep);
+ }
+ DCOUTVAR(epv);
+
+ // calc bound B
+ ex maxcoeff;
+ for ( int i=u.degree(x); i>=u.ldegree(x); --i ) {
+ maxcoeff += pow(abs(u.coeff(x, i)),2);
+ }
+ cl_I normmc = ceiling1(the<cl_R>(cln::sqrt(ex_to<numeric>(maxcoeff).to_cl_N())));
+ unsigned int maxdegree = 0;
+ for ( size_t i=0; i<factor_count; ++i ) {
+ if ( ufaclst[2*i+1].degree(x) > (int)maxdegree ) {
+ maxdegree = ufaclst[2*i+1].degree(x);
+ }
+ }
+ cl_I B = normmc * expt_pos(cl_I(2), maxdegree);
+ cl_I l = 1;
+ cl_I pl = prime;
+ while ( pl < B ) {
+ l = l + 1;
+ pl = pl * prime;
+ }
+
+ UniPolyVec uvec;
+ cl_modint_ring R = find_modint_ring(expt_pos(cl_I(prime),l));
+ for ( size_t i=0; i<(ufaclst.nops()-1)/2; ++i ) {
+ UniPoly newu(R, ufaclst.op(i*2+1), x);
+ uvec.push_back(newu);
+ }
+ DCOUTVAR(uvec);
+
+ ex res = hensel_multivar(ufaclst.op(0)*pp, x, epv, prime, l, uvec, C);
+ if ( res != lst() ) {
+ ex result = cont * ufaclst.op(0);
+ for ( size_t i=0; i<res.nops(); ++i ) {
+ result *= res.op(i).content(x) * res.op(i).unit(x);
+ result *= res.op(i).primpart(x);
+ }
+ DCOUTVAR(result);
+ DCOUT(END factor_multivariate);
+ return result;
+ }
+ }
}
static ex factor_sqrfree(const ex& poly)
}
if ( findsymbols.syms.size() == 1 ) {
+ // univariate case
const ex& x = *(findsymbols.syms.begin());
if ( poly.ldegree(x) > 0 ) {
int ld = poly.ldegree(x);
}
}
- // multivariate case not yet implemented!
- throw runtime_error("multivariate case not yet implemented!");
+ // multivariate case
+ ex res = factor_multivariate(poly, findsymbols.syms);
+ return res;
}
} // anonymous namespace