factor.cpp

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/*
 *  GiNaC Copyright (C) 1999-2022 Johannes Gutenberg University Mainz, Germany
 *
 *  This program is free software; you can redistribute it and/or modify
 *  it under the terms of the GNU General Public License as published by
 *  the Free Software Foundation; either version 2 of the License, or
 *  (at your option) any later version.
 *
 *  This program is distributed in the hope that it will be useful,
 *  but WITHOUT ANY WARRANTY; without even the implied warranty of
 *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *  GNU General Public License for more details.
 *
 *  You should have received a copy of the GNU General Public License
 *  along with this program; if not, write to the Free Software
 *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
 */
 
//#define DEBUGFACTOR
 
#include "factor.h"
 
#include "ex.h"
#include "numeric.h"
#include "operators.h"
#include "inifcns.h"
#include "symbol.h"
#include "relational.h"
#include "power.h"
#include "mul.h"
#include "normal.h"
#include "add.h"
 
#include <algorithm>
#include <limits>
#include <list>
#include <vector>
#include <stack>
#ifdef DEBUGFACTOR
#include <ostream>
#endif
using namespace std;
 
#include <cln/cln.h>
using namespace cln;
 
namespace GiNaC {
 
#ifdef DEBUGFACTOR
#define DCOUT(str) cout << #str << endl
#define DCOUTVAR(var) cout << #var << ": " << var << endl
#define DCOUT2(str,var) cout << #str << ": " << var << endl
ostream& operator<<(ostream& o, const vector<int>& v)
{
    auto i = v.begin(), end = v.end();
    while ( i != end ) {
        o << *i << " ";
        ++i;
    }
    return o;
}
static ostream& operator<<(ostream& o, const vector<cl_I>& v)
{
    auto i = v.begin(), end = v.end();
    while ( i != end ) {
        o << *i << "[" << i-v.begin() << "]" << " ";
        ++i;
    }
    return o;
}
static ostream& operator<<(ostream& o, const vector<cl_MI>& v)
{
    auto i = v.begin(), end = v.end();
    while ( i != end ) {
        o << *i << "[" << i-v.begin() << "]" << " ";
        ++i;
    }
    return o;
}
ostream& operator<<(ostream& o, const vector<numeric>& v)
{
    for ( size_t i=0; i<v.size(); ++i ) {
        o << v[i] << " ";
    }
    return o;
}
ostream& operator<<(ostream& o, const vector<vector<cl_MI>>& v)
{
    auto i = v.begin(), end = v.end();
    while ( i != end ) {
        o << i-v.begin() << ": " << *i << endl;
        ++i;
    }
    return o;
}
#else
#define DCOUT(str)
#define DCOUTVAR(var)
#define DCOUT2(str,var)
#endif // def DEBUGFACTOR
 
// anonymous namespace to hide all utility functions
namespace {
 
// modular univariate polynomial code
 
typedef std::vector<cln::cl_MI> umodpoly;
typedef std::vector<cln::cl_I> upoly;
typedef vector<umodpoly> upvec;
 
// COPY FROM UPOLY.HPP
 
// CHANGED size_t -> int !!!
template<typename T> static int degree(const T& p)
{
    return p.size() - 1;
}
 
template<typename T> static typename T::value_type lcoeff(const T& p)
{
    return p[p.size() - 1];
}
 
static bool normalize_in_field(umodpoly& a)
{
    if (a.size() == 0)
        return true;
    if ( lcoeff(a) == a[0].ring()->one() ) {
        return true;
    }
 
    const cln::cl_MI lc_1 = recip(lcoeff(a));
    for (std::size_t k = a.size(); k-- != 0; )
        a[k] = a[k]*lc_1;
    return false;
}
 
template<typename T> static void
canonicalize(T& p, const typename T::size_type hint = std::numeric_limits<typename T::size_type>::max())
{
    if (p.empty())
        return;
 
    std::size_t i = p.size() - 1;
    // Be fast if the polynomial is already canonicalized
    if (!zerop(p[i]))
        return;
 
    if (hint < p.size())
        i = hint;
 
    bool is_zero = false;
    do {
        if (!zerop(p[i])) {
            ++i;
            break;
        }
        if (i == 0) {
            is_zero = true;
            break;
        }
        --i;
    } while (true);
 
    if (is_zero) {
        p.clear();
        return;
    }
 
    p.erase(p.begin() + i, p.end());
}
 
// END COPY FROM UPOLY.HPP
 
static void expt_pos(umodpoly& a, unsigned int q)
{
    if ( a.empty() ) return;
    cl_MI zero = a[0].ring()->zero(); 
    int deg = degree(a);
    a.resize(degree(a)*q+1, zero);
    for ( int i=deg; i>0; --i ) {
        a[i*q] = a[i];
        a[i] = zero;
    }
}
 
template<bool COND, typename T = void> struct enable_if
{
    typedef T type;
};
 
template<typename T> struct enable_if<false, T> { /* empty */ };
 
template<typename T> struct uvar_poly_p
{
    static const bool value = false;
};
 
template<> struct uvar_poly_p<upoly>
{
    static const bool value = true;
};
 
template<> struct uvar_poly_p<umodpoly>
{
    static const bool value = true;
};
 
template<typename T>
// Don't define this for anything but univariate polynomials.
static typename enable_if<uvar_poly_p<T>::value, T>::type
operator+(const T& a, const T& b)
{
    int sa = a.size();
    int sb = b.size();
    if ( sa >= sb ) {
        T r(sa);
        int i = 0;
        for ( ; i<sb; ++i ) {
            r[i] = a[i] + b[i];
        }
        for ( ; i<sa; ++i ) {
            r[i] = a[i];
        }
        canonicalize(r);
        return r;
    }
    else {
        T r(sb);
        int i = 0;
        for ( ; i<sa; ++i ) {
            r[i] = a[i] + b[i];
        }
        for ( ; i<sb; ++i ) {
            r[i] = b[i];
        }
        canonicalize(r);
        return r;
    }
}
 
template<typename T>
// Don't define this for anything but univariate polynomials. Otherwise
// overload resolution might fail (this actually happens when compiling
// GiNaC with g++ 3.4).
static typename enable_if<uvar_poly_p<T>::value, T>::type
operator-(const T& a, const T& b)
{
    int sa = a.size();
    int sb = b.size();
    if ( sa >= sb ) {
        T r(sa);
        int i = 0;
        for ( ; i<sb; ++i ) {
            r[i] = a[i] - b[i];
        }
        for ( ; i<sa; ++i ) {
            r[i] = a[i];
        }
        canonicalize(r);
        return r;
    }
    else {
        T r(sb);
        int i = 0;
        for ( ; i<sa; ++i ) {
            r[i] = a[i] - b[i];
        }
        for ( ; i<sb; ++i ) {
            r[i] = -b[i];
        }
        canonicalize(r);
        return r;
    }
}
 
static upoly operator*(const upoly& a, const upoly& b)
{
    upoly c;
    if ( a.empty() || b.empty() ) return c;
 
    int n = degree(a) + degree(b);
    c.resize(n+1, 0);
    for ( int i=0 ; i<=n; ++i ) {
        for ( int j=0 ; j<=i; ++j ) {
            if ( j > degree(a) || (i-j) > degree(b) ) continue;
            c[i] = c[i] + a[j] * b[i-j];
        }
    }
    canonicalize(c);
    return c;
}
 
static umodpoly operator*(const umodpoly& a, const umodpoly& b)
{
    umodpoly c;
    if ( a.empty() || b.empty() ) return c;
 
    int n = degree(a) + degree(b);
    c.resize(n+1, a[0].ring()->zero());
    for ( int i=0 ; i<=n; ++i ) {
        for ( int j=0 ; j<=i; ++j ) {
            if ( j > degree(a) || (i-j) > degree(b) ) continue;
            c[i] = c[i] + a[j] * b[i-j];
        }
    }
    canonicalize(c);
    return c;
}
 
static upoly operator*(const upoly& a, const cl_I& x)
{
    if ( zerop(x) ) {
        upoly r;
        return r;
    }
    upoly r(a.size());
    for ( size_t i=0; i<a.size(); ++i ) {
        r[i] = a[i] * x;
    }
    return r;
}
 
static upoly operator/(const upoly& a, const cl_I& x)
{
    if ( zerop(x) ) {
        upoly r;
        return r;
    }
    upoly r(a.size());
    for ( size_t i=0; i<a.size(); ++i ) {
        r[i] = exquo(a[i],x);
    }
    return r;
}
 
static umodpoly operator*(const umodpoly& a, const cl_MI& x)
{
    umodpoly r(a.size());
    for ( size_t i=0; i<a.size(); ++i ) {
        r[i] = a[i] * x;
    }
    canonicalize(r);
    return r;
}
 
static void upoly_from_ex(upoly& up, const ex& e, const ex& x)
{
    // assert: e is in Z[x]
    int deg = e.degree(x);
    up.resize(deg+1);
    int ldeg = e.ldegree(x);
    for ( ; deg>=ldeg; --deg ) {
        up[deg] = the<cl_I>(ex_to<numeric>(e.coeff(x, deg)).to_cl_N());
    }
    for ( ; deg>=0; --deg ) {
        up[deg] = 0;
    }
    canonicalize(up);
}
 
static void umodpoly_from_upoly(umodpoly& ump, const upoly& e, const cl_modint_ring& R)
{
    int deg = degree(e);
    ump.resize(deg+1);
    for ( ; deg>=0; --deg ) {
        ump[deg] = R->canonhom(e[deg]);
    }
    canonicalize(ump);
}
 
static void umodpoly_from_ex(umodpoly& ump, const ex& e, const ex& x, const cl_modint_ring& R)
{
    // assert: e is in Z[x]
    int deg = e.degree(x);
    ump.resize(deg+1);
    int ldeg = e.ldegree(x);
    for ( ; deg>=ldeg; --deg ) {
        cl_I coeff = the<cl_I>(ex_to<numeric>(e.coeff(x, deg)).to_cl_N());
        ump[deg] = R->canonhom(coeff);
    }
    for ( ; deg>=0; --deg ) {
        ump[deg] = R->zero();
    }
    canonicalize(ump);
}
 
#ifdef DEBUGFACTOR
static void umodpoly_from_ex(umodpoly& ump, const ex& e, const ex& x, const cl_I& modulus)
{
    umodpoly_from_ex(ump, e, x, find_modint_ring(modulus));
}
#endif
 
static ex upoly_to_ex(const upoly& a, const ex& x)
{
    if ( a.empty() ) return 0;
    ex e;
    for ( int i=degree(a); i>=0; --i ) {
        e += numeric(a[i]) * pow(x, i);
    }
    return e;
}
 
static ex umodpoly_to_ex(const umodpoly& a, const ex& x)
{
    if ( a.empty() ) return 0;
    cl_modint_ring R = a[0].ring();
    cl_I mod = R->modulus;
    cl_I halfmod = (mod-1) >> 1;
    ex e;
    for ( int i=degree(a); i>=0; --i ) {
        cl_I n = R->retract(a[i]);
        if ( n > halfmod ) {
            e += numeric(n-mod) * pow(x, i);
        } else {
            e += numeric(n) * pow(x, i);
        }
    }
    return e;
}
 
static upoly umodpoly_to_upoly(const umodpoly& a)
{
    upoly e(a.size());
    if ( a.empty() ) return e;
    cl_modint_ring R = a[0].ring();
    cl_I mod = R->modulus;
    cl_I halfmod = (mod-1) >> 1;
    for ( int i=degree(a); i>=0; --i ) {
        cl_I n = R->retract(a[i]);
        if ( n > halfmod ) {
            e[i] = n-mod;
        } else {
            e[i] = n;
        }
    }
    return e;
}
 
static umodpoly umodpoly_to_umodpoly(const umodpoly& a, const cl_modint_ring& R, unsigned int m)
{
    umodpoly e;
    if ( a.empty() ) return e;
    cl_modint_ring oldR = a[0].ring();
    size_t sa = a.size();
    e.resize(sa+m, R->zero());
    for ( size_t i=0; i<sa; ++i ) {
        e[i+m] = R->canonhom(oldR->retract(a[i]));
    }
    canonicalize(e);
    return e;
}
 
static void reduce_coeff(umodpoly& a, const cl_I& x)
{
    if ( a.empty() ) return;
 
    cl_modint_ring R = a[0].ring();
    for (auto & i : a) {
        // cln cannot perform this division in the modular field
        cl_I c = R->retract(i);
        i = cl_MI(R, the<cl_I>(c / x));
    }
}
 
static void rem(const umodpoly& a, const umodpoly& b, umodpoly& r)
{
    int k, n;
    n = degree(b);
    k = degree(a) - n;
    r = a;
    if ( k < 0 ) return;
 
    do {
        cl_MI qk = div(r[n+k], b[n]);
        if ( !zerop(qk) ) {
            for ( int i=0; i<n; ++i ) {
                unsigned int j = n + k - 1 - i;
                r[j] = r[j] - qk * b[j-k];
            }
        }
    } while ( k-- );
 
    fill(r.begin()+n, r.end(), a[0].ring()->zero());
    canonicalize(r);
}
 
static void div(const umodpoly& a, const umodpoly& b, umodpoly& q)
{
    int k, n;
    n = degree(b);
    k = degree(a) - n;
    q.clear();
    if ( k < 0 ) return;
 
    umodpoly r = a;
    q.resize(k+1, a[0].ring()->zero());
    do {
        cl_MI qk = div(r[n+k], b[n]);
        if ( !zerop(qk) ) {
            q[k] = qk;
            for ( int i=0; i<n; ++i ) {
                unsigned int j = n + k - 1 - i;
                r[j] = r[j] - qk * b[j-k];
            }
        }
    } while ( k-- );
 
    canonicalize(q);
}
 
static void remdiv(const umodpoly& a, const umodpoly& b, umodpoly& r, umodpoly& q)
{
    int k, n;
    n = degree(b);
    k = degree(a) - n;
    q.clear();
    r = a;
    if ( k < 0 ) return;
 
    q.resize(k+1, a[0].ring()->zero());
    do {
        cl_MI qk = div(r[n+k], b[n]);
        if ( !zerop(qk) ) {
            q[k] = qk;
            for ( int i=0; i<n; ++i ) {
                unsigned int j = n + k - 1 - i;
                r[j] = r[j] - qk * b[j-k];
            }
        }
    } while ( k-- );
 
    fill(r.begin()+n, r.end(), a[0].ring()->zero());
    canonicalize(r);
    canonicalize(q);
}
 
static void gcd(const umodpoly& a, const umodpoly& b, umodpoly& c)
{
    if ( degree(a) < degree(b) ) return gcd(b, a, c);
 
    c = a;
    normalize_in_field(c);
    umodpoly d = b;
    normalize_in_field(d);
    umodpoly r;
    while ( !d.empty() ) {
        rem(c, d, r);
        c = d;
        d = r;
    }
    normalize_in_field(c);
}
 
static void deriv(const umodpoly& a, umodpoly& d)
{
    d.clear();
    if ( a.size() <= 1 ) return;
 
    d.insert(d.begin(), a.begin()+1, a.end());
    int max = d.size();
    for ( int i=1; i<max; ++i ) {
        d[i] = d[i] * (i+1);
    }
    canonicalize(d);
}
 
static bool unequal_one(const umodpoly& a)
{
    if ( a.empty() ) return true;
    return ( a.size() != 1 || a[0] != a[0].ring()->one() );
}
 
static bool equal_one(const umodpoly& a)
{
    return ( a.size() == 1 && a[0] == a[0].ring()->one() );
}
 
static bool squarefree(const umodpoly& a)
{
    umodpoly b;
    deriv(a, b);
    if ( b.empty() ) {
        return false;
    }
    umodpoly c;
    gcd(a, b, c);
    return equal_one(c);
}
 
// END modular univariate polynomial code
 
// modular matrix
 
typedef vector<cl_MI> mvec;
 
class modular_matrix
{
#ifdef DEBUGFACTOR
    friend ostream& operator<<(ostream& o, const modular_matrix& m);
#endif
public:
    modular_matrix(size_t r_, size_t c_, const cl_MI& init) : r(r_), c(c_)
    {
        m.resize(c*r, init);
    }
    size_t rowsize() const { return r; }
    size_t colsize() const { return c; }
    cl_MI& operator()(size_t row, size_t col) { return m[row*c + col]; }
    cl_MI operator()(size_t row, size_t col) const { return m[row*c + col]; }
    void mul_col(size_t col, const cl_MI x)
    {
        for ( size_t rc=0; rc<r; ++rc ) {
            std::size_t i = c*rc + col;
            m[i] = m[i] * x;
        }
    }
    void sub_col(size_t col1, size_t col2, const cl_MI fac)
    {
        for ( size_t rc=0; rc<r; ++rc ) {
            std::size_t i1 = col1 + c*rc;
            std::size_t i2 = col2 + c*rc;
            m[i1] = m[i1] - m[i2]*fac;
        }
    }
    void switch_col(size_t col1, size_t col2)
    {
        for ( size_t rc=0; rc<r; ++rc ) {
            std::size_t i1 = col1 + rc*c;
            std::size_t i2 = col2 + rc*c;
            std::swap(m[i1], m[i2]);
        }
    }
    void mul_row(size_t row, const cl_MI x)
    {
        for ( size_t cc=0; cc<c; ++cc ) {
            std::size_t i = row*c + cc; 
            m[i] = m[i] * x;
        }
    }
    void sub_row(size_t row1, size_t row2, const cl_MI fac)
    {
        for ( size_t cc=0; cc<c; ++cc ) {
            std::size_t i1 = row1*c + cc;
            std::size_t i2 = row2*c + cc;
            m[i1] = m[i1] - m[i2]*fac;
        }
    }
    void switch_row(size_t row1, size_t row2)
    {
        for ( size_t cc=0; cc<c; ++cc ) {
            std::size_t i1 = row1*c + cc;
            std::size_t i2 = row2*c + cc;
            std::swap(m[i1], m[i2]);
        }
    }
    bool is_col_zero(size_t col) const
    {
        for ( size_t rr=0; rr<r; ++rr ) {
            std::size_t i = col + rr*c;
            if ( !zerop(m[i]) ) {
                return false;
            }
        }
        return true;
    }
    bool is_row_zero(size_t row) const
    {
        for ( size_t cc=0; cc<c; ++cc ) {
            std::size_t i = row*c + cc;
            if ( !zerop(m[i]) ) {
                return false;
            }
        }
        return true;
    }
    void set_row(size_t row, const vector<cl_MI>& newrow)
    {
        for (std::size_t i2 = 0; i2 < newrow.size(); ++i2) {
            std::size_t i1 = row*c + i2;
            m[i1] = newrow[i2];
        }
    }
    mvec::const_iterator row_begin(size_t row) const { return m.begin()+row*c; }
    mvec::const_iterator row_end(size_t row) const { return m.begin()+row*c+r; }
private:
    size_t r, c;
    mvec m;
};
 
#ifdef DEBUGFACTOR
modular_matrix operator*(const modular_matrix& m1, const modular_matrix& m2)
{
    const unsigned int r = m1.rowsize();
    const unsigned int c = m2.colsize();
    modular_matrix o(r,c,m1(0,0));
 
    for ( size_t i=0; i<r; ++i ) {
        for ( size_t j=0; j<c; ++j ) {
            cl_MI buf;
            buf = m1(i,0) * m2(0,j);
            for ( size_t k=1; k<c; ++k ) {
                buf = buf + m1(i,k)*m2(k,j);
            }
            o(i,j) = buf;
        }
    }
    return o;
}
 
ostream& operator<<(ostream& o, const modular_matrix& m)
{
    cl_modint_ring R = m(0,0).ring();
    o << "{";
    for ( size_t i=0; i<m.rowsize(); ++i ) {
        o << "{";
        for ( size_t j=0; j<m.colsize()-1; ++j ) {
            o << R->retract(m(i,j)) << ",";
        }
        o << R->retract(m(i,m.colsize()-1)) << "}";
        if ( i != m.rowsize()-1 ) {
            o << ",";
        }
    }
    o << "}";
    return o;
}
#endif // def DEBUGFACTOR
 
// END modular matrix
 
static void q_matrix(const umodpoly& a_, modular_matrix& Q)
{
    umodpoly a = a_;
    normalize_in_field(a);
 
    int n = degree(a);
    unsigned int q = cl_I_to_uint(a[0].ring()->modulus);
    umodpoly r(n, a[0].ring()->zero());
    r[0] = a[0].ring()->one();
    Q.set_row(0, r);
    unsigned int max = (n-1) * q;
    for ( size_t m=1; m<=max; ++m ) {
        cl_MI rn_1 = r.back();
        for ( size_t i=n-1; i>0; --i ) {
            r[i] = r[i-1] - (rn_1 * a[i]);
        }
        r[0] = -rn_1 * a[0];
        if ( (m % q) == 0 ) {
            Q.set_row(m/q, r);
        }
    }
}
 
static void nullspace(modular_matrix& M, vector<mvec>& basis)
{
    const size_t n = M.rowsize();
    const cl_MI one = M(0,0).ring()->one();
    for ( size_t i=0; i<n; ++i ) {
        M(i,i) = M(i,i) - one;
    }
    for ( size_t r=0; r<n; ++r ) {
        size_t cc = 0;
        for ( ; cc<n; ++cc ) {
            if ( !zerop(M(r,cc)) ) {
                if ( cc < r ) {
                    if ( !zerop(M(cc,cc)) ) {
                        continue;
                    }
                    M.switch_col(cc, r);
                }
                else if ( cc > r ) {
                    M.switch_col(cc, r);
                }
                break;
            }
        }
        if ( cc < n ) {
            M.mul_col(r, recip(M(r,r)));
            for ( cc=0; cc<n; ++cc ) {
                if ( cc != r ) {
                    M.sub_col(cc, r, M(r,cc));
                }
            }
        }
    }
 
    for ( size_t i=0; i<n; ++i ) {
        M(i,i) = M(i,i) - one;
    }
    for ( size_t i=0; i<n; ++i ) {
        if ( !M.is_row_zero(i) ) {
            mvec nu(M.row_begin(i), M.row_end(i));
            basis.push_back(nu);
        }
    }
}
 
static void berlekamp(const umodpoly& a, upvec& upv)
{
    cl_modint_ring R = a[0].ring();
    umodpoly one(1, R->one());
 
    // find nullspace of Q matrix
    modular_matrix Q(degree(a), degree(a), R->zero());
    q_matrix(a, Q);
    vector<mvec> nu;
    nullspace(Q, nu);
 
    const unsigned int k = nu.size();
    if ( k == 1 ) {
        // irreducible
        return;
    }
 
    list<umodpoly> factors = {a};
    unsigned int size = 1;
    unsigned int r = 1;
    unsigned int q = cl_I_to_uint(R->modulus);
 
    list<umodpoly>::iterator u = factors.begin();
 
    // calculate all gcd's
    while ( true ) {
        for ( unsigned int s=0; s<q; ++s ) {
            umodpoly nur = nu[r];
            nur[0] = nur[0] - cl_MI(R, s);
            canonicalize(nur);
            umodpoly g;
            gcd(nur, *u, g);
            if ( unequal_one(g) && g != *u ) {
                umodpoly uo;
                div(*u, g, uo);
                if ( equal_one(uo) ) {
                    throw logic_error("berlekamp: unexpected divisor.");
                } else {
                    *u = uo;
                }
                factors.push_back(g);
                size = 0;
                for (auto & i : factors) {
                    if (degree(i))
                        ++size;
                }
                if ( size == k ) {
                    for (auto & i : factors) {
                        upv.push_back(i);
                    }
                    return;
                }
            }
        }
        if ( ++r == k ) {
            r = 1;
            ++u;
        }
    }
}
 
// modular square free factorization is not used at the moment so we deactivate
// the code
#if 0
 
static void expt_1_over_p(const umodpoly& a, unsigned int prime, umodpoly& ap)
{
    size_t newdeg = degree(a)/prime;
    ap.resize(newdeg+1);
    ap[0] = a[0];
    for ( size_t i=1; i<=newdeg; ++i ) {
        ap[i] = a[i*prime];
    }
}
 
static void modsqrfree(const umodpoly& a, upvec& factors, vector<int>& mult)
{
    const unsigned int prime = cl_I_to_uint(a[0].ring()->modulus);
    int i = 1;
    umodpoly b;
    deriv(a, b);
    if ( b.size() ) {
        umodpoly c;
        gcd(a, b, c);
        umodpoly w;
        div(a, c, w);
        while ( unequal_one(w) ) {
            umodpoly y;
            gcd(w, c, y);
            umodpoly z;
            div(w, y, z);
            factors.push_back(z);
            mult.push_back(i);
            ++i;
            w = y;
            umodpoly buf;
            div(c, y, buf);
            c = buf;
        }
        if ( unequal_one(c) ) {
            umodpoly cp;
            expt_1_over_p(c, prime, cp);
            size_t previ = mult.size();
            modsqrfree(cp, factors, mult);
            for ( size_t i=previ; i<mult.size(); ++i ) {
                mult[i] *= prime;
            }
        }
    } else {
        umodpoly ap;
        expt_1_over_p(a, prime, ap);
        size_t previ = mult.size();
        modsqrfree(ap, factors, mult);
        for ( size_t i=previ; i<mult.size(); ++i ) {
            mult[i] *= prime;
        }
    }
}
 
#endif // deactivation of square free factorization
 
static void distinct_degree_factor(const umodpoly& a_, vector<int>& degrees, upvec& ddfactors)
{
    umodpoly a = a_;
 
    cl_modint_ring R = a[0].ring();
    int q = cl_I_to_int(R->modulus);
    int nhalf = degree(a)/2;
 
    int i = 1;
    umodpoly w(2);
    w[0] = R->zero();
    w[1] = R->one();
    umodpoly x = w;
 
    while ( i <= nhalf ) {
        expt_pos(w, q);
        umodpoly buf;
        rem(w, a, buf);
        w = buf;
        umodpoly wx = w - x;
        gcd(a, wx, buf);
        if ( unequal_one(buf) ) {
            degrees.push_back(i);
            ddfactors.push_back(buf);
        }
        if ( unequal_one(buf) ) {
            umodpoly buf2;
            div(a, buf, buf2);
            a = buf2;
            nhalf = degree(a)/2;
            rem(w, a, buf);
            w = buf;
        }
        ++i;
    }
    if ( unequal_one(a) ) {
        degrees.push_back(degree(a));
        ddfactors.push_back(a);
    }
}
 
static void same_degree_factor(const umodpoly& a, upvec& upv)
{
    cl_modint_ring R = a[0].ring();
 
    vector<int> degrees;
    upvec ddfactors;
    distinct_degree_factor(a, degrees, ddfactors);
 
    for ( size_t i=0; i<degrees.size(); ++i ) {
        if ( degrees[i] == degree(ddfactors[i]) ) {
            upv.push_back(ddfactors[i]);
        } else {
            berlekamp(ddfactors[i], upv);
        }
    }
}
 
// Yes, we can (choose).
#define USE_SAME_DEGREE_FACTOR
 
static void factor_modular(const umodpoly& p, upvec& upv)
{
#ifdef USE_SAME_DEGREE_FACTOR
    same_degree_factor(p, upv);
#else
    berlekamp(p, upv);
#endif
}
 
static void exteuclid(const umodpoly& a, const umodpoly& b, umodpoly& s, umodpoly& t)
{
    if ( degree(a) < degree(b) ) {
        exteuclid(b, a, t, s);
        return;
    }
 
    umodpoly one(1, a[0].ring()->one());
    umodpoly c = a; normalize_in_field(c);
    umodpoly d = b; normalize_in_field(d);
    s = one;
    t.clear();
    umodpoly d1;
    umodpoly d2 = one;
    umodpoly q;
    while ( true ) {
        div(c, d, q);
        umodpoly r = c - q * d;
        umodpoly r1 = s - q * d1;
        umodpoly r2 = t - q * d2;
        c = d;
        s = d1;
        t = d2;
        if ( r.empty() ) break;
        d = r;
        d1 = r1;
        d2 = r2;
    }
    cl_MI fac = recip(lcoeff(a) * lcoeff(c));
    for (auto & i : s) {
        i = i * fac;
    }
    canonicalize(s);
    fac = recip(lcoeff(b) * lcoeff(c));
    for (auto & i : t) {
        i = i * fac;
    }
    canonicalize(t);
}
 
static upoly replace_lc(const upoly& poly, const cl_I& lc)
{
    if ( poly.empty() ) return poly;
    upoly r = poly;
    r.back() = lc;
    return r;
}
 
static inline cl_I calc_bound(const ex& a, const ex& x, int maxdeg)
{
    cl_I maxcoeff = 0;
    cl_R coeff = 0;
    for ( int i=a.degree(x); i>=a.ldegree(x); --i ) {
        cl_I aa = abs(the<cl_I>(ex_to<numeric>(a.coeff(x, i)).to_cl_N()));
        if ( aa > maxcoeff ) maxcoeff = aa;
        coeff = coeff + square(aa);
    }
    cl_I coeffnorm = ceiling1(the<cl_R>(cln::sqrt(coeff)));
    cl_I B = coeffnorm * expt_pos(cl_I(2), cl_I(maxdeg));
    return ( B > maxcoeff ) ? B : maxcoeff;
}
 
static inline cl_I calc_bound(const upoly& a, int maxdeg)
{
    cl_I maxcoeff = 0;
    cl_R coeff = 0;
    for ( int i=degree(a); i>=0; --i ) {
        cl_I aa = abs(a[i]);
        if ( aa > maxcoeff ) maxcoeff = aa;
        coeff = coeff + square(aa);
    }
    cl_I coeffnorm = ceiling1(the<cl_R>(cln::sqrt(coeff)));
    cl_I B = coeffnorm * expt_pos(cl_I(2), cl_I(maxdeg));
    return ( B > maxcoeff ) ? B : maxcoeff;
}
 
static void hensel_univar(const upoly& a_, unsigned int p, const umodpoly& u1_, const umodpoly& w1_, upoly& u, upoly& w)
{
    upoly a = a_;
    const cl_modint_ring& R = u1_[0].ring();
 
    // calc bound B
    int maxdeg = (degree(u1_) > degree(w1_)) ? degree(u1_) : degree(w1_);
    cl_I maxmodulus = 2*calc_bound(a, maxdeg);
 
    // step 1
    cl_I alpha = lcoeff(a);
    a = a * alpha;
    umodpoly nu1 = u1_;
    normalize_in_field(nu1);
    umodpoly nw1 = w1_;
    normalize_in_field(nw1);
    upoly phi;
    phi = umodpoly_to_upoly(nu1) * alpha;
    umodpoly u1;
    umodpoly_from_upoly(u1, phi, R);
    phi = umodpoly_to_upoly(nw1) * alpha;
    umodpoly w1;
    umodpoly_from_upoly(w1, phi, R);
 
    // step 2
    umodpoly s;
    umodpoly t;
    exteuclid(u1, w1, s, t);
 
    // step 3
    u = replace_lc(umodpoly_to_upoly(u1), alpha);
    w = replace_lc(umodpoly_to_upoly(w1), alpha);
    upoly e = a - u * w;
    cl_I modulus = p;
 
    // step 4
    while ( !e.empty() && modulus < maxmodulus ) {
        upoly c = e / modulus;
        phi = umodpoly_to_upoly(s) * c;
        umodpoly sigmatilde;
        umodpoly_from_upoly(sigmatilde, phi, R);
        phi = umodpoly_to_upoly(t) * c;
        umodpoly tautilde;
        umodpoly_from_upoly(tautilde, phi, R);
        umodpoly r, q;
        remdiv(sigmatilde, w1, r, q);
        umodpoly sigma = r;
        phi = umodpoly_to_upoly(tautilde) + umodpoly_to_upoly(q) * umodpoly_to_upoly(u1);
        umodpoly tau;
        umodpoly_from_upoly(tau, phi, R);
        u = u + umodpoly_to_upoly(tau) * modulus;
        w = w + umodpoly_to_upoly(sigma) * modulus;
        e = a - u * w;
        modulus = modulus * p;
    }
 
    // step 5
    if ( e.empty() ) {
        cl_I g = u[0];
        for ( size_t i=1; i<u.size(); ++i ) {
            g = gcd(g, u[i]);
            if ( g == 1 ) break;
        }
        if ( g != 1 ) {
            u = u / g;
            w = w * g;
        }
        if ( alpha != 1 ) {
            w = w / alpha;
        }
    } else {
        u.clear();
    }
}
 
static unsigned int next_prime(unsigned int p)
{
    static vector<unsigned int> primes;
    if (primes.empty()) {
        primes = {3, 5, 7};
    }
    if ( p >= primes.back() ) {
        unsigned int candidate = primes.back() + 2;
        while ( true ) {
            size_t n = primes.size()/2;
            for ( size_t i=0; i<n; ++i ) {
                if (candidate % primes[i])
                    continue;
                candidate += 2;
                i=-1;
            }
            primes.push_back(candidate);
            if (candidate > p)
                break;
        }
        return candidate;
    }
    for (auto & it : primes) {
        if ( it > p ) {
            return it;
        }
    }
    throw logic_error("next_prime: should not reach this point!");
}
 
class factor_partition
{
public:
    factor_partition(const upvec& factors_) : factors(factors_)
    {
        n = factors.size();
        k.resize(n, 0);
        k[0] = 1;
        cache.resize(n-1);
        one.resize(1, factors.front()[0].ring()->one());
        len = 1;
        last = 0;
        split();
    }
    int operator[](size_t i) const { return k[i]; }
    size_t size() const { return n; }
    size_t size_left() const { return n-len; }
    size_t size_right() const { return len; }
    bool next()
    {
        if ( last == n-1 ) {
            int rem = len - 1;
            int p = last - 1;
            while ( rem ) {
                if ( k[p] ) {
                    --rem;
                    --p;
                    continue;
                }
                last = p - 1;
                while ( k[last] == 0 ) { --last; }
                if ( last == 0 && n == 2*len ) return false;
                k[last++] = 0;
                for ( size_t i=0; i<=len-rem; ++i ) {
                    k[last] = 1;
                    ++last;
                }
                fill(k.begin()+last, k.end(), 0);
                --last;
                split();
                return true;
            }
            last = len;
            ++len;
            if ( len > n/2 ) return false;
            fill(k.begin(), k.begin()+len, 1);
            fill(k.begin()+len+1, k.end(), 0);
        } else {
            k[last++] = 0;
            k[last] = 1;
        }
        split();
        return true;
    }
    umodpoly& left() { return lr[0]; }
    umodpoly& right() { return lr[1]; }
private:
    void split_cached()
    {
        size_t i = 0;
        do {
            size_t pos = i;
            int group = k[i++];
            size_t d = 0;
            while ( i < n && k[i] == group ) { ++d; ++i; }
            if ( d ) {
                if ( cache[pos].size() >= d ) {
                    lr[group] = lr[group] * cache[pos][d-1];
                } else {
                    if ( cache[pos].size() == 0 ) {
                        cache[pos].push_back(factors[pos] * factors[pos+1]);
                    }
                    size_t j = pos + cache[pos].size() + 1;
                    d -= cache[pos].size();
                    while ( d ) {
                        umodpoly buf = cache[pos].back() * factors[j];
                        cache[pos].push_back(buf);
                        --d;
                        ++j;
                    }
                    lr[group] = lr[group] * cache[pos].back();
                }
            } else {
                lr[group] = lr[group] * factors[pos];
            }
        } while ( i < n );
    }
    void split()
    {
        lr[0] = one;
        lr[1] = one;
        if ( n > 6 ) {
            split_cached();
        } else {
            for ( size_t i=0; i<n; ++i ) {
                lr[k[i]] = lr[k[i]] * factors[i];
            }
        }
    }
private:
    umodpoly lr[2];
    vector<vector<umodpoly>> cache;
    upvec factors;
    umodpoly one;
    size_t n;
    size_t len;
    size_t last;
    vector<int> k;
};
 
struct ModFactors
{
    upoly poly;
    upvec factors;
};
 
static ex factor_univariate(const ex& poly, const ex& x, unsigned int& prime)
{
    ex unit, cont, prim_ex;
    poly.unitcontprim(x, unit, cont, prim_ex);
    upoly prim;
    upoly_from_ex(prim, prim_ex, x);
    if (prim_ex.is_equal(1)) {
        return poly;
    }
 
    // determine proper prime and minimize number of modular factors
    prime = 3;
    unsigned int lastp = prime;
    cl_modint_ring R;
    unsigned int trials = 0;
    unsigned int minfactors = 0;
 
    const numeric& cont_n = ex_to<numeric>(cont);
    cl_I i_cont;
    if (cont_n.is_integer()) {
        i_cont = the<cl_I>(cont_n.to_cl_N());
    } else {
        // poly \in Q[x] => poly = q ipoly, ipoly \in Z[x], q \in Q
        // factor(poly) \equiv q factor(ipoly)
        i_cont = cl_I(1);
    }
    cl_I lc = lcoeff(prim)*i_cont;
    upvec factors;
    while ( trials < 2 ) {
        umodpoly modpoly;
        while ( true ) {
            prime = next_prime(prime);
            if ( !zerop(rem(lc, prime)) ) {
                R = find_modint_ring(prime);
                umodpoly_from_upoly(modpoly, prim, R);
                if ( squarefree(modpoly) ) break;
            }
        }
 
        // do modular factorization
        upvec trialfactors;
        factor_modular(modpoly, trialfactors);
        if ( trialfactors.size() <= 1 ) {
            // irreducible for sure
            return poly;
        }
 
        if ( minfactors == 0 || trialfactors.size() < minfactors ) {
            factors = trialfactors;
            minfactors = trialfactors.size();
            lastp = prime;
            trials = 1;
        } else {
            ++trials;
        }
    }
    prime = lastp;
    R = find_modint_ring(prime);
 
    // lift all factor combinations
    stack<ModFactors> tocheck;
    ModFactors mf;
    mf.poly = prim;
    mf.factors = factors;
    tocheck.push(mf);
    upoly f1, f2;
    ex result = 1;
    while ( tocheck.size() ) {
        const size_t n = tocheck.top().factors.size();
        factor_partition part(tocheck.top().factors);
        while ( true ) {
            // call Hensel lifting
            hensel_univar(tocheck.top().poly, prime, part.left(), part.right(), f1, f2);
            if ( !f1.empty() ) {
                // successful, update the stack and the result
                if ( part.size_left() == 1 ) {
                    if ( part.size_right() == 1 ) {
                        result *= upoly_to_ex(f1, x) * upoly_to_ex(f2, x);
                        tocheck.pop();
                        break;
                    }
                    result *= upoly_to_ex(f1, x);
                    tocheck.top().poly = f2;
                    for ( size_t i=0; i<n; ++i ) {
                        if ( part[i] == 0 ) {
                            tocheck.top().factors.erase(tocheck.top().factors.begin()+i);
                            break;
                        }
                    }
                    break;
                }
                else if ( part.size_right() == 1 ) {
                    if ( part.size_left() == 1 ) {
                        result *= upoly_to_ex(f1, x) * upoly_to_ex(f2, x);
                        tocheck.pop();
                        break;
                    }
                    result *= upoly_to_ex(f2, x);
                    tocheck.top().poly = f1;
                    for ( size_t i=0; i<n; ++i ) {
                        if ( part[i] == 1 ) {
                            tocheck.top().factors.erase(tocheck.top().factors.begin()+i);
                            break;
                        }
                    }
                    break;
                } else {
                    upvec newfactors1(part.size_left()), newfactors2(part.size_right());
                    auto i1 = newfactors1.begin(), i2 = newfactors2.begin();
                    for ( size_t i=0; i<n; ++i ) {
                        if ( part[i] ) {
                            *i2++ = tocheck.top().factors[i];
                        } else {
                            *i1++ = tocheck.top().factors[i];
                        }
                    }
                    tocheck.top().factors = newfactors1;
                    tocheck.top().poly = f1;
                    ModFactors mf;
                    mf.factors = newfactors2;
                    mf.poly = f2;
                    tocheck.push(mf);
                    break;
                }
            } else {
                // not successful
                if ( !part.next() ) {
                    // if no more combinations left, return polynomial as
                    // irreducible
                    result *= upoly_to_ex(tocheck.top().poly, x);
                    tocheck.pop();
                    break;
                }
            }
        }
    }
 
    return unit * cont * result;
}
 
static inline ex factor_univariate(const ex& poly, const ex& x)
{
    unsigned int prime;
    return factor_univariate(poly, x, prime);
}
 
struct EvalPoint
{
    ex x;
    int evalpoint;
};
 
#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
 
// forward declaration
static vector<ex> multivar_diophant(const vector<ex>& a_, const ex& x, const ex& c, const vector<EvalPoint>& I, unsigned int d, unsigned int p, unsigned int k);
 
static upvec multiterm_eea_lift(const upvec& a, const ex& x, unsigned int p, unsigned int k)
{
    const size_t r = a.size();
    cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
    upvec q(r-1);
    q[r-2] = a[r-1];
    for ( size_t j=r-2; j>=1; --j ) {
        q[j-1] = a[j] * q[j];
    }
    umodpoly beta(1, R->one());
    upvec s;
    for ( size_t j=1; j<r; ++j ) {
        vector<ex> mdarg(2);
        mdarg[0] = umodpoly_to_ex(q[j-1], x);
        mdarg[1] = umodpoly_to_ex(a[j-1], x);
        vector<EvalPoint> empty;
        vector<ex> exsigma = multivar_diophant(mdarg, x, umodpoly_to_ex(beta, x), empty, 0, p, k);
        umodpoly sigma1;
        umodpoly_from_ex(sigma1, exsigma[0], x, R);
        umodpoly sigma2;
        umodpoly_from_ex(sigma2, exsigma[1], x, R);
        beta = sigma1;
        s.push_back(sigma2);
    }
    s.push_back(beta);
    return s;
}
 
static void change_modulus(const cl_modint_ring& R, umodpoly& a)
{
    if ( a.empty() ) return;
    cl_modint_ring oldR = a[0].ring();
    for (auto & i : a) {
        i = R->canonhom(oldR->retract(i));
    }
    canonicalize(a);
}
 
static void eea_lift(const umodpoly& a, const umodpoly& b, const ex& x, unsigned int p, unsigned int k, umodpoly& s_, umodpoly& t_)
{
    cl_modint_ring R = find_modint_ring(p);
    umodpoly amod = a;
    change_modulus(R, amod);
    umodpoly bmod = b;
    change_modulus(R, bmod);
 
    umodpoly smod;
    umodpoly tmod;
    exteuclid(amod, bmod, smod, tmod);
 
    cl_modint_ring Rpk = find_modint_ring(expt_pos(cl_I(p),k));
    umodpoly s = smod;
    change_modulus(Rpk, s);
    umodpoly t = tmod;
    change_modulus(Rpk, t);
 
    cl_I modulus(p);
    umodpoly one(1, Rpk->one());
    for ( size_t j=1; j<k; ++j ) {
        umodpoly e = one - a * s - b * t;
        reduce_coeff(e, modulus);
        umodpoly c = e;
        change_modulus(R, c);
        umodpoly sigmabar = smod * c;
        umodpoly taubar = tmod * c;
        umodpoly sigma, q;
        remdiv(sigmabar, bmod, sigma, q);
        umodpoly tau = taubar + q * amod;
        umodpoly sadd = sigma;
        change_modulus(Rpk, sadd);
        cl_MI modmodulus(Rpk, modulus);
        s = s + sadd * modmodulus;
        umodpoly tadd = tau;
        change_modulus(Rpk, tadd);
        t = t + tadd * modmodulus;
        modulus = modulus * p;
    }
 
    s_ = s; t_ = t;
}
 
static upvec univar_diophant(const upvec& a, const ex& x, unsigned int m, unsigned int p, unsigned int k)
{
    cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
 
    const size_t r = a.size();
    upvec result;
    if ( r > 2 ) {
        upvec s = multiterm_eea_lift(a, x, p, k);
        for ( size_t j=0; j<r; ++j ) {
            umodpoly bmod = umodpoly_to_umodpoly(s[j], R, m);
            umodpoly buf;
            rem(bmod, a[j], buf);
            result.push_back(buf);
        }
    } else {
        umodpoly s, t;
        eea_lift(a[1], a[0], x, p, k, s, t);
        umodpoly bmod = umodpoly_to_umodpoly(s, R, m);
        umodpoly buf, q;
        remdiv(bmod, a[0], buf, q);
        result.push_back(buf);
        umodpoly t1mod = umodpoly_to_umodpoly(t, R, m);
        buf = t1mod + q * a[1];
        result.push_back(buf);
    }
 
    return result;
}
 
struct make_modular_map : public map_function {
    cl_modint_ring R;
    make_modular_map(const cl_modint_ring& R_) : R(R_) { }
    ex operator()(const ex& e) override
    {
        if ( is_a<add>(e) || is_a<mul>(e) ) {
            return e.map(*this);
        }
        else if ( is_a<numeric>(e) ) {
            numeric mod(R->modulus);
            numeric halfmod = (mod-1)/2;
            cl_MI emod = R->canonhom(the<cl_I>(ex_to<numeric>(e).to_cl_N()));
            numeric n(R->retract(emod));
            if ( n > halfmod ) {
                return n-mod;
            } else {
                return n;
            }
        }
        return e;
    }
};
 
static ex make_modular(const ex& e, const cl_modint_ring& R)
{
    make_modular_map map(R);
    return map(e.expand());
}
 
static vector<ex> multivar_diophant(const vector<ex>& a_, const ex& x, const ex& c, const vector<EvalPoint>& I,
                                    unsigned int d, unsigned int p, unsigned int k)
{
    vector<ex> a = a_;
 
    const cl_I modulus = expt_pos(cl_I(p),k);
    const cl_modint_ring R = find_modint_ring(modulus);
    const size_t r = a.size();
    const size_t nu = I.size() + 1;
 
    vector<ex> sigma;
    if ( nu > 1 ) {
        ex xnu = I.back().x;
        int alphanu = I.back().evalpoint;
 
        ex A = 1;
        for ( size_t i=0; i<r; ++i ) {
            A *= a[i];
        }
        vector<ex> b(r);
        for ( size_t i=0; i<r; ++i ) {
            b[i] = normal(A / a[i]);
        }
 
        vector<ex> anew = a;
        for ( size_t i=0; i<r; ++i ) {
            anew[i] = anew[i].subs(xnu == alphanu);
        }
        ex cnew = c.subs(xnu == alphanu);
        vector<EvalPoint> Inew = I;
        Inew.pop_back();
        sigma = multivar_diophant(anew, x, cnew, Inew, d, p, k);
 
        ex buf = c;
        for ( size_t i=0; i<r; ++i ) {
            buf -= sigma[i] * b[i];
        }
        ex e = make_modular(buf, R);
 
        ex monomial = 1;
        for ( size_t m=1; !e.is_zero() && e.has(xnu) && m<=d; ++m ) {
            monomial *= (xnu - alphanu);
            monomial = expand(monomial);
            ex cm = e.diff(ex_to<symbol>(xnu), m).subs(xnu==alphanu) / factorial(m);
            cm = make_modular(cm, R);
            if ( !cm.is_zero() ) {
                vector<ex> delta_s = multivar_diophant(anew, x, cm, Inew, d, p, k);
                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 = make_modular(buf, R);
            }
        }
    } else {
        upvec amod;
        for ( size_t i=0; i<a.size(); ++i ) {
            umodpoly up;
            umodpoly_from_ex(up, a[i], x, R);
            amod.push_back(up);
        }
 
        sigma.insert(sigma.begin(), r, 0);
        size_t nterms;
        ex z;
        if ( is_a<add>(c) ) {
            nterms = c.nops();
            z = c.op(0);
        } else {
            nterms = 1;
            z = c;
        }
        for ( size_t i=0; i<nterms; ++i ) {
            int m = z.degree(x);
            cl_I cm = the<cl_I>(ex_to<numeric>(z.lcoeff(x)).to_cl_N());
            upvec delta_s = univar_diophant(amod, x, m, p, k);
            cl_MI modcm;
            cl_I poscm = plusp(cm) ? cm : mod(cm, modulus);
            modcm = cl_MI(R, poscm);
            for ( size_t j=0; j<delta_s.size(); ++j ) {
                delta_s[j] = delta_s[j] * modcm;
                sigma[j] = sigma[j] + umodpoly_to_ex(delta_s[j], x);
            }
            if ( nterms > 1 && i+1 != nterms ) {
                z = c.op(i+1);
            }
        }
    }
 
    for ( size_t i=0; i<sigma.size(); ++i ) {
        sigma[i] = make_modular(sigma[i], R);
    }
 
    return sigma;
}
 
static ex hensel_multivar(const ex& a, const ex& x, const vector<EvalPoint>& I,
                          unsigned int p, const cl_I& l, const upvec& u, const vector<ex>& lcU)
{
    const size_t nu = I.size() + 1;
    const cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),l));
 
    vector<ex> A(nu);
    A[nu-1] = a;
 
    for ( size_t j=nu; j>=2; --j ) {
        ex x = I[j-2].x;
        int alpha = I[j-2].evalpoint;
        A[j-2] = A[j-1].subs(x==alpha);
        A[j-2] = make_modular(A[j-2], R);
    }
 
    int maxdeg = a.degree(I.front().x);
    for ( size_t i=1; i<I.size(); ++i ) {
        int maxdeg2 = a.degree(I[i].x);
        if ( maxdeg2 > maxdeg ) maxdeg = maxdeg2;
    }
 
    const size_t n = u.size();
    vector<ex> U(n);
    for ( size_t i=0; i<n; ++i ) {
        U[i] = umodpoly_to_ex(u[i], x);
    }
 
    for ( size_t j=2; j<=nu; ++j ) {
        vector<ex> U1 = U;
        ex monomial = 1;
        for ( size_t m=0; m<n; ++m) {
            if ( lcU[m] != 1 ) {
                ex coef = lcU[m];
                for ( size_t i=j-1; i<nu-1; ++i ) {
                    coef = coef.subs(I[i].x == I[i].evalpoint);
                }
                coef = make_modular(coef, R);
                int deg = U[m].degree(x);
                U[m] = U[m] - U[m].lcoeff(x) * pow(x,deg) + coef * pow(x,deg);
            }
        }
        ex Uprod = 1;
        for ( size_t i=0; i<n; ++i ) {
            Uprod *= U[i];
        }
        ex e = expand(A[j-1] - Uprod);
 
        vector<EvalPoint> newI;
        for ( size_t i=1; i<=j-2; ++i ) {
            newI.push_back(I[i-1]);
        }
 
        ex xj = I[j-2].x;
        int alphaj = I[j-2].evalpoint;
        size_t deg = A[j-1].degree(xj);
        for ( size_t k=1; k<=deg; ++k ) {
            if ( !e.is_zero() ) {
                monomial *= (xj - alphaj);
                monomial = expand(monomial);
                ex dif = e.diff(ex_to<symbol>(xj), k);
                ex c = dif.subs(xj==alphaj) / factorial(k);
                if ( !c.is_zero() ) {
                    vector<ex> deltaU = multivar_diophant(U1, x, c, newI, maxdeg, p, cl_I_to_uint(l));
                    for ( size_t i=0; i<n; ++i ) {
                        deltaU[i] *= monomial;
                        U[i] += deltaU[i];
                        U[i] = make_modular(U[i], R);
                    }
                    ex Uprod = 1;
                    for ( size_t i=0; i<n; ++i ) {
                        Uprod *= U[i];
                    }
                    e = A[j-1] - Uprod;
                    e = make_modular(e, R);
                }
            }
        }
    }
 
    ex acand = 1;
    for ( size_t i=0; i<U.size(); ++i ) {
        acand *= U[i];
    }
    if ( expand(a-acand).is_zero() ) {
        return lst(U.begin(), U.end());
    } else {
        return lst{};
    }
}
 
static ex put_factors_into_lst(const ex& e)
{
    lst result;
    if ( is_a<numeric>(e) ) {
        result.append(e);
        return result;
    }
    if ( is_a<power>(e) ) {
        result.append(1);
        result.append(e.op(0));
        return result;
    }
    if ( is_a<symbol>(e) || is_a<add>(e) ) {
        ex icont(e.integer_content());
        result.append(icont);
        result.append(e/icont);
        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));
            }
            if ( is_a<symbol>(op) || is_a<add>(op) ) {
                result.append(op);
            }
        }
        result.prepend(nfac);
        return result;
    }
    throw runtime_error("put_factors_into_lst: bad term.");
}
 
static bool checkdivisors(const lst& f)
{
    const int k = f.nops();
    numeric q, r;
    vector<numeric> d(k);
    d[0] = ex_to<numeric>(abs(f.op(0)));
    for ( int i=1; i<k; ++i ) {
        q = ex_to<numeric>(abs(f.op(i)));
        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 lst& f,
                         numeric& modulus, ex& u0, vector<numeric>& a)
{
    const ex& x = *syms.begin();
    while ( true ) {
        ++modulus;
        // generate a set of integers ...
        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]);
                // ... for which the leading coefficient doesn't vanish ...
            } while ( vnatry == 0 );
            vna = vnatry;
            u0 = u0.subs(*s == a[i]);
            ++s;
        }
        // ... for which u0 is square free ...
        ex g = gcd(u0, u0.diff(ex_to<symbol>(x)));
        if ( !is_a<numeric>(g) ) {
            continue;
        }
        if ( !is_a<numeric>(vn) ) {
            // ... and for which the evaluated factors have each an unique prime factor
            lst fnum = f;
            fnum.let_op(0) = fnum.op(0) * u0.content(x);
            for ( size_t i=1; i<fnum.nops(); ++i ) {
                if ( !is_a<numeric>(fnum.op(i)) ) {
                    s = syms.begin();
                    ++s;
                    for ( size_t j=0; j<a.size(); ++j, ++s ) {
                        fnum.let_op(i) = fnum.op(i).subs(*s == a[j]);
                    }
                }
            }
            if ( checkdivisors(fnum) ) {
                continue;
            }
        }
        // ok, we have a valid set now
        return;
    }
}
 
// forward declaration
static ex factor_sqrfree(const ex& poly);
 
static ex factor_multivariate(const ex& poly, const exset& syms)
{
    const ex& x = *syms.begin();
 
    // make polynomial primitive
    ex unit, cont, pp;
    poly.unitcontprim(x, unit, cont, pp);
    if ( !is_a<numeric>(cont) ) {
        return unit * factor_sqrfree(cont) * factor_sqrfree(pp);
    }
 
    // factor leading coefficient
    ex vn = pp.collect(x).lcoeff(x);
    ex vnlst;
    if ( is_a<numeric>(vn) ) {
        vnlst = lst{vn};
    }
    else {
        ex vnfactors = factor(vn);
        vnlst = put_factors_into_lst(vnfactors);
    }
 
    const unsigned int maxtrials = 3;
    numeric modulus = (vnlst.nops() > 3) ? vnlst.nops() : 3;
    vector<numeric> a(syms.size()-1, 0);
 
    // try now to factorize until we are successful
    while ( true ) {
 
        unsigned int trialcount = 0;
        unsigned int prime;
        int factor_count = 0;
        int min_factor_count = -1;
        ex u, delta;
        ex ufac, ufaclst;
 
        // try several evaluation points to reduce the number of factors
        while ( trialcount < maxtrials ) {
 
            // generate a set of valid evaluation points
            generate_set(pp, vn, syms, ex_to<lst>(vnlst), modulus, u, a);
 
            ufac = factor_univariate(u, x, prime);
            ufaclst = put_factors_into_lst(ufac);
            factor_count = ufaclst.nops()-1;
            delta = ufaclst.op(0);
 
            if ( factor_count <= 1 ) {
                // irreducible
                return poly;
            }
            if ( min_factor_count < 0 ) {
                // first time here
                min_factor_count = factor_count;
            }
            else if ( min_factor_count == factor_count ) {
                // one less to try
                ++trialcount;
            }
            else if ( min_factor_count > factor_count ) {
                // new minimum, reset trial counter
                min_factor_count = factor_count;
                trialcount = 0;
            }
        }
 
        // determine true leading coefficients for the Hensel lifting
        vector<ex> C(factor_count);
        if ( is_a<numeric>(vn) ) {
            // easy case
            for ( size_t i=1; i<ufaclst.nops(); ++i ) {
                C[i-1] = ufaclst.op(i).lcoeff(x);
            }
        } else {
            // difficult case.
            // we use the property of the ftilde having a unique prime factor.
            // details can be found in [Wan].
            // calculate ftilde
            vector<numeric> ftilde(vnlst.nops()-1);
            for ( size_t i=0; i<ftilde.size(); ++i ) {
                ex ft = vnlst.op(i+1);
                auto 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);
            }
            // calculate D and C
            vector<bool> used_flag(ftilde.size(), false);
            vector<ex> D(factor_count, 1);
            if ( delta == 1 ) {
                for ( int i=0; i<factor_count; ++i ) {
                    numeric prefac = ex_to<numeric>(ufaclst.op(i+1).lcoeff(x));
                    for ( int j=ftilde.size()-1; j>=0; --j ) {
                        int count = 0;
                        while ( irem(prefac, ftilde[j]) == 0 ) {
                            prefac = iquo(prefac, ftilde[j]);
                            ++count;
                        }
                        if ( count ) {
                            used_flag[j] = true;
                            D[i] = D[i] * pow(vnlst.op(j+1), count);
                        }
                    }
                    C[i] = D[i] * prefac;
                }
            } else {
                for ( int i=0; i<factor_count; ++i ) {
                    numeric prefac = ex_to<numeric>(ufaclst.op(i+1).lcoeff(x));
                    for ( int j=ftilde.size()-1; j>=0; --j ) {
                        int count = 0;
                        while ( irem(prefac, ftilde[j]) == 0 ) {
                            prefac = iquo(prefac, ftilde[j]);
                            ++count;
                        }
                        while ( irem(ex_to<numeric>(delta)*prefac, ftilde[j]) == 0 ) {
                            numeric g = gcd(prefac, ex_to<numeric>(ftilde[j]));
                            prefac = iquo(prefac, g);
                            delta = delta / (ftilde[j]/g);
                            ufaclst.let_op(i+1) = ufaclst.op(i+1) * (ftilde[j]/g);
                            ++count;
                        }
                        if ( count ) {
                            used_flag[j] = true;
                            D[i] = D[i] * pow(vnlst.op(j+1), count);
                        }
                    }
                    C[i] = D[i] * prefac;
                }
            }
            // check if something went wrong
            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 ) {
                continue;
            }
        }
        
        // multiply the remaining content of the univariate polynomial into the
        // first factor
        if ( delta != 1 ) {
            C[0] = C[0] * delta;
            ufaclst.let_op(1) = ufaclst.op(1) * delta;
        }
 
        // set up evaluation points
        EvalPoint ep;
        vector<EvalPoint> epv;
        auto 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 p^l
        int maxdeg = 0;
        for ( int i=1; i<=factor_count; ++i ) {
            if ( ufaclst.op(i).degree(x) > maxdeg ) {
                maxdeg = ufaclst[i].degree(x);
            }
        }
        cl_I B = 2*calc_bound(u, x, maxdeg);
        cl_I l = 1;
        cl_I pl = prime;
        while ( pl < B ) {
            l = l + 1;
            pl = pl * prime;
        }
        
        // set up modular factors (mod p^l)
        cl_modint_ring R = find_modint_ring(expt_pos(cl_I(prime),l));
        upvec modfactors(ufaclst.nops()-1);
        for ( size_t i=1; i<ufaclst.nops(); ++i ) {
            umodpoly_from_ex(modfactors[i-1], ufaclst.op(i), x, R);
        }
 
        // try Hensel lifting
        ex res = hensel_multivar(pp, x, epv, prime, l, modfactors, C);
        if ( res != lst{} ) {
            ex result = cont * unit;
            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);
            }
            return result;
        }
    }
}
 
struct find_symbols_map : public map_function {
    exset syms;
    ex operator()(const ex& e) override
    {
        if ( is_a<symbol>(e) ) {
            syms.insert(e);
            return e;
        }
        return e.map(*this);
    }
};
 
static ex factor_sqrfree(const ex& poly)
{
    // determine all symbols in poly
    find_symbols_map findsymbols;
    findsymbols(poly);
    if ( findsymbols.syms.size() == 0 ) {
        return poly;
    }
 
    if ( findsymbols.syms.size() == 1 ) {
        // univariate case
        const ex& x = *(findsymbols.syms.begin());
        int ld = poly.ldegree(x);
        if ( ld > 0 ) {
            // pull out direct factors
            ex res = factor_univariate(expand(poly/pow(x, ld)), x);
            return res * pow(x,ld);
        } else {
            ex res = factor_univariate(poly, x);
            return res;
        }
    }
 
    // multivariate case
    ex res = factor_multivariate(poly, findsymbols.syms);
    return res;
}
 
struct apply_factor_map : public map_function {
    unsigned options;
    apply_factor_map(unsigned options_) : options(options_) { }
    ex operator()(const ex& e) override
    {
        if ( e.info(info_flags::polynomial) ) {
            return factor(e, options);
        }
        if ( is_a<add>(e) ) {
            ex s1, s2;
            for ( size_t i=0; i<e.nops(); ++i ) {
                if ( e.op(i).info(info_flags::polynomial) ) {
                    s1 += e.op(i);
                } else {
                    s2 += e.op(i);
                }
            }
            return factor(s1, options) + s2.map(*this);
        }
        return e.map(*this);
    }
};
 
template <typename F> void
factor_iter(const ex &e, F yield)
{
    if (is_a<mul>(e)) {
        for (const auto &f : e) {
            if (is_a<power>(f)) {
                yield(f.op(0), f.op(1));
            } else {
                yield(f, ex(1));
            }
        }
    } else {
        if (is_a<power>(e)) {
            yield(e.op(0), e.op(1));
        } else {
            yield(e, ex(1));
        }
    }
}
 
static ex factor1(const ex& poly, unsigned options)
{
    // check arguments
    if ( !poly.info(info_flags::polynomial) ) {
        if ( options & factor_options::all ) {
            options &= ~factor_options::all;
            apply_factor_map factor_map(options);
            return factor_map(poly);
        }
        return poly;
    }
 
    // determine all symbols in poly
    find_symbols_map findsymbols;
    findsymbols(poly);
    if ( findsymbols.syms.size() == 0 ) {
        return poly;
    }
    lst syms;
    for (auto & i : findsymbols.syms ) {
        syms.append(i);
    }
 
    // make poly square free
    ex sfpoly = sqrfree(poly.expand(), syms);
 
    // factorize the square free components
    ex res = 1;
    factor_iter(sfpoly,
        [&](const ex &f, const ex &k) {
            if ( is_a<add>(f) ) {
                res *= pow(factor_sqrfree(f), k);
            } else {
                // simple case: (monomial)^exponent
                res *= pow(f, k);
            }
        });
    return res;
}
 
} // anonymous namespace
 
ex factor(const ex& poly, unsigned options)
{
    ex result = 1;
    factor_iter(poly,
        [&](const ex &f1, const ex &k1) {
            factor_iter(factor1(f1, options),
                [&](const ex &f2, const ex &k2) {
                    result *= pow(f2, k1*k2);
                });
        });
    return result;
}
 
} // namespace GiNaC