Fixed lots of bugs in factor_multivariate().
[ginac.git] / ginac / factor.cpp
index f7dded5e1566e1bebe33470b6800c0e37583fc88..fb73897218db06ed31b40f32bf52aabca7e33acf 100644 (file)
@@ -1,6 +1,6 @@
 /** @file factor.cpp
  *
- *  Polynomial factorization code (Implementation).
+ *  Polynomial factorization code (implementation).
  *
  *  Algorithms used can be found in
  *    [W1]  An Improved Multivariate Polynomial Factoring Algorithm,
 
 //#define DEBUGFACTOR
 
-#ifdef DEBUGFACTOR
-#include <ostream>
-#include <ginac/ginac.h>
-using namespace GiNaC;
-#else
 #include "factor.h"
 
 #include "ex.h"
@@ -46,21 +41,21 @@ using namespace GiNaC;
 #include "mul.h"
 #include "normal.h"
 #include "add.h"
-#endif
 
 #include <algorithm>
+#include <cmath>
+#include <limits>
 #include <list>
 #include <vector>
+#ifdef DEBUGFACTOR
+#include <ostream>
+#endif
 using namespace std;
 
 #include <cln/cln.h>
 using namespace cln;
 
-#ifdef DEBUGFACTOR
-namespace Factor {
-#else
 namespace GiNaC {
-#endif
 
 #ifdef DEBUGFACTOR
 #define DCOUT(str) cout << #str << endl
@@ -72,127 +67,330 @@ namespace GiNaC {
 #define DCOUT2(str,var)
 #endif
 
-// forward declaration
-ex factor(const ex& poly, unsigned options);
-
 // anonymous namespace to hide all utility functions
 namespace {
 
 typedef vector<cl_MI> mvec;
-
 #ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const mvec& v)
+ostream& operator<<(ostream& o, const vector<int>& v)
 {
-       mvec::const_iterator i = v.begin(), end = v.end();
+       vector<int>::const_iterator i = v.begin(), end = v.end();
        while ( i != end ) {
                o << *i++ << " ";
        }
        return o;
 }
-#endif // def DEBUGFACTOR
-
-#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const vector<mvec>& v)
+ostream& operator<<(ostream& o, const vector<cl_I>& v)
 {
-       vector<mvec>::const_iterator i = v.begin(), end = v.end();
+       vector<cl_I>::const_iterator i = v.begin(), end = v.end();
        while ( i != end ) {
-               o << *i++ << endl;
+               o << *i << "[" << i-v.begin() << "]" << " ";
+               ++i;
        }
        return o;
 }
-#endif // def DEBUGFACTOR
+ostream& operator<<(ostream& o, const vector<cl_MI>& v)
+{
+       vector<cl_MI>::const_iterator i = v.begin(), end = v.end();
+       while ( i != end ) {
+               o << *i << "[" << i-v.begin() << "]" << " ";
+               ++i;
+       }
+       return o;
+}
+ostream& operator<<(ostream& o, const vector< vector<cl_MI> >& v)
+{
+       vector< vector<cl_MI> >::const_iterator i = v.begin(), end = v.end();
+       while ( i != end ) {
+               o << i-v.begin() << ": " << *i << endl;
+               ++i;
+       }
+       return o;
+}
+#endif
 
 ////////////////////////////////////////////////////////////////////////////////
 // modular univariate polynomial code
 
-typedef cl_UP_MI umod;
-typedef vector<umod> umodvec;
+typedef std::vector<cln::cl_MI> umodpoly;
+typedef std::vector<cln::cl_I> upoly;
+typedef vector<umodpoly> upvec;
 
-#define COPY(to,from) from.ring()->create(degree(from)); \
-       for ( int II=0; II<=degree(from); ++II ) to.set_coeff(II, coeff(from, II)); \
-       to.finalize()
+// COPY FROM UPOLY.HPP
 
-#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const umodvec& v)
+// CHANGED size_t -> int !!!
+template<typename T> static int degree(const T& p)
 {
-       umodvec::const_iterator i = v.begin(), end = v.end();
-       while ( i != end ) {
-               o << *i++ << " , " << endl;
+       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;
        }
-       return o;
+
+       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<typename T>
+static T 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>
+static T 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;
 }
-#endif // def DEBUGFACTOR
 
-static umod umod_from_ex(const ex& e, const ex& x, const cl_univpoly_modint_ring& UPR)
+static void upoly_from_ex(upoly& up, const ex& e, const ex& x)
 {
        // assert: e is in Z[x]
        int deg = e.degree(x);
-       umod p = UPR->create(deg);
+       up.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());
-               p.set_coeff(deg, UPR->basering()->canonhom(coeff));
+               up[deg] = the<cl_I>(ex_to<numeric>(e.coeff(x, deg)).to_cl_N());
        }
        for ( ; deg>=0; --deg ) {
-               p.set_coeff(deg, UPR->basering()->zero());
+               up[deg] = 0;
        }
-       p.finalize();
-       return p;
+       canonicalize(up);
 }
 
-static umod umod_from_ex(const ex& e, const ex& x, const cl_modint_ring& R)
+static void umodpoly_from_upoly(umodpoly& ump, const upoly& e, const cl_modint_ring& R)
 {
-       return umod_from_ex(e, x, find_univpoly_ring(R));
+       int deg = degree(e);
+       ump.resize(deg+1);
+       for ( ; deg>=0; --deg ) {
+               ump[deg] = R->canonhom(e[deg]);
+       }
+       canonicalize(ump);
 }
 
-static umod umod_from_ex(const ex& e, const ex& x, const cl_I& modulus)
+static void umodpoly_from_ex(umodpoly& ump, const ex& e, const ex& x, const cl_modint_ring& R)
 {
-       return umod_from_ex(e, x, find_modint_ring(modulus));
+       // 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);
 }
 
-static umod umod_from_modvec(const mvec& mv)
+#ifdef DEBUGFACTOR
+static void umodpoly_from_ex(umodpoly& ump, const ex& e, const ex& x, const cl_I& modulus)
 {
-       size_t n = mv.size(); // assert: n>0
-       while ( n && zerop(mv[n-1]) ) --n;
-       cl_univpoly_modint_ring UPR = find_univpoly_ring(mv.front().ring());
-       if ( n == 0 ) {
-               umod p = UPR->create(-1);
-               p.finalize();
-               return p;
-       }
-       umod p = UPR->create(n-1);
-       for ( size_t i=0; i<n; ++i ) {
-               p.set_coeff(i, mv[i]);
-       }
-       p.finalize();
-       return p;
+       umodpoly_from_ex(ump, e, x, find_modint_ring(modulus));
 }
+#endif
 
-static umod divide(const umod& a, const cl_I& x)
+static ex upoly_to_ex(const upoly& a, const ex& x)
 {
-       DCOUT(divide);
-       DCOUTVAR(a);
-       cl_univpoly_modint_ring UPR = a.ring();
-       cl_modint_ring R = UPR->basering();
-       int deg = degree(a);
-       umod newa = UPR->create(deg);
-       for ( int i=0; i<=deg; ++i ) {
-               cl_I c = R->retract(coeff(a, i));
-               newa.set_coeff(i, cl_MI(R, the<cl_I>(c / x)));
+       if ( a.empty() ) return 0;
+       ex e;
+       for ( int i=degree(a); i>=0; --i ) {
+               e += numeric(a[i]) * pow(x, i);
        }
-       newa.finalize();
-       DCOUT(END divide);
-       return newa;
+       return e;
 }
 
-static ex umod_to_ex(const umod& a, const ex& x)
+static ex umodpoly_to_ex(const umodpoly& a, const ex& x)
 {
-       ex e;
-       cl_modint_ring R = a.ring()->basering();
+       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(coeff(a, i));
+               cl_I n = R->retract(a[i]);
                if ( n > halfmod ) {
                        e += numeric(n-mod) * pow(x, i);
                } else {
@@ -202,147 +400,219 @@ static ex umod_to_ex(const umod& a, const ex& x)
        return e;
 }
 
-static void unit_normal(umod& a)
+static upoly umodpoly_to_upoly(const umodpoly& a)
 {
-       int deg = degree(a);
-       if ( deg >= 0 ) {
-               cl_MI lc = coeff(a, deg);
-               cl_MI one = a.ring()->basering()->one();
-               if ( lc != one ) {
-                       umod newa = a.ring()->create(deg);
-                       newa.set_coeff(deg, one);
-                       for ( --deg; deg>=0; --deg ) {
-                               cl_MI nc = div(coeff(a, deg), lc);
-                               newa.set_coeff(deg, nc);
-                       }
-                       newa.finalize();
-                       a = newa;
+       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;
+}
+
+/** Divides all coefficients of the polynomial a by the integer x.
+ *  All coefficients are supposed to be divisible by x. If they are not, the
+ *  the<cl_I> cast will raise an exception.
+ *
+ *  @param[in,out] a  polynomial of which the coefficients will be reduced by x
+ *  @param[in]     x  integer that divides the coefficients
+ */
+static void reduce_coeff(umodpoly& a, const cl_I& x)
+{
+       if ( a.empty() ) return;
+
+       cl_modint_ring R = a[0].ring();
+       umodpoly::iterator i = a.begin(), end = a.end();
+       for ( ; i!=end; ++i ) {
+               // 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 umod rem(const umod& a, const umod& b)
+/** Calculates remainder of a/b.
+ *  Assertion: a and b not empty.
+ *
+ *  @param[in]  a  polynomial dividend
+ *  @param[in]  b  polynomial divisor
+ *  @param[out] r  polynomial remainder
+ */
+static void rem(const umodpoly& a, const umodpoly& b, umodpoly& r)
 {
        int k, n;
        n = degree(b);
        k = degree(a) - n;
-       if ( k < 0 ) {
-               umod c = COPY(c, a);
-               return c;
-       }
+       r = a;
+       if ( k < 0 ) return;
 
-       umod c = COPY(c, a);
        do {
-               cl_MI qk = div(coeff(c, n+k), coeff(b, n));
+               cl_MI qk = div(r[n+k], b[n]);
                if ( !zerop(qk) ) {
-                       unsigned int j;
                        for ( int i=0; i<n; ++i ) {
-                               j = n + k - 1 - i;
-                               c.set_coeff(j, coeff(c, j) - qk * coeff(b, j-k));
+                               unsigned int j = n + k - 1 - i;
+                               r[j] = r[j] - qk * b[j-k];
                        }
                }
        } while ( k-- );
 
-       cl_MI zero = a.ring()->basering()->zero();
-       for ( int i=degree(a); i>=n; --i ) {
-               c.set_coeff(i, zero);
-       }
-
-       c.finalize();
-       return c;
+       fill(r.begin()+n, r.end(), a[0].ring()->zero());
+       canonicalize(r);
 }
 
-static umod div(const umod& a, const umod& b)
+/** Calculates quotient of a/b.
+ *  Assertion: a and b not empty.
+ *
+ *  @param[in]  a  polynomial dividend
+ *  @param[in]  b  polynomial divisor
+ *  @param[out] q  polynomial quotient
+ */
+static void div(const umodpoly& a, const umodpoly& b, umodpoly& q)
 {
        int k, n;
        n = degree(b);
        k = degree(a) - n;
-       if ( k < 0 ) {
-               umod q = a.ring()->create(-1);
-               q.finalize();
-               return q;
-       }
+       q.clear();
+       if ( k < 0 ) return;
 
-       umod c = COPY(c, a);
-       umod q = a.ring()->create(k);
+       umodpoly r = a;
+       q.resize(k+1, a[0].ring()->zero());
        do {
-               cl_MI qk = div(coeff(c, n+k), coeff(b, n));
+               cl_MI qk = div(r[n+k], b[n]);
                if ( !zerop(qk) ) {
-                       q.set_coeff(k, qk);
-                       unsigned int j;
+                       q[k] = qk;
                        for ( int i=0; i<n; ++i ) {
-                               j = n + k - 1 - i;
-                               c.set_coeff(j, coeff(c, j) - qk * coeff(b, j-k));
+                               unsigned int j = n + k - 1 - i;
+                               r[j] = r[j] - qk * b[j-k];
                        }
                }
        } while ( k-- );
 
-       q.finalize();
-       return q;
+       canonicalize(q);
 }
 
-static umod remdiv(const umod& a, const umod& b, umod& q)
+/** Calculates quotient and remainder of a/b.
+ *  Assertion: a and b not empty.
+ *
+ *  @param[in]  a  polynomial dividend
+ *  @param[in]  b  polynomial divisor
+ *  @param[out] r  polynomial remainder
+ *  @param[out] q  polynomial quotient
+ */
+static void remdiv(const umodpoly& a, const umodpoly& b, umodpoly& r, umodpoly& q)
 {
        int k, n;
        n = degree(b);
        k = degree(a) - n;
-       if ( k < 0 ) {
-               q = a.ring()->create(-1);
-               q.finalize();
-               umod c = COPY(c, a);
-               return c;
-       }
+       q.clear();
+       r = a;
+       if ( k < 0 ) return;
 
-       umod c = COPY(c, a);
-       q = a.ring()->create(k);
+       q.resize(k+1, a[0].ring()->zero());
        do {
-               cl_MI qk = div(coeff(c, n+k), coeff(b, n));
+               cl_MI qk = div(r[n+k], b[n]);
                if ( !zerop(qk) ) {
-                       q.set_coeff(k, qk);
-                       unsigned int j;
+                       q[k] = qk;
                        for ( int i=0; i<n; ++i ) {
-                               j = n + k - 1 - i;
-                               c.set_coeff(j, coeff(c, j) - qk * coeff(b, j-k));
+                               unsigned int j = n + k - 1 - i;
+                               r[j] = r[j] - qk * b[j-k];
                        }
                }
        } while ( k-- );
 
-       cl_MI zero = a.ring()->basering()->zero();
-       for ( int i=degree(a); i>=n; --i ) {
-               c.set_coeff(i, zero);
-       }
+       fill(r.begin()+n, r.end(), a[0].ring()->zero());
+       canonicalize(r);
+       canonicalize(q);
+}
 
-       q.finalize();
-       c.finalize();
-       return c;
+/** Calculates the GCD of polynomial a and b.
+ *
+ *  @param[in]  a  polynomial
+ *  @param[in]  b  polynomial
+ *  @param[out] c  GCD
+ */
+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 umod gcd(const umod& a, const umod& b)
+/** Calculates the derivative of the polynomial a.
+ *  
+ *  @param[in]  a  polynomial of which to take the derivative
+ *  @param[out] d  result/derivative
+ */
+static void deriv(const umodpoly& a, umodpoly& d)
 {
-       if ( degree(a) < degree(b) ) return gcd(b, a);
+       d.clear();
+       if ( a.size() <= 1 ) return;
 
-       umod c = COPY(c, a);
-       unit_normal(c);
-       umod d = COPY(d, b);
-       unit_normal(d);
-       while ( !zerop(d) ) {
-               umod r = rem(c, d);
-               c = COPY(c, d);
-               d = COPY(d, r);
+       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);
        }
-       unit_normal(c);
-       return c;
+       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 squarefree(const umod& a)
+static bool equal_one(const umodpoly& a)
+{
+       return ( a.size() == 1 && a[0] == a[0].ring()->one() );
+}
+
+/** Returns true if polynomial a is square free.
+ *
+ *  @param[in] a  polynomial to check
+ *  @return       true if polynomial is square free, false otherwise
+ */
+static bool squarefree(const umodpoly& a)
 {
-       umod b = deriv(a);
-       if ( zerop(b) ) {
+       umodpoly b;
+       deriv(a, b);
+       if ( b.empty() ) {
                return false;
        }
-       umod one = a.ring()->one();
-       umod c = gcd(a, b);
-       return c == one;
+       umodpoly c;
+       gcd(a, b, c);
+       return equal_one(c);
 }
 
 // END modular univariate polynomial code
@@ -480,15 +750,19 @@ modular_matrix operator*(const modular_matrix& m1, const modular_matrix& m2)
 
 ostream& operator<<(ostream& o, const modular_matrix& m)
 {
-       vector<cl_MI>::const_iterator i = m.m.begin(), end = m.m.end();
-       size_t wrap = 1;
-       for ( ; i != end; ++i ) {
-               o << *i << " ";
-               if ( !(wrap++ % m.c) ) {
-                       o << endl;
+       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 << endl;
+       o << "}";
        return o;
 }
 #endif // def DEBUGFACTOR
@@ -496,37 +770,26 @@ ostream& operator<<(ostream& o, const modular_matrix& m)
 // END modular matrix
 ////////////////////////////////////////////////////////////////////////////////
 
-static void q_matrix(const umod& a, modular_matrix& Q)
+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.ring()->basering()->modulus);
-// fast and buggy
-//     vector<cl_MI> r(n, a.R->zero());
-//     r[0] = a.R->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);
-//             }
-//     }
-// slow and (hopefully) correct
-       cl_MI one = a.ring()->basering()->one();
-       for ( int i=0; i<n; ++i ) {
-               umod qk = a.ring()->create(i*q);
-               qk.set_coeff(i*q, one);
-               qk.finalize();
-               umod r = rem(qk, a);
-               mvec rvec;
-               for ( int j=0; j<n; ++j ) {
-                       rvec.push_back(coeff(r, j));
-               }
-               Q.set_row(i, rvec);
+       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);
+               }
        }
 }
 
@@ -574,52 +837,54 @@ static void nullspace(modular_matrix& M, vector<mvec>& basis)
        }
 }
 
-static void berlekamp(const umod& a, umodvec& upv)
+static void berlekamp(const umodpoly& a, upvec& upv)
 {
-       cl_modint_ring R = a.ring()->basering();
-       const umod one = a.ring()->one();
+       cl_modint_ring R = a[0].ring();
+       umodpoly one(1, R->one());
 
        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 ) {
                return;
        }
 
-       list<umod> factors;
+       list<umodpoly> factors;
        factors.push_back(a);
        unsigned int size = 1;
        unsigned int r = 1;
        unsigned int q = cl_I_to_uint(R->modulus);
 
-       list<umod>::iterator u = factors.begin();
+       list<umodpoly>::iterator u = factors.begin();
 
        while ( true ) {
                for ( unsigned int s=0; s<q; ++s ) {
-                       umod nur = umod_from_modvec(nu[r]);
-                       cl_MI buf = coeff(nur, 0) - cl_MI(R, s);
-                       nur.set_coeff(0, buf);
-                       nur.finalize();
-                       umod g = gcd(nur, *u);
-                       if ( g != one && g != *u ) {
-                               umod uo = div(*u, g);
-                               if ( uo == one ) {
+                       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 = COPY((*u), uo);
+                                       *u = uo;
                                }
                                factors.push_back(g);
                                size = 0;
-                               list<umod>::const_iterator i = factors.begin(), end = factors.end();
+                               list<umodpoly>::const_iterator i = factors.begin(), end = factors.end();
                                while ( i != end ) {
                                        if ( degree(*i) ) ++size; 
                                        ++i;
                                }
                                if ( size == k ) {
-                                       list<umod>::const_iterator i = factors.begin(), end = factors.end();
+                                       list<umodpoly>::const_iterator i = factors.begin(), end = factors.end();
                                        while ( i != end ) {
                                                upv.push_back(*i++);
                                        }
@@ -634,128 +899,290 @@ static void berlekamp(const umod& a, umodvec& upv)
        }
 }
 
-static void factor_modular(const umod& p, umodvec& upv)
+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;
+               }
+       }
+}
+
+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);
+               }
+       }
+}
+
+#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);
-       return;
+#endif
 }
 
-static void exteuclid(const umod& a, const umod& b, umod& g, umod& s, umod& t)
+/** Calculates polynomials s and t such that a*s+b*t==1.
+ *  Assertion: a and b are relatively prime and not zero.
+ *
+ *  @param[in]  a  polynomial
+ *  @param[in]  b  polynomial
+ *  @param[out] s  polynomial
+ *  @param[out] t  polynomial
+ */
+static void exteuclid(const umodpoly& a, const umodpoly& b, umodpoly& s, umodpoly& t)
 {
        if ( degree(a) < degree(b) ) {
-               exteuclid(b, a, g, t, s);
+               exteuclid(b, a, t, s);
                return;
        }
-       umod c = COPY(c, a); unit_normal(c);
-       umod d = COPY(d, b); unit_normal(d);
-       umod c1 = a.ring()->one();
-       umod c2 = a.ring()->create(-1);
-       umod d1 = a.ring()->create(-1);
-       umod d2 = a.ring()->one();
-       while ( !zerop(d) ) {
-               umod q = div(c, d);
-               umod r = c - q * d;
-               umod r1 = c1 - q * d1;
-               umod r2 = c2 - q * d2;
-               c = COPY(c, d);
-               c1 = COPY(c1, d1);
-               c2 = COPY(c2, d2);
-               d = COPY(d, r);
-               d1 = COPY(d1, r1);
-               d2 = COPY(d2, r2);
-       }
-       g = COPY(g, c); unit_normal(g);
-       s = COPY(s, c1);
-       for ( int i=0; i<=degree(s); ++i ) {
-               s.set_coeff(i, coeff(s, i) * recip(coeff(a, degree(a)) * coeff(c, degree(c))));
-       }
-       s.finalize();
-       t = COPY(t, c2);
-       for ( int i=0; i<=degree(t); ++i ) {
-               t.set_coeff(i, coeff(t, i) * recip(coeff(b, degree(b)) * coeff(c, degree(c))));
-       }
-       t.finalize();
-}
-
-static ex replace_lc(const ex& poly, const ex& x, const ex& lc)
-{
-       ex r = expand(poly + (lc - poly.lcoeff(x)) * pow(x, poly.degree(x)));
-       return r;
+
+       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));
+       umodpoly::iterator i = s.begin(), end = s.end();
+       for ( ; i!=end; ++i ) {
+               *i = *i * fac;
+       }
+       canonicalize(s);
+       fac = recip(lcoeff(b) * lcoeff(c));
+       i = t.begin(), end = t.end();
+       for ( ; i!=end; ++i ) {
+               *i = *i * fac;
+       }
+       canonicalize(t);
 }
 
-static ex hensel_univar(const ex& a_, const ex& x, unsigned int p, const umod& u1_, const umod& w1_, const ex& gamma_ = 0)
+static upoly replace_lc(const upoly& poly, const cl_I& lc)
 {
-       ex a = a_;
-       const cl_univpoly_modint_ring& UPR = u1_.ring();
-       const cl_modint_ring& R = UPR->basering();
+       if ( poly.empty() ) return poly;
+       upoly r = poly;
+       r.back() = lc;
+       return r;
+}
 
-       // calc bound B
-       ex maxcoeff;
+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 ) {
-               maxcoeff += pow(abs(a.coeff(x, i)),2);
+               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 normmc = ceiling1(the<cl_R>(cln::sqrt(ex_to<numeric>(maxcoeff).to_cl_N())));
-       cl_I maxdegree = (degree(u1_) > degree(w1_)) ? degree(u1_) : degree(w1_);
-       cl_I B = normmc * expt_pos(cl_I(2), maxdegree);
+       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
-       ex alpha = a.lcoeff(x);
-       ex gamma = gamma_;
-       if ( gamma == 0 ) {
-               gamma = alpha;
-       }
-       numeric gamma_ui = ex_to<numeric>(abs(gamma));
-       a = a * gamma;
-       umod nu1 = COPY(nu1, u1_);
-       unit_normal(nu1);
-       umod nw1 = COPY(nw1, w1_);
-       unit_normal(nw1);
-       ex phi;
-       phi = gamma * umod_to_ex(nu1, x);
-       umod u1 = umod_from_ex(phi, x, R);
-       phi = alpha * umod_to_ex(nw1, x);
-       umod w1 = umod_from_ex(phi, x, R);
+       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
-       umod g = UPR->create(-1);
-       umod s = UPR->create(-1);
-       umod t = UPR->create(-1);
-       exteuclid(u1, w1, g, s, t);
+       umodpoly s;
+       umodpoly t;
+       exteuclid(u1, w1, s, t);
 
        // step 3
-       ex u = replace_lc(umod_to_ex(u1, x), x, gamma);
-       ex w = replace_lc(umod_to_ex(w1, x), x, alpha);
-       ex e = expand(a - u * w);
-       numeric modulus = p;
-       const numeric maxmodulus = 2*numeric(B)*gamma_ui;
+       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.is_zero() && modulus < maxmodulus ) {
-               ex c = e / modulus;
-               phi = expand(umod_to_ex(s, x) * c);
-               umod sigmatilde = umod_from_ex(phi, x, R);
-               phi = expand(umod_to_ex(t, x) * c);
-               umod tautilde = umod_from_ex(phi, x, R);
-               umod q = div(sigmatilde, w1);
-               umod r = rem(sigmatilde, w1);
-               umod sigma = COPY(sigma, r);
-               phi = expand(umod_to_ex(tautilde, x) + umod_to_ex(q, x) * umod_to_ex(u1, x));
-               umod tau = umod_from_ex(phi, x, R);
-               u = expand(u + umod_to_ex(tau, x) * modulus);
-               w = expand(w + umod_to_ex(sigma, x) * modulus);
-               e = expand(a - u * w);
+       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.is_zero() ) {
-               ex delta = u.content(x);
-               u = u / delta;
-               w = w / gamma * delta;
-               return lst(u, w);
+       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 {
-               return lst();
+               u.clear();
        }
 }
 
@@ -789,91 +1216,157 @@ static unsigned int next_prime(unsigned int p)
        throw logic_error("next_prime: should not reach this point!");
 }
 
-class Partition
+class factor_partition
 {
 public:
-       Partition(size_t n_) : n(n_)
+       factor_partition(const upvec& factors_) : factors(factors_)
        {
-               k.resize(n, 1);
-               k[0] = 0;
-               sum = n-1;
+               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_first() const { return n-sum; }
-       size_t size_second() const { return sum; }
+       size_t size_left() const { return n-len; }
+       size_t size_right() const { return len; }
 #ifdef DEBUGFACTOR
-       void get() const
-       {
-               for ( size_t i=0; i<k.size(); ++i ) {
-                       cout << k[i] << " ";
-               }
-               cout << endl;
-       }
+       void get() const { DCOUTVAR(k); }
 #endif
        bool next()
        {
-               for ( size_t i=n-1; i>=1; --i ) {
-                       if ( k[i] ) {
-                               --k[i];
-                               --sum;
-                               return sum > 0;
+               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;
                        }
-                       ++k[i];
-                       ++sum;
+                       last = len;
+                       ++len;
+                       if ( len > n/2 ) return false;
+                       fill(k.begin(), k.begin()+len, 1);
+                       fill(k.begin()+len+1, k.end(), 0);
                }
-               return false;
+               else {
+                       k[last++] = 0;
+                       k[last] = 1;
+               }
+               split();
+               return true;
        }
+       umodpoly& left() { return lr[0]; }
+       umodpoly& right() { return lr[1]; }
 private:
-       size_t n, sum;
-       vector<int> k;
-};
-
-static void split(const umodvec& factors, const Partition& part, umod& a, umod& b)
-{
-       a = factors.front().ring()->one();
-       b = factors.front().ring()->one();
-       for ( size_t i=0; i<part.size(); ++i ) {
-               if ( part[i] ) {
-                       b = b * factors[i];
+       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 {
-                       a = a * factors[i];
+                       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
 {
-       ex poly;
-       umodvec factors;
+       upoly poly;
+       upvec factors;
 };
 
-static ex factor_univariate(const ex& poly, const ex& x)
+static ex factor_univariate(const ex& poly, const ex& x, unsigned int& prime)
 {
-       ex unit, cont, prim;
-       poly.unitcontprim(x, unit, cont, prim);
+       ex unit, cont, prim_ex;
+       poly.unitcontprim(x, unit, cont, prim_ex);
+       upoly prim;
+       upoly_from_ex(prim, prim_ex, x);
 
        // determine proper prime and minimize number of modular factors
-       unsigned int p = 3, lastp;
+       prime = 3;
+       unsigned int lastp = prime;
        cl_modint_ring R;
        unsigned int trials = 0;
        unsigned int minfactors = 0;
-       numeric lcoeff = ex_to<numeric>(prim.lcoeff(x));
-       umodvec factors;
+       cl_I lc = lcoeff(prim) * the<cl_I>(ex_to<numeric>(cont).to_cl_N());
+       upvec factors;
        while ( trials < 2 ) {
+               umodpoly modpoly;
                while ( true ) {
-                       p = next_prime(p);
-                       if ( irem(lcoeff, p) != 0 ) {
-                               R = find_modint_ring(p);
-                               umod modpoly = umod_from_ex(prim, x, R);
+                       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
-               umod modpoly = umod_from_ex(prim, x, R);
-               umodvec trialfactors;
+               upvec trialfactors;
                factor_modular(modpoly, trialfactors);
                if ( trialfactors.size() <= 1 ) {
                        // irreducible for sure
@@ -882,17 +1375,16 @@ static ex factor_univariate(const ex& poly, const ex& x)
 
                if ( minfactors == 0 || trialfactors.size() < minfactors ) {
                        factors = trialfactors;
-                       minfactors = factors.size();
-                       lastp = p;
+                       minfactors = trialfactors.size();
+                       lastp = prime;
                        trials = 1;
                }
                else {
                        ++trials;
                }
        }
-       p = lastp;
-       R = find_modint_ring(p);
-       cl_univpoly_modint_ring UPR = find_univpoly_ring(R);
+       prime = lastp;
+       R = find_modint_ring(prime);
 
        // lift all factor combinations
        stack<ModFactors> tocheck;
@@ -900,25 +1392,22 @@ static ex factor_univariate(const ex& poly, const ex& x)
        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();
-               Partition part(n);
+               factor_partition part(tocheck.top().factors);
                while ( true ) {
-                       umod a = UPR->create(-1);
-                       umod b = UPR->create(-1);
-                       split(tocheck.top().factors, part, a, b);
-
-                       ex answer = hensel_univar(tocheck.top().poly, x, p, a, b);
-                       if ( answer != lst() ) {
-                               if ( part.size_first() == 1 ) {
-                                       if ( part.size_second() == 1 ) {
-                                               result *= answer.op(0) * answer.op(1);
+                       hensel_univar(tocheck.top().poly, prime, part.left(), part.right(), f1, f2);
+                       if ( !f1.empty() ) {
+                               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 *= answer.op(0);
-                                       tocheck.top().poly = answer.op(1);
+                                       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);
@@ -927,14 +1416,14 @@ static ex factor_univariate(const ex& poly, const ex& x)
                                        }
                                        break;
                                }
-                               else if ( part.size_second() == 1 ) {
-                                       if ( part.size_first() == 1 ) {
-                                               result *= answer.op(0) * answer.op(1);
+                               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 *= answer.op(1);
-                                       tocheck.top().poly = answer.op(0);
+                                       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);
@@ -944,8 +1433,8 @@ static ex factor_univariate(const ex& poly, const ex& x)
                                        break;
                                }
                                else {
-                                       umodvec newfactors1(part.size_first(), UPR->create(-1)), newfactors2(part.size_second(), UPR->create(-1));
-                                       umodvec::iterator i1 = newfactors1.begin(), i2 = newfactors2.begin();
+                                       upvec newfactors1(part.size_left()), newfactors2(part.size_right());
+                                       upvec::iterator i1 = newfactors1.begin(), i2 = newfactors2.begin();
                                        for ( size_t i=0; i<n; ++i ) {
                                                if ( part[i] ) {
                                                        *i2++ = tocheck.top().factors[i];
@@ -955,17 +1444,17 @@ static ex factor_univariate(const ex& poly, const ex& x)
                                                }
                                        }
                                        tocheck.top().factors = newfactors1;
-                                       tocheck.top().poly = answer.op(0);
+                                       tocheck.top().poly = f1;
                                        ModFactors mf;
                                        mf.factors = newfactors2;
-                                       mf.poly = answer.op(1);
+                                       mf.poly = f2;
                                        tocheck.push(mf);
                                        break;
                                }
                        }
                        else {
                                if ( !part.next() ) {
-                                       result *= tocheck.top().poly;
+                                       result *= upoly_to_ex(tocheck.top().poly, x);
                                        tocheck.pop();
                                        break;
                                }
@@ -976,185 +1465,133 @@ static ex factor_univariate(const ex& poly, const ex& x)
        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;
 };
 
-// MARK
-
 // forward declaration
-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 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);
 
-umodvec multiterm_eea_lift(const umodvec& a, const ex& x, unsigned int p, unsigned int k)
+static upvec multiterm_eea_lift(const upvec& a, const ex& x, unsigned int p, unsigned int k)
 {
-       DCOUT(multiterm_eea_lift);
-       DCOUTVAR(a);
-       DCOUTVAR(p);
-       DCOUTVAR(k);
-
        const size_t r = a.size();
-       DCOUTVAR(r);
        cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
-       cl_univpoly_modint_ring UPR = find_univpoly_ring(R);
-       umodvec q(r-1, UPR->create(-1));
+       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];
        }
-       DCOUTVAR(q);
-       umod beta = UPR->one();
-       umodvec s;
+       umodpoly beta(1, R->one());
+       upvec s;
        for ( size_t j=1; j<r; ++j ) {
-               DCOUTVAR(j);
-               DCOUTVAR(beta);
                vector<ex> mdarg(2);
-               mdarg[0] = umod_to_ex(q[j-1], x);
-               mdarg[1] = umod_to_ex(a[j-1], x);
+               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, umod_to_ex(beta, x), empty, 0, p, k);
-               umod sigma1 = umod_from_ex(exsigma[0], x, R);
-               umod sigma2 = umod_from_ex(exsigma[1], x, R);
-               beta = COPY(beta, sigma1);
+               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);
-
-       DCOUTVAR(s);
-       DCOUT(END multiterm_eea_lift);
        return s;
 }
 
-void change_modulus(umod& out, const umod& in)
+/**
+ *  Assert: a not empty.
+ */
+static void change_modulus(const cl_modint_ring& R, umodpoly& a)
 {
-       // ASSERT: out and in have same degree
-       if ( out.ring() == in.ring() ) {
-               out = COPY(out, in);
-       }
-       else {
-               for ( int i=0; i<=degree(in); ++i ) {
-                       out.set_coeff(i, out.ring()->basering()->canonhom(in.ring()->basering()->retract(coeff(in, i))));
-               }
-               out.finalize();
+       if ( a.empty() ) return;
+       cl_modint_ring oldR = a[0].ring();
+       umodpoly::iterator i = a.begin(), end = a.end();
+       for ( ; i!=end; ++i ) {
+               *i = R->canonhom(oldR->retract(*i));
        }
+       canonicalize(a);
 }
 
-void eea_lift(const umod& a, const umod& b, const ex& x, unsigned int p, unsigned int k, umod& s_, umod& t_)
+static void eea_lift(const umodpoly& a, const umodpoly& b, const ex& x, unsigned int p, unsigned int k, umodpoly& s_, umodpoly& t_)
 {
-       DCOUT(eea_lift);
-       DCOUTVAR(a);
-       DCOUTVAR(b);
-       DCOUTVAR(x);
-       DCOUTVAR(p);
-       DCOUTVAR(k);
-
        cl_modint_ring R = find_modint_ring(p);
-       cl_univpoly_modint_ring UPR = find_univpoly_ring(R);
-       umod amod = UPR->create(degree(a));
-       DCOUTVAR(a);
-       change_modulus(amod, a);
-       umod bmod = UPR->create(degree(b));
-       change_modulus(bmod, b);
-       DCOUTVAR(amod);
-       DCOUTVAR(bmod);
-
-       umod g = UPR->create(-1);
-       umod smod = UPR->create(-1);
-       umod tmod = UPR->create(-1);
-       exteuclid(amod, bmod, g, smod, tmod);
-       
-       DCOUTVAR(smod);
-       DCOUTVAR(tmod);
-       DCOUTVAR(g);
-       DCOUTVAR(a);
+       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));
-       cl_univpoly_modint_ring UPRpk = find_univpoly_ring(Rpk);
-       umod s = UPRpk->create(degree(smod));
-       change_modulus(s, smod);
-       umod t = UPRpk->create(degree(tmod));
-       change_modulus(t, tmod);
-       DCOUTVAR(s);
-       DCOUTVAR(t);
+       umodpoly s = smod;
+       change_modulus(Rpk, s);
+       umodpoly t = tmod;
+       change_modulus(Rpk, t);
 
        cl_I modulus(p);
-       DCOUTVAR(a);
-
-       umod one = UPRpk->one();
+       umodpoly one(1, Rpk->one());
        for ( size_t j=1; j<k; ++j ) {
-               DCOUTVAR(a);
-               umod e = one - a * s - b * t;
-               DCOUTVAR(one);
-               DCOUTVAR(a*s);
-               DCOUTVAR(b*t);
-               DCOUTVAR(e);
-               e = divide(e, modulus);
-               umod c = UPR->create(degree(e));
-               change_modulus(c, e);
-               umod sigmabar = smod * c;
-               umod taubar = tmod * c;
-               umod q = div(sigmabar, bmod);
-               umod sigma = rem(sigmabar, bmod);
-               umod tau = taubar + q * amod;
-               umod sadd = UPRpk->create(degree(sigma));
-               change_modulus(sadd, sigma);
+               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;
-               umod tadd = UPRpk->create(degree(tau));
-               change_modulus(tadd, tau);
+               umodpoly tadd = tau;
+               change_modulus(Rpk, tadd);
                t = t + tadd * modmodulus;
                modulus = modulus * p;
        }
 
        s_ = s; t_ = t;
-
-       DCOUTVAR(s);
-       DCOUTVAR(t);
-       DCOUT2(check, a*s + b*t);
-       DCOUT(END eea_lift);
 }
 
-umodvec univar_diophant(const umodvec& a, const ex& x, unsigned int m, unsigned int p, unsigned int k)
+static upvec univar_diophant(const upvec& a, const ex& x, unsigned int m, unsigned int p, unsigned int k)
 {
-       DCOUT(univar_diophant);
-       DCOUTVAR(a);
-       DCOUTVAR(x);
-       DCOUTVAR(m);
-       DCOUTVAR(p);
-       DCOUTVAR(k);
-
        cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
-       cl_univpoly_modint_ring UPR = find_univpoly_ring(R);
 
        const size_t r = a.size();
-       umodvec result;
+       upvec result;
        if ( r > 2 ) {
-               umodvec s = multiterm_eea_lift(a, x, p, k);
+               upvec s = multiterm_eea_lift(a, x, p, k);
                for ( size_t j=0; j<r; ++j ) {
-                       ex phi = expand(pow(x,m) * umod_to_ex(s[j], x));
-                       umod bmod = umod_from_ex(phi, x, R);
-                       umod buf = rem(bmod, a[j]);
+                       umodpoly bmod = umodpoly_to_umodpoly(s[j], R, m);
+                       umodpoly buf;
+                       rem(bmod, a[j], buf);
                        result.push_back(buf);
                }
        }
        else {
-               umod s = UPR->create(-1);
-               umod t = UPR->create(-1);
+               umodpoly s, t;
                eea_lift(a[1], a[0], x, p, k, s, t);
-               ex phi = expand(pow(x,m) * umod_to_ex(s, x));
-               umod bmod = umod_from_ex(phi, x, R);
-               umod buf = rem(bmod, a[0]);
+               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);
-               umod q = div(bmod, a[0]);
-               phi = expand(pow(x,m) * umod_to_ex(t, x));
-               umod t1mod = umod_from_ex(phi, x, R);
-               umod buf2 = t1mod + q * a[1];
-               result.push_back(buf2);
        }
 
-       DCOUTVAR(result);
-       DCOUT(END univar_diophant);
        return result;
 }
 
@@ -1185,39 +1622,17 @@ struct make_modular_map : public map_function {
 static ex make_modular(const ex& e, const cl_modint_ring& R)
 {
        make_modular_map map(R);
-       return map(e);
+       return map(e.expand());
 }
 
-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 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_;
 
-       DCOUT(multivar_diophant);
-#ifdef DEBUGFACTOR
-       cout << "a ";
-       for ( size_t i=0; i<a.size(); ++i ) {
-               cout << a[i] << " ";
-       }
-       cout << endl;
-#endif
-       DCOUTVAR(x);
-       DCOUTVAR(c);
-#ifdef DEBUGFACTOR
-       cout << "I ";
-       for ( size_t i=0; i<I.size(); ++i ) {
-               cout << I[i].x << "=" << I[i].evalpoint << " ";
-       }
-       cout << endl;
-#endif
-       DCOUTVAR(d);
-       DCOUTVAR(p);
-       DCOUTVAR(k);
-
        const cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
        const size_t r = a.size();
        const size_t nu = I.size() + 1;
-       DCOUTVAR(r);
-       DCOUTVAR(nu);
 
        vector<ex> sigma;
        if ( nu > 1 ) {
@@ -1241,48 +1656,36 @@ vector<ex> multivar_diophant(const vector<ex>& a_, const ex& x, const ex& c, con
                vector<EvalPoint> Inew = I;
                Inew.pop_back();
                sigma = multivar_diophant(anew, x, cnew, Inew, d, p, k);
-               DCOUTVAR(sigma);
 
                ex buf = c;
                for ( size_t i=0; i<r; ++i ) {
                        buf -= sigma[i] * b[i];
                }
-               ex e = buf;
-               e = make_modular(e, R);
+               ex e = make_modular(buf, R);
 
-               e = e.expand();
-               DCOUTVAR(e);
-               DCOUTVAR(d);
                ex monomial = 1;
-               for ( size_t m=1; m<=d; ++m ) {
-                       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.expand();
-                                       e = make_modular(e, R);
+               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 {
-               DCOUT(uniterm left);
-               umodvec amod;
+               upvec amod;
                for ( size_t i=0; i<a.size(); ++i ) {
-                       umod up = umod_from_ex(a[i], x, R);
+                       umodpoly up;
+                       umodpoly_from_ex(up, a[i], x, R);
                        amod.push_back(up);
                }
 
@@ -1297,58 +1700,30 @@ vector<ex> multivar_diophant(const vector<ex>& a_, const ex& x, const ex& c, con
                        nterms = 1;
                        z = c;
                }
-               DCOUTVAR(nterms);
                for ( size_t i=0; i<nterms; ++i ) {
-                       DCOUTVAR(z);
                        int m = z.degree(x);
-                       DCOUTVAR(m);
                        cl_I cm = the<cl_I>(ex_to<numeric>(z.lcoeff(x)).to_cl_N());
-                       DCOUTVAR(cm);
-                       umodvec delta_s = univar_diophant(amod, x, m, p, k);
+                       upvec delta_s = univar_diophant(amod, x, m, p, k);
                        cl_MI modcm;
                        cl_I poscm = cm;
                        while ( poscm < 0 ) {
                                poscm = poscm + expt_pos(cl_I(p),k);
                        }
                        modcm = cl_MI(R, poscm);
-                       DCOUTVAR(modcm);
                        for ( size_t j=0; j<delta_s.size(); ++j ) {
                                delta_s[j] = delta_s[j] * modcm;
-                               sigma[j] = sigma[j] + umod_to_ex(delta_s[j], x);
-                       }
-                       DCOUTVAR(delta_s);
-#ifdef DEBUGFACTOR
-                       cout << "STEP " << i << " sigma ";
-                       for ( size_t p=0; p<sigma.size(); ++p ) {
-                               cout << sigma[p] << " ";
+                               sigma[j] = sigma[j] + umodpoly_to_ex(delta_s[j], x);
                        }
-                       cout << endl;
-#endif
                        if ( nterms > 1 ) {
                                z = c.op(i+1);
                        }
                }
        }
-#ifdef DEBUGFACTOR
-       cout << "sigma ";
-       for ( size_t i=0; i<sigma.size(); ++i ) {
-               cout << sigma[i] << " ";
-       }
-       cout << endl;
-#endif
 
        for ( size_t i=0; i<sigma.size(); ++i ) {
                sigma[i] = make_modular(sigma[i], R);
        }
 
-#ifdef DEBUGFACTOR
-       cout << "sigma ";
-       for ( size_t i=0; i<sigma.size(); ++i ) {
-               cout << sigma[i] << " ";
-       }
-       cout << endl;
-#endif
-       DCOUT(END multivar_diophant);
        return sigma;
 }
 
@@ -1362,22 +1737,11 @@ ostream& operator<<(ostream& o, const vector<EvalPoint>& v)
 }
 #endif // def DEBUGFACTOR
 
-
-ex hensel_multivar(const ex& a, const ex& x, const vector<EvalPoint>& I, unsigned int p, const cl_I& l, const umodvec& u, const vector<ex>& lcU)
+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)
 {
-       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(nu);
-       
        vector<ex> A(nu);
        A[nu-1] = a;
 
@@ -1388,101 +1752,65 @@ ex hensel_multivar(const ex& a, const ex& x, const vector<EvalPoint>& I, unsigne
                A[j-2] = make_modular(A[j-2], R);
        }
 
-#ifdef DEBUGFACTOR
-       cout << "A ";
-       for ( size_t i=0; i<A.size(); ++i) cout << A[i] << " ";
-       cout << endl;
-#endif
-
        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;
        }
-       DCOUTVAR(maxdeg);
 
        const size_t n = u.size();
-       DCOUTVAR(n);
        vector<ex> U(n);
        for ( size_t i=0; i<n; ++i ) {
-               U[i] = umod_to_ex(u[i], x);
+               U[i] = umodpoly_to_ex(u[i], x);
        }
-#ifdef DEBUGFACTOR
-       cout << "U ";
-       for ( size_t i=0; i<U.size(); ++i) cout << U[i] << " ";
-       cout << endl;
-#endif
 
        for ( size_t j=2; j<=nu; ++j ) {
-               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];
                                for ( size_t i=j-1; i<nu-1; ++i ) {
                                        coef = coef.subs(I[i].x == I[i].evalpoint);
                                }
-                               coef = expand(coef);
                                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);
                        }
                }
-               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);
-               DCOUTVAR(deg);
                for ( size_t k=1; k<=deg; ++k ) {
-                       DCOUTVAR(k);
                        if ( !e.is_zero() ) {
-                               DCOUTVAR(xj);
-                               DCOUTVAR(alphaj);
                                monomial *= (xj - alphaj);
                                monomial = expand(monomial);
-                               DCOUTVAR(monomial);
                                ex dif = e.diff(ex_to<symbol>(xj), k);
-                               DCOUTVAR(dif);
                                ex c = dif.subs(xj==alphaj) / factorial(k);
-                               DCOUTVAR(c);
                                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 ) {
-                                               DCOUTVAR(i);
-                                               DCOUTVAR(deltaU[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 = A[j-1] - Uprod;
                                        e = make_modular(e, R);
-                                       DCOUTVAR(e);
-                               }
-                               else {
-                                       break;
                                }
                        }
                }
@@ -1492,51 +1820,35 @@ ex hensel_multivar(const ex& a, const ex& x, const vector<EvalPoint>& I, unsigne
        for ( size_t i=0; i<U.size(); ++i ) {
                acand *= U[i];
        }
-       DCOUTVAR(acand);
        if ( expand(a-acand).is_zero() ) {
                lst res;
                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 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) || 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) ) {
@@ -1548,16 +1860,12 @@ static ex put_factors_into_lst(const ex& e)
                        }
                        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.");
@@ -1573,370 +1881,257 @@ ostream& operator<<(ostream& o, const vector<numeric>& v)
 }
 #endif // def DEBUGFACTOR
 
-static bool checkdivisors(const lst& f, vector<numeric>& d)
+/** Checks whether in a set of numbers each has a unique prime factor.
+ *
+ *  @param[in]  f  list of numbers to check
+ *  @return        true: if number set is bad, false: otherwise
+ */
+static bool checkdivisors(const lst& f)
 {
-       DCOUT(checkdivisors);
-       const int k = f.nops()-2;
-       DCOUTVAR(k);
-       DCOUTVAR(d.size());
+       const int k = f.nops();
        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)));
+       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)));
-               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)
+/** Generates a set of evaluation points for a multivariate polynomial.
+ *  The set fulfills the following conditions:
+ *  1. lcoeff(evaluated_polynomial) does not vanish
+ *  2. factors of lcoeff(evaluated_polynomial) have each a unique prime factor
+ *  3. evaluated_polynomial is square free
+ *  See [W1] for more details.
+ *
+ *  @param[in]     u        multivariate polynomial to be factored
+ *  @param[in]     vn       leading coefficient of u in x (x==first symbol in syms)
+ *  @param[in]     syms     set of symbols that appear in u
+ *  @param[in]     f        lst containing the factors of the leading coefficient vn
+ *  @param[in,out] modulus  integer modulus for random number generation (i.e. |a_i| < modulus)
+ *  @param[out]    u0       returns the evaluated (univariate) polynomial
+ *  @param[out]    a        returns the valid evaluation points. must have initial size equal
+ *                          number of symbols-1 before calling generate_set
+ */
+static void generate_set(const ex& u, const ex& vn, const exset& syms, const lst& f,
+                         numeric& modulus, ex& u0, vector<numeric>& a)
 {
-       // 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;
+       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 ) {
-                       DCOUTVAR(*s);
                        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;
                }
-               DCOUTVAR(a);
-               DCOUTVAR(u0);
-               if ( gcd(u0,u0.diff(ex_to<symbol>(x))) != 1 ) {
+               /* ... 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) ) {
-                       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) ) {
+               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 ) {
-                                               fs = fs.subs(*s == a[j]);
-                                               ++s;
-                                       }
-                                       if ( abs(fs) == 1 ) {
-                                               problem = true;
-                                               break;
+                                       for ( size_t j=0; j<a.size(); ++j, ++s ) {
+                                               fnum.let_op(i) = fnum.op(i).subs(*s == a[j]);
                                        }
                                }
-                               fnum.append(fs);
-                               ++i; ++i;
                        }
-                       if ( problem ) {
-                               return true;
+                       if ( checkdivisors(fnum) ) {
+                               continue;
                        }
-                       ex con = u0.content(x);
-                       fnum.append(con);
-                       DCOUTVAR(fnum);
-                       trying = checkdivisors(fnum, d);
                }
-       } while ( trying );
-       DCOUT(END generate_set);
-       return false;
+               /* ok, we have a valid set now */
+               return;
+       }
 }
 
+// forward declaration
+static ex factor_sqrfree(const ex& poly);
+
+/**
+ *  ASSERT: poly is expanded
+ */
 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 p = poly.collect(x);
        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()));
+       for ( int i=p.degree(x)-1; i>=p.ldegree(x); --i ) {
+               cont = gcd(cont, p.coeff(x,i));
                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
+               return factor_sqrfree(cont) * factor_sqrfree(pp);
        }
 
        /* factor leading coefficient */
-       pp = pp.collect(x);
-       ex vn = pp.lcoeff(x);
-       pp = pp.expand();
+       ex vn = pp.collect(x).lcoeff(x);
        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;
+       const unsigned int maxtrials = 3;
+       numeric modulus = (vnlst.nops() > 3) ? vnlst.nops() : 3;
        vector<numeric> a(syms.size()-1, 0);
-       vector<numeric> d((vnlst.nops()-1)/2+1, 0);
 
+       /* try now to factorize until we are successful */
        while ( true ) {
-               numeric trialcount = 0;
-               ex u, delta;
+
+               unsigned int trialcount = 0;
                unsigned int prime;
-               size_t factor_count;
-               ex ufac;
-               ex ufaclst;
+               int factor_count = 0;
+               int min_factor_count = -1;
+               ex u, delta;
+               ex ufac, ufaclst;
+
+               /* try several evaluation points to reduce the number of modular factors */
                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 ) {
-                                       umod modpoly = umod_from_ex(u, x, R);
-                                       if ( squarefree(modpoly) ) break;
-                               }
-                               prime = next_prime(prime);
-                               DCOUTVAR(prime);
-                               R = find_modint_ring(prime);
-                       }
 
-#ifdef DEBUGFACTOR
-                       ufac = ::factor(u);
-#else
-                       ufac = factor(u);
-#endif
-                       DCOUTVAR(ufac);
+                       /* 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);
-                       DCOUTVAR(ufaclst);
-                       factor_count = (ufaclst.nops()-1)/2;
-                       DCOUTVAR(factor_count);
+                       factor_count = ufaclst.nops()-1;
+                       delta = ufaclst.op(0);
 
                        if ( factor_count <= 1 ) {
-                               DCOUTVAR(poly);
-                               DCOUT(END factor_multivariate);
+                               /* irreducible */
                                return poly;
                        }
-
-                       if ( minimalr < 0 ) {
-                               minimalr = factor_count;
+                       if ( min_factor_count < 0 ) {
+                               /* first time here */
+                               min_factor_count = factor_count;
                        }
-                       else if ( minimalr == factor_count ) {
+                       else if ( min_factor_count == factor_count ) {
+                               /* one less to try */
                                ++trialcount;
-                               ++modulus;
                        }
-                       else if ( minimalr > factor_count ) {
-                               minimalr = factor_count;
+                       else if ( min_factor_count > factor_count ) {
+                               /* new minimum, reset trial counter */
+                               min_factor_count = 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;
                }
 
+               // determine true leading coefficients for the Hensel lifting
                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);
+               if ( is_a<numeric>(vn) ) {
+                       for ( size_t i=1; i<ufaclst.nops(); ++i ) {
+                               C[i-1] = ufaclst.op(i).lcoeff(x);
                        }
                }
                else {
-                       for ( size_t i=0; i<D.size(); ++i ) {
-                               ex Dtilde = D[i];
+                       vector<numeric> ftilde(vnlst.nops()-1);
+                       for ( size_t i=0; i<ftilde.size(); ++i ) {
+                               ex ft = vnlst.op(i+1);
                                s = syms.begin();
                                ++s;
                                for ( size_t j=0; j<a.size(); ++j ) {
-                                       Dtilde = Dtilde.subs(*s == a[j]);
+                                       ft = ft.subs(*s == a[j]);
                                        ++s;
                                }
-                               ex ui;
-                               if ( i == 0 ) {
-                                       ui = ufaclst.op(0);
+                               ftilde[i] = ex_to<numeric>(ft);
+                       }
+
+                       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 {
-                                       ui = ufaclst.op(2*(i-1)+1);
+                       }
+                       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;
                                }
-                               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);
+                       }
+
+                       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;
+                       }
+               }
+
+               if ( delta != 1 ) {
+                       C[0] = C[0] * delta;
+                       ufaclst.let_op(1) = ufaclst.op(1) * delta;
                }
-               DCOUTVAR(C);
 
                EvalPoint ep;
                vector<EvalPoint> epv;
@@ -1947,45 +2142,34 @@ static ex factor_multivariate(const ex& poly, const exset& syms)
                        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);
+               // 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 = normmc * expt_pos(cl_I(2), maxdegree);
+               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;
                }
-
-               umodvec 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 ) {
-                       umod newu = umod_from_ex(ufaclst.op(i*2+1), x, R);
-                       uvec.push_back(newu);
+               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);
                }
-               DCOUTVAR(uvec);
 
-               ex res = hensel_multivar(ufaclst.op(0)*pp, x, epv, prime, l, uvec, C);
+               ex res = hensel_multivar(pp, x, epv, prime, l, modfactors, C);
                if ( res != lst() ) {
-                       ex result = cont * ufaclst.op(0);
+                       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;
                }
        }
@@ -2037,11 +2221,7 @@ struct apply_factor_map : public map_function {
        ex operator()(const ex& e)
        {
                if ( e.info(info_flags::polynomial) ) {
-#ifdef DEBUGFACTOR
-                       return ::factor(e, options);
-#else
                        return factor(e, options);
-#endif
                }
                if ( is_a<add>(e) ) {
                        ex s1, s2;
@@ -2055,11 +2235,7 @@ struct apply_factor_map : public map_function {
                        }
                        s1 = s1.eval();
                        s2 = s2.eval();
-#ifdef DEBUGFACTOR
-                       return ::factor(s1, options) + s2.map(*this);
-#else
                        return factor(s1, options) + s2.map(*this);
-#endif
                }
                return e.map(*this);
        }
@@ -2067,11 +2243,7 @@ struct apply_factor_map : public map_function {
 
 } // anonymous namespace
 
-#ifdef DEBUGFACTOR
-ex factor(const ex& poly, unsigned options = 0)
-#else
 ex factor(const ex& poly, unsigned options)
-#endif
 {
        // check arguments
        if ( !poly.info(info_flags::polynomial) ) {
@@ -2096,7 +2268,7 @@ ex factor(const ex& poly, unsigned options)
        }
 
        // make poly square free
-       ex sfpoly = sqrfree(poly, syms);
+       ex sfpoly = sqrfree(poly.expand(), syms);
 
        // factorize the square free components
        if ( is_a<power>(sfpoly) ) {
@@ -2143,3 +2315,7 @@ ex factor(const ex& poly, unsigned options)
 }
 
 } // namespace GiNaC
+
+#ifdef DEBUGFACTOR
+#include "test.h"
+#endif