index 12405dd696e870fa4f0e001e860ac764f1866095..3afaec54727d32b19e70772827963ce7cb62fa0b 100644 (file)
@@ -1660,14 +1660,27 @@ vector<ex> multivar_diophant(const vector<ex>& a_, const ex& x, const ex& c, con
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));

ex hensel_multivar(const ex& a, const ex& x, const vector<EvalPoint>& I, unsigne
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)
+{
+       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) ) {
+               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;
+       }
+}
+
+static bool checkdivisors(const lst& f, vector<numeric>& d)
+{
+       const int k = f.nops()-2;
+       numeric q, r;
+       d[0] = ex_to<numeric>(f.op(0) * f.op(f.nops()-1));
+       for ( int i=1; i<=k; ++i ) {
+               q = ex_to<numeric>(abs(f.op(i-1)));
+               for ( int j=i-1; j>=0; --j ) {
+                       r = d[j];
+                       do {
+                               r = gcd(r, q);
+                               q = q/r;
+                       } while ( r != 1 );
+                       if ( q == 1 ) {
+                               return true;
+                       }
+               }
+               d[i] = q;
+       }
+       return false;
+}
+
+static void generate_set(const ex& u, const ex& vn, const exset& syms, const ex& f, const numeric& modulus, vector<numeric>& a, vector<numeric>& d)
+{
+       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 ) {
+                       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]);
+               }
+               if ( gcd(u0,u0.diff(ex_to<symbol>(x))) != 1 ) {
+                       continue;
+               }
+               if ( is_a<numeric>(vn) ) {
+                       d = a;
+                       trying = false;
+               }
+               else {
+                       lst fnum;
+                       lst::const_iterator i = ex_to<lst>(f).begin();
+                       fnum.append(*i++);
+                       while ( i!=ex_to<lst>(f).end() ) {
+                               ex fs = *i;
+                               s = syms.begin();
+                               ++s;
+                               for ( size_t j=0; j<a.size(); ++j ) {
+                                       fs = fs.subs(*s == a[j]);
+                               }
+                               fnum.append(fs);
+                               ++i; ++i;
+                       }
+                       ex con = u0.content(x);
+                       fnum.append(con);
+                       trying = checkdivisors(fnum, d);
+               }
+       } while ( trying );
+}
+
+#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 ex factor_multivariate(const ex& poly, const exset& syms)
{
-       // TODO
-       return 666;
+       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) ) {
+               return factor(cont) * factor(pp);
+       }
+
+       /* factor leading coefficient */
+       pp = pp.collect(x);
+       ex vn = p.lcoeff(x);
+       ex vnlst;
+       if ( is_a<numeric>(vn) ) {
+               vnlst = lst(vn);
+       }
+       else {
+               ex vnfactors = factor(vn);
+               vnlst = put_factors_into_lst(vnfactors);
+       }
+       DCOUTVAR(vnlst);
+
+       const numeric maxtrials = 3;
+       numeric modulus = (vnlst.nops()-1 > 3) ? vnlst.nops()-1 : 3;
+       numeric minimalr = -1;
+       vector<numeric> a(syms.size()-1);
+       vector<numeric> d(syms.size()-1);
+
+       while ( true ) {
+               numeric trialcount = 0;
+               ex u, delta;
+               unsigned int prime;
+               UniPolyVec uvec;
+               while ( trialcount < maxtrials ) {
+                       uvec.clear();
+                       generate_set(pp, vn, syms, vnlst, modulus, a, d);
+                       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);
+
+                       // determine proper prime
+                       prime = 3;
+                       cl_modint_ring R = find_modint_ring(prime);
+                       while ( true ) {
+                               if ( irem(ex_to<numeric>(u.lcoeff(x)), prime) != 0 ) {
+                                       UniPoly modpoly(R, u, x);
+                                       UniFactorVec sqrfree_ufv;
+                                       squarefree(modpoly, sqrfree_ufv);
+                                       if ( sqrfree_ufv.factors.size() == 1 && sqrfree_ufv.factors.front().exp == 1 ) break;
+                               }
+                               prime = next_prime(prime);
+                               R = find_modint_ring(prime);
+                       }
+
+                       UniPoly umod(R, u, x);
+                       DCOUTVAR(u);
+                       factor_modular(umod, uvec);
+                       DCOUTVAR(uvec);
+
+                       if ( uvec.size() == 1 ) {
+                               DCOUTVAR(poly);
+                               DCOUT(END factor_multivariate);
+                               return poly;
+                       }
+
+                       if ( minimalr < 0 ) {
+                               minimalr = uvec.size();
+                       }
+                       else if ( minimalr == uvec.size() ) {
+                               ++trialcount;
+                               ++modulus;
+                       }
+                       else if ( minimalr > uvec.size() ) {
+                               minimalr = uvec.size();
+                               trialcount = 0;
+                       }
+                       DCOUTVAR(trialcount);
+                       DCOUTVAR(minimalr);
+                       if ( minimalr == 0 ) {
+                               DCOUTVAR(poly);
+                               DCOUT(END factor_multivariate);
+                               return poly;
+                       }
+               }
+
+               vector<ex> C;
+               if ( vnlst.nops() == 1 ) {
+                       C.resize(uvec.size(), 1);
+               }
+               else {
+
+                       vector<numeric> ftilde((vnlst.nops()-1)/2);
+                       for ( size_t i=0; i<ftilde.size(); ++i ) {
+                               ex ft = vnlst.op(i*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<ex> D;
+                       vector<bool> fflag(ftilde.size(), false);
+                       for ( size_t i=0; i<uvec.size(); ++i ) {
+                               ex ui = uvec[i].to_ex(x);
+                               ex Di = 1;
+                               numeric coeff = ex_to<numeric>(ui.lcoeff(x));
+                               for ( size_t j=0; j<ftilde.size(); ++j ) {
+                                       if ( numeric(coeff / ftilde[j]).is_integer() ) {
+                                               coeff = coeff / ftilde[j];
+                                               Di *= ftilde[j];
+                                               fflag[j] = true;
+                                               --j;
+                                       }
+                               }
+                               D.push_back(Di.expand());
+                       }
+                       for ( size_t i=0; i<fflag.size(); ++i ) {
+                               if ( !fflag[i] ) {
+                                       --minimalr;
+                                       continue;
+                               }
+                       }
+                       DCOUTVAR(D);
+
+                       C.resize(D.size());
+                       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;
+                                       }
+                                       ex Ci = D[i] * (uvec[i].to_ex(x).lcoeff(x) / Dtilde);
+                                       C.push_back(Ci);
+                               }
+                       }
+                       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 = uvec[i].to_ex(x);
+                                       ex Ci;
+                                       while ( true ) {
+                                               ex d = gcd(ui.lcoeff(x), Dtilde);
+                                               Ci = D[i] * ( ui.lcoeff(x) / d );
+                                               ui = ui * ( Dtilde[i] / d );
+                                               delta = delta / ( Dtilde[i] / d );
+                                               if ( delta == 1 ) break;
+                                               ui = delta * ui;
+                                               Ci = delta * Ci;
+                                               pp = pp * pow(delta, D.size()-1);
+                                       }
+                               }
+                       }
+
+               }
+
+               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);
+               }
+
+               // 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<uvec.size(); ++i ) {
+                       if ( uvec[i].degree() > maxdegree ) {
+                               maxdegree = uvec[i].degree();
+                       }
+               }
+               unsigned int B = cl_I_to_uint(normmc * expt_pos(cl_I(2), maxdegree));
+
+               ex res = hensel_multivar(poly, x, epv, prime, B, uvec, C);
+               if ( res != lst() ) {
+                       ex result = cont;
+                       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)
@@ -1806,6 +2177,7 @@ 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);
@@ -1818,8 +2190,9 @@ static ex factor_sqrfree(const ex& poly)
}
}

-       // 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