3 * Polynomial factorization (implementation).
5 * The interface function factor() at the end of this file is defined in the
6 * GiNaC namespace. All other utility functions and classes are defined in an
7 * additional anonymous namespace.
9 * Factorization starts by doing a square free factorization and making the
10 * coefficients integer. Then, depending on the number of free variables it
11 * proceeds either in dedicated univariate or multivariate factorization code.
13 * Univariate factorization does a modular factorization via Berlekamp's
14 * algorithm and distinct degree factorization. Hensel lifting is used at the
17 * Multivariate factorization uses the univariate factorization (applying a
18 * evaluation homomorphism first) and Hensel lifting raises the answer to the
19 * multivariate domain. The Hensel lifting code is completely distinct from the
20 * code used by the univariate factorization.
22 * Algorithms used can be found in
23 * [Wan] An Improved Multivariate Polynomial Factoring Algorithm,
25 * Mathematics of Computation, Vol. 32, No. 144 (1978) 1215--1231.
26 * [GCL] Algorithms for Computer Algebra,
27 * K.O.Geddes, S.R.Czapor, G.Labahn,
28 * Springer Verlag, 1992.
29 * [Mig] Some Useful Bounds,
31 * In "Computer Algebra, Symbolic and Algebraic Computation" (B.Buchberger et al., eds.),
32 * pp. 259-263, Springer-Verlag, New York, 1982.
36 * GiNaC Copyright (C) 1999-2010 Johannes Gutenberg University Mainz, Germany
38 * This program is free software; you can redistribute it and/or modify
39 * it under the terms of the GNU General Public License as published by
40 * the Free Software Foundation; either version 2 of the License, or
41 * (at your option) any later version.
43 * This program is distributed in the hope that it will be useful,
44 * but WITHOUT ANY WARRANTY; without even the implied warranty of
45 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
46 * GNU General Public License for more details.
48 * You should have received a copy of the GNU General Public License
49 * along with this program; if not, write to the Free Software
50 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
59 #include "operators.h"
62 #include "relational.h"
84 #define DCOUT(str) cout << #str << endl
85 #define DCOUTVAR(var) cout << #var << ": " << var << endl
86 #define DCOUT2(str,var) cout << #str << ": " << var << endl
87 ostream& operator<<(ostream& o, const vector<int>& v)
89 vector<int>::const_iterator i = v.begin(), end = v.end();
95 static ostream& operator<<(ostream& o, const vector<cl_I>& v)
97 vector<cl_I>::const_iterator i = v.begin(), end = v.end();
99 o << *i << "[" << i-v.begin() << "]" << " ";
104 static ostream& operator<<(ostream& o, const vector<cl_MI>& v)
106 vector<cl_MI>::const_iterator i = v.begin(), end = v.end();
108 o << *i << "[" << i-v.begin() << "]" << " ";
113 ostream& operator<<(ostream& o, const vector<numeric>& v)
115 for ( size_t i=0; i<v.size(); ++i ) {
120 ostream& operator<<(ostream& o, const vector< vector<cl_MI> >& v)
122 vector< vector<cl_MI> >::const_iterator i = v.begin(), end = v.end();
124 o << i-v.begin() << ": " << *i << endl;
131 #define DCOUTVAR(var)
132 #define DCOUT2(str,var)
133 #endif // def DEBUGFACTOR
135 // anonymous namespace to hide all utility functions
138 ////////////////////////////////////////////////////////////////////////////////
139 // modular univariate polynomial code
141 typedef std::vector<cln::cl_MI> umodpoly;
142 typedef std::vector<cln::cl_I> upoly;
143 typedef vector<umodpoly> upvec;
145 // COPY FROM UPOLY.HPP
147 // CHANGED size_t -> int !!!
148 template<typename T> static int degree(const T& p)
153 template<typename T> static typename T::value_type lcoeff(const T& p)
155 return p[p.size() - 1];
158 static bool normalize_in_field(umodpoly& a)
162 if ( lcoeff(a) == a[0].ring()->one() ) {
166 const cln::cl_MI lc_1 = recip(lcoeff(a));
167 for (std::size_t k = a.size(); k-- != 0; )
172 template<typename T> static void
173 canonicalize(T& p, const typename T::size_type hint = std::numeric_limits<typename T::size_type>::max())
178 std::size_t i = p.size() - 1;
179 // Be fast if the polynomial is already canonicalized
186 bool is_zero = false;
204 p.erase(p.begin() + i, p.end());
207 // END COPY FROM UPOLY.HPP
209 static void expt_pos(umodpoly& a, unsigned int q)
211 if ( a.empty() ) return;
212 cl_MI zero = a[0].ring()->zero();
214 a.resize(degree(a)*q+1, zero);
215 for ( int i=deg; i>0; --i ) {
221 template<bool COND, typename T = void> struct enable_if
226 template<typename T> struct enable_if<false, T> { /* empty */ };
228 template<typename T> struct uvar_poly_p
230 static const bool value = false;
233 template<> struct uvar_poly_p<upoly>
235 static const bool value = true;
238 template<> struct uvar_poly_p<umodpoly>
240 static const bool value = true;
244 // Don't define this for anything but univariate polynomials.
245 static typename enable_if<uvar_poly_p<T>::value, T>::type
246 operator+(const T& a, const T& b)
253 for ( ; i<sb; ++i ) {
256 for ( ; i<sa; ++i ) {
265 for ( ; i<sa; ++i ) {
268 for ( ; i<sb; ++i ) {
277 // Don't define this for anything but univariate polynomials. Otherwise
278 // overload resolution might fail (this actually happens when compiling
279 // GiNaC with g++ 3.4).
280 static typename enable_if<uvar_poly_p<T>::value, T>::type
281 operator-(const T& a, const T& b)
288 for ( ; i<sb; ++i ) {
291 for ( ; i<sa; ++i ) {
300 for ( ; i<sa; ++i ) {
303 for ( ; i<sb; ++i ) {
311 static upoly operator*(const upoly& a, const upoly& b)
314 if ( a.empty() || b.empty() ) return c;
316 int n = degree(a) + degree(b);
318 for ( int i=0 ; i<=n; ++i ) {
319 for ( int j=0 ; j<=i; ++j ) {
320 if ( j > degree(a) || (i-j) > degree(b) ) continue;
321 c[i] = c[i] + a[j] * b[i-j];
328 static umodpoly operator*(const umodpoly& a, const umodpoly& b)
331 if ( a.empty() || b.empty() ) return c;
333 int n = degree(a) + degree(b);
334 c.resize(n+1, a[0].ring()->zero());
335 for ( int i=0 ; i<=n; ++i ) {
336 for ( int j=0 ; j<=i; ++j ) {
337 if ( j > degree(a) || (i-j) > degree(b) ) continue;
338 c[i] = c[i] + a[j] * b[i-j];
345 static upoly operator*(const upoly& a, const cl_I& x)
352 for ( size_t i=0; i<a.size(); ++i ) {
358 static upoly operator/(const upoly& a, const cl_I& x)
365 for ( size_t i=0; i<a.size(); ++i ) {
366 r[i] = exquo(a[i],x);
371 static umodpoly operator*(const umodpoly& a, const cl_MI& x)
373 umodpoly r(a.size());
374 for ( size_t i=0; i<a.size(); ++i ) {
381 static void upoly_from_ex(upoly& up, const ex& e, const ex& x)
383 // assert: e is in Z[x]
384 int deg = e.degree(x);
386 int ldeg = e.ldegree(x);
387 for ( ; deg>=ldeg; --deg ) {
388 up[deg] = the<cl_I>(ex_to<numeric>(e.coeff(x, deg)).to_cl_N());
390 for ( ; deg>=0; --deg ) {
396 static void umodpoly_from_upoly(umodpoly& ump, const upoly& e, const cl_modint_ring& R)
400 for ( ; deg>=0; --deg ) {
401 ump[deg] = R->canonhom(e[deg]);
406 static void umodpoly_from_ex(umodpoly& ump, const ex& e, const ex& x, const cl_modint_ring& R)
408 // assert: e is in Z[x]
409 int deg = e.degree(x);
411 int ldeg = e.ldegree(x);
412 for ( ; deg>=ldeg; --deg ) {
413 cl_I coeff = the<cl_I>(ex_to<numeric>(e.coeff(x, deg)).to_cl_N());
414 ump[deg] = R->canonhom(coeff);
416 for ( ; deg>=0; --deg ) {
417 ump[deg] = R->zero();
423 static void umodpoly_from_ex(umodpoly& ump, const ex& e, const ex& x, const cl_I& modulus)
425 umodpoly_from_ex(ump, e, x, find_modint_ring(modulus));
429 static ex upoly_to_ex(const upoly& a, const ex& x)
431 if ( a.empty() ) return 0;
433 for ( int i=degree(a); i>=0; --i ) {
434 e += numeric(a[i]) * pow(x, i);
439 static ex umodpoly_to_ex(const umodpoly& a, const ex& x)
441 if ( a.empty() ) return 0;
442 cl_modint_ring R = a[0].ring();
443 cl_I mod = R->modulus;
444 cl_I halfmod = (mod-1) >> 1;
446 for ( int i=degree(a); i>=0; --i ) {
447 cl_I n = R->retract(a[i]);
449 e += numeric(n-mod) * pow(x, i);
451 e += numeric(n) * pow(x, i);
457 static upoly umodpoly_to_upoly(const umodpoly& a)
460 if ( a.empty() ) return e;
461 cl_modint_ring R = a[0].ring();
462 cl_I mod = R->modulus;
463 cl_I halfmod = (mod-1) >> 1;
464 for ( int i=degree(a); i>=0; --i ) {
465 cl_I n = R->retract(a[i]);
475 static umodpoly umodpoly_to_umodpoly(const umodpoly& a, const cl_modint_ring& R, unsigned int m)
478 if ( a.empty() ) return e;
479 cl_modint_ring oldR = a[0].ring();
480 size_t sa = a.size();
481 e.resize(sa+m, R->zero());
482 for ( size_t i=0; i<sa; ++i ) {
483 e[i+m] = R->canonhom(oldR->retract(a[i]));
489 /** Divides all coefficients of the polynomial a by the integer x.
490 * All coefficients are supposed to be divisible by x. If they are not, the
491 * the<cl_I> cast will raise an exception.
493 * @param[in,out] a polynomial of which the coefficients will be reduced by x
494 * @param[in] x integer that divides the coefficients
496 static void reduce_coeff(umodpoly& a, const cl_I& x)
498 if ( a.empty() ) return;
500 cl_modint_ring R = a[0].ring();
501 umodpoly::iterator i = a.begin(), end = a.end();
502 for ( ; i!=end; ++i ) {
503 // cln cannot perform this division in the modular field
504 cl_I c = R->retract(*i);
505 *i = cl_MI(R, the<cl_I>(c / x));
509 /** Calculates remainder of a/b.
510 * Assertion: a and b not empty.
512 * @param[in] a polynomial dividend
513 * @param[in] b polynomial divisor
514 * @param[out] r polynomial remainder
516 static void rem(const umodpoly& a, const umodpoly& b, umodpoly& r)
525 cl_MI qk = div(r[n+k], b[n]);
527 for ( int i=0; i<n; ++i ) {
528 unsigned int j = n + k - 1 - i;
529 r[j] = r[j] - qk * b[j-k];
534 fill(r.begin()+n, r.end(), a[0].ring()->zero());
538 /** Calculates quotient of a/b.
539 * Assertion: a and b not empty.
541 * @param[in] a polynomial dividend
542 * @param[in] b polynomial divisor
543 * @param[out] q polynomial quotient
545 static void div(const umodpoly& a, const umodpoly& b, umodpoly& q)
554 q.resize(k+1, a[0].ring()->zero());
556 cl_MI qk = div(r[n+k], b[n]);
559 for ( int i=0; i<n; ++i ) {
560 unsigned int j = n + k - 1 - i;
561 r[j] = r[j] - qk * b[j-k];
569 /** Calculates quotient and remainder of a/b.
570 * Assertion: a and b not empty.
572 * @param[in] a polynomial dividend
573 * @param[in] b polynomial divisor
574 * @param[out] r polynomial remainder
575 * @param[out] q polynomial quotient
577 static void remdiv(const umodpoly& a, const umodpoly& b, umodpoly& r, umodpoly& q)
586 q.resize(k+1, a[0].ring()->zero());
588 cl_MI qk = div(r[n+k], b[n]);
591 for ( int i=0; i<n; ++i ) {
592 unsigned int j = n + k - 1 - i;
593 r[j] = r[j] - qk * b[j-k];
598 fill(r.begin()+n, r.end(), a[0].ring()->zero());
603 /** Calculates the GCD of polynomial a and b.
605 * @param[in] a polynomial
606 * @param[in] b polynomial
609 static void gcd(const umodpoly& a, const umodpoly& b, umodpoly& c)
611 if ( degree(a) < degree(b) ) return gcd(b, a, c);
614 normalize_in_field(c);
616 normalize_in_field(d);
618 while ( !d.empty() ) {
623 normalize_in_field(c);
626 /** Calculates the derivative of the polynomial a.
628 * @param[in] a polynomial of which to take the derivative
629 * @param[out] d result/derivative
631 static void deriv(const umodpoly& a, umodpoly& d)
634 if ( a.size() <= 1 ) return;
636 d.insert(d.begin(), a.begin()+1, a.end());
638 for ( int i=1; i<max; ++i ) {
644 static bool unequal_one(const umodpoly& a)
646 if ( a.empty() ) return true;
647 return ( a.size() != 1 || a[0] != a[0].ring()->one() );
650 static bool equal_one(const umodpoly& a)
652 return ( a.size() == 1 && a[0] == a[0].ring()->one() );
655 /** Returns true if polynomial a is square free.
657 * @param[in] a polynomial to check
658 * @return true if polynomial is square free, false otherwise
660 static bool squarefree(const umodpoly& a)
672 // END modular univariate polynomial code
673 ////////////////////////////////////////////////////////////////////////////////
675 ////////////////////////////////////////////////////////////////////////////////
678 typedef vector<cl_MI> mvec;
682 friend ostream& operator<<(ostream& o, const modular_matrix& m);
684 modular_matrix(size_t r_, size_t c_, const cl_MI& init) : r(r_), c(c_)
688 size_t rowsize() const { return r; }
689 size_t colsize() const { return c; }
690 cl_MI& operator()(size_t row, size_t col) { return m[row*c + col]; }
691 cl_MI operator()(size_t row, size_t col) const { return m[row*c + col]; }
692 void mul_col(size_t col, const cl_MI x)
694 for ( size_t rc=0; rc<r; ++rc ) {
695 std::size_t i = c*rc + col;
699 void sub_col(size_t col1, size_t col2, const cl_MI fac)
701 for ( size_t rc=0; rc<r; ++rc ) {
702 std::size_t i1 = col1 + c*rc;
703 std::size_t i2 = col2 + c*rc;
704 m[i1] = m[i1] - m[i2]*fac;
707 void switch_col(size_t col1, size_t col2)
709 for ( size_t rc=0; rc<r; ++rc ) {
710 std::size_t i1 = col1 + rc*c;
711 std::size_t i2 = col2 + rc*c;
712 std::swap(m[i1], m[i2]);
715 void mul_row(size_t row, const cl_MI x)
717 for ( size_t cc=0; cc<c; ++cc ) {
718 std::size_t i = row*c + cc;
722 void sub_row(size_t row1, size_t row2, const cl_MI fac)
724 for ( size_t cc=0; cc<c; ++cc ) {
725 std::size_t i1 = row1*c + cc;
726 std::size_t i2 = row2*c + cc;
727 m[i1] = m[i1] - m[i2]*fac;
730 void switch_row(size_t row1, size_t row2)
732 for ( size_t cc=0; cc<c; ++cc ) {
733 std::size_t i1 = row1*c + cc;
734 std::size_t i2 = row2*c + cc;
735 std::swap(m[i1], m[i2]);
738 bool is_col_zero(size_t col) const
740 mvec::const_iterator i = m.begin() + col;
741 for ( size_t rr=0; rr<r; ++rr ) {
742 std::size_t i = col + rr*c;
743 if ( !zerop(m[i]) ) {
749 bool is_row_zero(size_t row) const
751 for ( size_t cc=0; cc<c; ++cc ) {
752 std::size_t i = row*c + cc;
753 if ( !zerop(m[i]) ) {
759 void set_row(size_t row, const vector<cl_MI>& newrow)
761 for (std::size_t i2 = 0; i2 < newrow.size(); ++i2) {
762 std::size_t i1 = row*c + i2;
766 mvec::const_iterator row_begin(size_t row) const { return m.begin()+row*c; }
767 mvec::const_iterator row_end(size_t row) const { return m.begin()+row*c+r; }
774 modular_matrix operator*(const modular_matrix& m1, const modular_matrix& m2)
776 const unsigned int r = m1.rowsize();
777 const unsigned int c = m2.colsize();
778 modular_matrix o(r,c,m1(0,0));
780 for ( size_t i=0; i<r; ++i ) {
781 for ( size_t j=0; j<c; ++j ) {
783 buf = m1(i,0) * m2(0,j);
784 for ( size_t k=1; k<c; ++k ) {
785 buf = buf + m1(i,k)*m2(k,j);
793 ostream& operator<<(ostream& o, const modular_matrix& m)
795 cl_modint_ring R = m(0,0).ring();
797 for ( size_t i=0; i<m.rowsize(); ++i ) {
799 for ( size_t j=0; j<m.colsize()-1; ++j ) {
800 o << R->retract(m(i,j)) << ",";
802 o << R->retract(m(i,m.colsize()-1)) << "}";
803 if ( i != m.rowsize()-1 ) {
810 #endif // def DEBUGFACTOR
812 // END modular matrix
813 ////////////////////////////////////////////////////////////////////////////////
815 /** Calculates the Q matrix for a polynomial. Used by Berlekamp's algorithm.
817 * @param[in] a_ modular polynomial
818 * @param[out] Q Q matrix
820 static void q_matrix(const umodpoly& a_, modular_matrix& Q)
823 normalize_in_field(a);
826 unsigned int q = cl_I_to_uint(a[0].ring()->modulus);
827 umodpoly r(n, a[0].ring()->zero());
828 r[0] = a[0].ring()->one();
830 unsigned int max = (n-1) * q;
831 for ( size_t m=1; m<=max; ++m ) {
832 cl_MI rn_1 = r.back();
833 for ( size_t i=n-1; i>0; --i ) {
834 r[i] = r[i-1] - (rn_1 * a[i]);
837 if ( (m % q) == 0 ) {
843 /** Determine the nullspace of a matrix M-1.
845 * @param[in,out] M matrix, will be modified
846 * @param[out] basis calculated nullspace of M-1
848 static void nullspace(modular_matrix& M, vector<mvec>& basis)
850 const size_t n = M.rowsize();
851 const cl_MI one = M(0,0).ring()->one();
852 for ( size_t i=0; i<n; ++i ) {
853 M(i,i) = M(i,i) - one;
855 for ( size_t r=0; r<n; ++r ) {
857 for ( ; cc<n; ++cc ) {
858 if ( !zerop(M(r,cc)) ) {
860 if ( !zerop(M(cc,cc)) ) {
872 M.mul_col(r, recip(M(r,r)));
873 for ( cc=0; cc<n; ++cc ) {
875 M.sub_col(cc, r, M(r,cc));
881 for ( size_t i=0; i<n; ++i ) {
882 M(i,i) = M(i,i) - one;
884 for ( size_t i=0; i<n; ++i ) {
885 if ( !M.is_row_zero(i) ) {
886 mvec nu(M.row_begin(i), M.row_end(i));
892 /** Berlekamp's modular factorization.
894 * The implementation follows the algorithm in chapter 8 of [GCL].
896 * @param[in] a modular polynomial
897 * @param[out] upv vector containing modular factors. if upv was not empty the
898 * new elements are added at the end
900 static void berlekamp(const umodpoly& a, upvec& upv)
902 cl_modint_ring R = a[0].ring();
903 umodpoly one(1, R->one());
905 // find nullspace of Q matrix
906 modular_matrix Q(degree(a), degree(a), R->zero());
911 const unsigned int k = nu.size();
917 list<umodpoly> factors;
918 factors.push_back(a);
919 unsigned int size = 1;
921 unsigned int q = cl_I_to_uint(R->modulus);
923 list<umodpoly>::iterator u = factors.begin();
925 // calculate all gcd's
927 for ( unsigned int s=0; s<q; ++s ) {
928 umodpoly nur = nu[r];
929 nur[0] = nur[0] - cl_MI(R, s);
933 if ( unequal_one(g) && g != *u ) {
936 if ( equal_one(uo) ) {
937 throw logic_error("berlekamp: unexpected divisor.");
942 factors.push_back(g);
944 list<umodpoly>::const_iterator i = factors.begin(), end = factors.end();
946 if ( degree(*i) ) ++size;
950 list<umodpoly>::const_iterator i = factors.begin(), end = factors.end();
965 // modular square free factorization is not used at the moment so we deactivate
969 /** Calculates a^(1/prime).
971 * @param[in] a polynomial
972 * @param[in] prime prime number -> exponent 1/prime
973 * @param[in] ap resulting polynomial
975 static void expt_1_over_p(const umodpoly& a, unsigned int prime, umodpoly& ap)
977 size_t newdeg = degree(a)/prime;
980 for ( size_t i=1; i<=newdeg; ++i ) {
985 /** Modular square free factorization.
987 * @param[in] a polynomial
988 * @param[out] factors modular factors
989 * @param[out] mult corresponding multiplicities (exponents)
991 static void modsqrfree(const umodpoly& a, upvec& factors, vector<int>& mult)
993 const unsigned int prime = cl_I_to_uint(a[0].ring()->modulus);
1002 while ( unequal_one(w) ) {
1007 factors.push_back(z);
1015 if ( unequal_one(c) ) {
1017 expt_1_over_p(c, prime, cp);
1018 size_t previ = mult.size();
1019 modsqrfree(cp, factors, mult);
1020 for ( size_t i=previ; i<mult.size(); ++i ) {
1027 expt_1_over_p(a, prime, ap);
1028 size_t previ = mult.size();
1029 modsqrfree(ap, factors, mult);
1030 for ( size_t i=previ; i<mult.size(); ++i ) {
1036 #endif // deactivation of square free factorization
1038 /** Distinct degree factorization (DDF).
1040 * The implementation follows the algorithm in chapter 8 of [GCL].
1042 * @param[in] a_ modular polynomial
1043 * @param[out] degrees vector containing the degrees of the factors of the
1044 * corresponding polynomials in ddfactors.
1045 * @param[out] ddfactors vector containing polynomials which factors have the
1046 * degree given in degrees.
1048 static void distinct_degree_factor(const umodpoly& a_, vector<int>& degrees, upvec& ddfactors)
1052 cl_modint_ring R = a[0].ring();
1053 int q = cl_I_to_int(R->modulus);
1054 int nhalf = degree(a)/2;
1062 while ( i <= nhalf ) {
1067 umodpoly wx = w - x;
1069 if ( unequal_one(buf) ) {
1070 degrees.push_back(i);
1071 ddfactors.push_back(buf);
1073 if ( unequal_one(buf) ) {
1077 nhalf = degree(a)/2;
1083 if ( unequal_one(a) ) {
1084 degrees.push_back(degree(a));
1085 ddfactors.push_back(a);
1089 /** Modular same degree factorization.
1090 * Same degree factorization is a kind of misnomer. It performs distinct degree
1091 * factorization, but instead of using the Cantor-Zassenhaus algorithm it
1092 * (sub-optimally) uses Berlekamp's algorithm for the factors of the same
1095 * @param[in] a modular polynomial
1096 * @param[out] upv vector containing modular factors. if upv was not empty the
1097 * new elements are added at the end
1099 static void same_degree_factor(const umodpoly& a, upvec& upv)
1101 cl_modint_ring R = a[0].ring();
1103 vector<int> degrees;
1105 distinct_degree_factor(a, degrees, ddfactors);
1107 for ( size_t i=0; i<degrees.size(); ++i ) {
1108 if ( degrees[i] == degree(ddfactors[i]) ) {
1109 upv.push_back(ddfactors[i]);
1112 berlekamp(ddfactors[i], upv);
1117 // Yes, we can (choose).
1118 #define USE_SAME_DEGREE_FACTOR
1120 /** Modular univariate factorization.
1122 * In principle, we have two algorithms at our disposal: Berlekamp's algorithm
1123 * and same degree factorization (SDF). SDF seems to be slightly faster in
1124 * almost all cases so it is activated as default.
1126 * @param[in] p modular polynomial
1127 * @param[out] upv vector containing modular factors. if upv was not empty the
1128 * new elements are added at the end
1130 static void factor_modular(const umodpoly& p, upvec& upv)
1132 #ifdef USE_SAME_DEGREE_FACTOR
1133 same_degree_factor(p, upv);
1139 /** Calculates modular polynomials s and t such that a*s+b*t==1.
1140 * Assertion: a and b are relatively prime and not zero.
1142 * @param[in] a polynomial
1143 * @param[in] b polynomial
1144 * @param[out] s polynomial
1145 * @param[out] t polynomial
1147 static void exteuclid(const umodpoly& a, const umodpoly& b, umodpoly& s, umodpoly& t)
1149 if ( degree(a) < degree(b) ) {
1150 exteuclid(b, a, t, s);
1154 umodpoly one(1, a[0].ring()->one());
1155 umodpoly c = a; normalize_in_field(c);
1156 umodpoly d = b; normalize_in_field(d);
1164 umodpoly r = c - q * d;
1165 umodpoly r1 = s - q * d1;
1166 umodpoly r2 = t - q * d2;
1170 if ( r.empty() ) break;
1175 cl_MI fac = recip(lcoeff(a) * lcoeff(c));
1176 umodpoly::iterator i = s.begin(), end = s.end();
1177 for ( ; i!=end; ++i ) {
1181 fac = recip(lcoeff(b) * lcoeff(c));
1182 i = t.begin(), end = t.end();
1183 for ( ; i!=end; ++i ) {
1189 /** Replaces the leading coefficient in a polynomial by a given number.
1191 * @param[in] poly polynomial to change
1192 * @param[in] lc new leading coefficient
1193 * @return changed polynomial
1195 static upoly replace_lc(const upoly& poly, const cl_I& lc)
1197 if ( poly.empty() ) return poly;
1203 /** Calculates the bound for the modulus.
1206 static inline cl_I calc_bound(const ex& a, const ex& x, int maxdeg)
1210 for ( int i=a.degree(x); i>=a.ldegree(x); --i ) {
1211 cl_I aa = abs(the<cl_I>(ex_to<numeric>(a.coeff(x, i)).to_cl_N()));
1212 if ( aa > maxcoeff ) maxcoeff = aa;
1213 coeff = coeff + square(aa);
1215 cl_I coeffnorm = ceiling1(the<cl_R>(cln::sqrt(coeff)));
1216 cl_I B = coeffnorm * expt_pos(cl_I(2), cl_I(maxdeg));
1217 return ( B > maxcoeff ) ? B : maxcoeff;
1220 /** Calculates the bound for the modulus.
1223 static inline cl_I calc_bound(const upoly& a, int maxdeg)
1227 for ( int i=degree(a); i>=0; --i ) {
1228 cl_I aa = abs(a[i]);
1229 if ( aa > maxcoeff ) maxcoeff = aa;
1230 coeff = coeff + square(aa);
1232 cl_I coeffnorm = ceiling1(the<cl_R>(cln::sqrt(coeff)));
1233 cl_I B = coeffnorm * expt_pos(cl_I(2), cl_I(maxdeg));
1234 return ( B > maxcoeff ) ? B : maxcoeff;
1237 /** Hensel lifting as used by factor_univariate().
1239 * The implementation follows the algorithm in chapter 6 of [GCL].
1241 * @param[in] a_ primitive univariate polynomials
1242 * @param[in] p prime number that does not divide lcoeff(a)
1243 * @param[in] u1_ modular factor of a (mod p)
1244 * @param[in] w1_ modular factor of a (mod p), relatively prime to u1_,
1245 * fulfilling u1_*w1_ == a mod p
1246 * @param[out] u lifted factor
1247 * @param[out] w lifted factor, u*w = a
1249 static void hensel_univar(const upoly& a_, unsigned int p, const umodpoly& u1_, const umodpoly& w1_, upoly& u, upoly& w)
1252 const cl_modint_ring& R = u1_[0].ring();
1255 int maxdeg = (degree(u1_) > degree(w1_)) ? degree(u1_) : degree(w1_);
1256 cl_I maxmodulus = 2*calc_bound(a, maxdeg);
1259 cl_I alpha = lcoeff(a);
1262 normalize_in_field(nu1);
1264 normalize_in_field(nw1);
1266 phi = umodpoly_to_upoly(nu1) * alpha;
1268 umodpoly_from_upoly(u1, phi, R);
1269 phi = umodpoly_to_upoly(nw1) * alpha;
1271 umodpoly_from_upoly(w1, phi, R);
1276 exteuclid(u1, w1, s, t);
1279 u = replace_lc(umodpoly_to_upoly(u1), alpha);
1280 w = replace_lc(umodpoly_to_upoly(w1), alpha);
1281 upoly e = a - u * w;
1285 while ( !e.empty() && modulus < maxmodulus ) {
1286 upoly c = e / modulus;
1287 phi = umodpoly_to_upoly(s) * c;
1288 umodpoly sigmatilde;
1289 umodpoly_from_upoly(sigmatilde, phi, R);
1290 phi = umodpoly_to_upoly(t) * c;
1292 umodpoly_from_upoly(tautilde, phi, R);
1294 remdiv(sigmatilde, w1, r, q);
1296 phi = umodpoly_to_upoly(tautilde) + umodpoly_to_upoly(q) * umodpoly_to_upoly(u1);
1298 umodpoly_from_upoly(tau, phi, R);
1299 u = u + umodpoly_to_upoly(tau) * modulus;
1300 w = w + umodpoly_to_upoly(sigma) * modulus;
1302 modulus = modulus * p;
1308 for ( size_t i=1; i<u.size(); ++i ) {
1310 if ( g == 1 ) break;
1325 /** Returns a new prime number.
1327 * @param[in] p prime number
1328 * @return next prime number after p
1330 static unsigned int next_prime(unsigned int p)
1332 static vector<unsigned int> primes;
1333 if ( primes.size() == 0 ) {
1334 primes.push_back(3); primes.push_back(5); primes.push_back(7);
1336 vector<unsigned int>::const_iterator it = primes.begin();
1337 if ( p >= primes.back() ) {
1338 unsigned int candidate = primes.back() + 2;
1340 size_t n = primes.size()/2;
1341 for ( size_t i=0; i<n; ++i ) {
1342 if ( candidate % primes[i] ) continue;
1346 primes.push_back(candidate);
1347 if ( candidate > p ) break;
1351 vector<unsigned int>::const_iterator end = primes.end();
1352 for ( ; it!=end; ++it ) {
1357 throw logic_error("next_prime: should not reach this point!");
1360 /** Manages the splitting a vector of of modular factors into two partitions.
1362 class factor_partition
1365 /** Takes the vector of modular factors and initializes the first partition */
1366 factor_partition(const upvec& factors_) : factors(factors_)
1372 one.resize(1, factors.front()[0].ring()->one());
1377 int operator[](size_t i) const { return k[i]; }
1378 size_t size() const { return n; }
1379 size_t size_left() const { return n-len; }
1380 size_t size_right() const { return len; }
1381 /** Initializes the next partition.
1382 Returns true, if there is one, false otherwise. */
1385 if ( last == n-1 ) {
1395 while ( k[last] == 0 ) { --last; }
1396 if ( last == 0 && n == 2*len ) return false;
1398 for ( size_t i=0; i<=len-rem; ++i ) {
1402 fill(k.begin()+last, k.end(), 0);
1409 if ( len > n/2 ) return false;
1410 fill(k.begin(), k.begin()+len, 1);
1411 fill(k.begin()+len+1, k.end(), 0);
1420 /** Get first partition */
1421 umodpoly& left() { return lr[0]; }
1422 /** Get second partition */
1423 umodpoly& right() { return lr[1]; }
1432 while ( i < n && k[i] == group ) { ++d; ++i; }
1434 if ( cache[pos].size() >= d ) {
1435 lr[group] = lr[group] * cache[pos][d-1];
1438 if ( cache[pos].size() == 0 ) {
1439 cache[pos].push_back(factors[pos] * factors[pos+1]);
1441 size_t j = pos + cache[pos].size() + 1;
1442 d -= cache[pos].size();
1444 umodpoly buf = cache[pos].back() * factors[j];
1445 cache[pos].push_back(buf);
1449 lr[group] = lr[group] * cache[pos].back();
1453 lr[group] = lr[group] * factors[pos];
1465 for ( size_t i=0; i<n; ++i ) {
1466 lr[k[i]] = lr[k[i]] * factors[i];
1472 vector< vector<umodpoly> > cache;
1481 /** Contains a pair of univariate polynomial and its modular factors.
1482 * Used by factor_univariate().
1490 /** Univariate polynomial factorization.
1492 * Modular factorization is tried for several primes to minimize the number of
1493 * modular factors. Then, Hensel lifting is performed.
1495 * @param[in] poly expanded square free univariate polynomial
1496 * @param[in] x symbol
1497 * @param[in,out] prime prime number to start trying modular factorization with,
1498 * output value is the prime number actually used
1500 static ex factor_univariate(const ex& poly, const ex& x, unsigned int& prime)
1502 ex unit, cont, prim_ex;
1503 poly.unitcontprim(x, unit, cont, prim_ex);
1505 upoly_from_ex(prim, prim_ex, x);
1507 // determine proper prime and minimize number of modular factors
1509 unsigned int lastp = prime;
1511 unsigned int trials = 0;
1512 unsigned int minfactors = 0;
1513 cl_I lc = lcoeff(prim) * the<cl_I>(ex_to<numeric>(cont).to_cl_N());
1515 while ( trials < 2 ) {
1518 prime = next_prime(prime);
1519 if ( !zerop(rem(lc, prime)) ) {
1520 R = find_modint_ring(prime);
1521 umodpoly_from_upoly(modpoly, prim, R);
1522 if ( squarefree(modpoly) ) break;
1526 // do modular factorization
1528 factor_modular(modpoly, trialfactors);
1529 if ( trialfactors.size() <= 1 ) {
1530 // irreducible for sure
1534 if ( minfactors == 0 || trialfactors.size() < minfactors ) {
1535 factors = trialfactors;
1536 minfactors = trialfactors.size();
1545 R = find_modint_ring(prime);
1547 // lift all factor combinations
1548 stack<ModFactors> tocheck;
1551 mf.factors = factors;
1555 while ( tocheck.size() ) {
1556 const size_t n = tocheck.top().factors.size();
1557 factor_partition part(tocheck.top().factors);
1559 // call Hensel lifting
1560 hensel_univar(tocheck.top().poly, prime, part.left(), part.right(), f1, f2);
1561 if ( !f1.empty() ) {
1562 // successful, update the stack and the result
1563 if ( part.size_left() == 1 ) {
1564 if ( part.size_right() == 1 ) {
1565 result *= upoly_to_ex(f1, x) * upoly_to_ex(f2, x);
1569 result *= upoly_to_ex(f1, x);
1570 tocheck.top().poly = f2;
1571 for ( size_t i=0; i<n; ++i ) {
1572 if ( part[i] == 0 ) {
1573 tocheck.top().factors.erase(tocheck.top().factors.begin()+i);
1579 else if ( part.size_right() == 1 ) {
1580 if ( part.size_left() == 1 ) {
1581 result *= upoly_to_ex(f1, x) * upoly_to_ex(f2, x);
1585 result *= upoly_to_ex(f2, x);
1586 tocheck.top().poly = f1;
1587 for ( size_t i=0; i<n; ++i ) {
1588 if ( part[i] == 1 ) {
1589 tocheck.top().factors.erase(tocheck.top().factors.begin()+i);
1596 upvec newfactors1(part.size_left()), newfactors2(part.size_right());
1597 upvec::iterator i1 = newfactors1.begin(), i2 = newfactors2.begin();
1598 for ( size_t i=0; i<n; ++i ) {
1600 *i2++ = tocheck.top().factors[i];
1603 *i1++ = tocheck.top().factors[i];
1606 tocheck.top().factors = newfactors1;
1607 tocheck.top().poly = f1;
1609 mf.factors = newfactors2;
1617 if ( !part.next() ) {
1618 // if no more combinations left, return polynomial as
1620 result *= upoly_to_ex(tocheck.top().poly, x);
1628 return unit * cont * result;
1631 /** Second interface to factor_univariate() to be used if the information about
1632 * the prime is not needed.
1634 static inline ex factor_univariate(const ex& poly, const ex& x)
1637 return factor_univariate(poly, x, prime);
1640 /** Represents an evaluation point (<symbol>==<integer>).
1649 ostream& operator<<(ostream& o, const vector<EvalPoint>& v)
1651 for ( size_t i=0; i<v.size(); ++i ) {
1652 o << "(" << v[i].x << "==" << v[i].evalpoint << ") ";
1656 #endif // def DEBUGFACTOR
1658 // forward declaration
1659 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);
1661 /** Utility function for multivariate Hensel lifting.
1663 * Solves the equation
1664 * s_1*b_1 + ... + s_r*b_r == 1 mod p^k
1665 * with deg(s_i) < deg(a_i)
1666 * and with given b_1 = a_1 * ... * a_{i-1} * a_{i+1} * ... * a_r
1668 * The implementation follows the algorithm in chapter 6 of [GCL].
1670 * @param[in] a vector of modular univariate polynomials
1671 * @param[in] x symbol
1672 * @param[in] p prime number
1673 * @param[in] k p^k is modulus
1674 * @return vector of polynomials (s_i)
1676 static upvec multiterm_eea_lift(const upvec& a, const ex& x, unsigned int p, unsigned int k)
1678 const size_t r = a.size();
1679 cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
1682 for ( size_t j=r-2; j>=1; --j ) {
1683 q[j-1] = a[j] * q[j];
1685 umodpoly beta(1, R->one());
1687 for ( size_t j=1; j<r; ++j ) {
1688 vector<ex> mdarg(2);
1689 mdarg[0] = umodpoly_to_ex(q[j-1], x);
1690 mdarg[1] = umodpoly_to_ex(a[j-1], x);
1691 vector<EvalPoint> empty;
1692 vector<ex> exsigma = multivar_diophant(mdarg, x, umodpoly_to_ex(beta, x), empty, 0, p, k);
1694 umodpoly_from_ex(sigma1, exsigma[0], x, R);
1696 umodpoly_from_ex(sigma2, exsigma[1], x, R);
1698 s.push_back(sigma2);
1704 /** Changes the modulus of a modular polynomial. Used by eea_lift().
1706 * @param[in] R new modular ring
1707 * @param[in,out] a polynomial to change (in situ)
1709 static void change_modulus(const cl_modint_ring& R, umodpoly& a)
1711 if ( a.empty() ) return;
1712 cl_modint_ring oldR = a[0].ring();
1713 umodpoly::iterator i = a.begin(), end = a.end();
1714 for ( ; i!=end; ++i ) {
1715 *i = R->canonhom(oldR->retract(*i));
1720 /** Utility function for multivariate Hensel lifting.
1722 * Solves s*a + t*b == 1 mod p^k given a,b.
1724 * The implementation follows the algorithm in chapter 6 of [GCL].
1726 * @param[in] a polynomial
1727 * @param[in] b polynomial
1728 * @param[in] x symbol
1729 * @param[in] p prime number
1730 * @param[in] k p^k is modulus
1731 * @param[out] s_ output polynomial
1732 * @param[out] t_ output polynomial
1734 static void eea_lift(const umodpoly& a, const umodpoly& b, const ex& x, unsigned int p, unsigned int k, umodpoly& s_, umodpoly& t_)
1736 cl_modint_ring R = find_modint_ring(p);
1738 change_modulus(R, amod);
1740 change_modulus(R, bmod);
1744 exteuclid(amod, bmod, smod, tmod);
1746 cl_modint_ring Rpk = find_modint_ring(expt_pos(cl_I(p),k));
1748 change_modulus(Rpk, s);
1750 change_modulus(Rpk, t);
1753 umodpoly one(1, Rpk->one());
1754 for ( size_t j=1; j<k; ++j ) {
1755 umodpoly e = one - a * s - b * t;
1756 reduce_coeff(e, modulus);
1758 change_modulus(R, c);
1759 umodpoly sigmabar = smod * c;
1760 umodpoly taubar = tmod * c;
1762 remdiv(sigmabar, bmod, sigma, q);
1763 umodpoly tau = taubar + q * amod;
1764 umodpoly sadd = sigma;
1765 change_modulus(Rpk, sadd);
1766 cl_MI modmodulus(Rpk, modulus);
1767 s = s + sadd * modmodulus;
1768 umodpoly tadd = tau;
1769 change_modulus(Rpk, tadd);
1770 t = t + tadd * modmodulus;
1771 modulus = modulus * p;
1777 /** Utility function for multivariate Hensel lifting.
1779 * Solves the equation
1780 * s_1*b_1 + ... + s_r*b_r == x^m mod p^k
1781 * with given b_1 = a_1 * ... * a_{i-1} * a_{i+1} * ... * a_r
1783 * The implementation follows the algorithm in chapter 6 of [GCL].
1785 * @param a vector with univariate polynomials mod p^k
1787 * @param m exponent of x^m in the equation to solve
1788 * @param p prime number
1789 * @param k p^k is modulus
1790 * @return vector of polynomials (s_i)
1792 static upvec univar_diophant(const upvec& a, const ex& x, unsigned int m, unsigned int p, unsigned int k)
1794 cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
1796 const size_t r = a.size();
1799 upvec s = multiterm_eea_lift(a, x, p, k);
1800 for ( size_t j=0; j<r; ++j ) {
1801 umodpoly bmod = umodpoly_to_umodpoly(s[j], R, m);
1803 rem(bmod, a[j], buf);
1804 result.push_back(buf);
1809 eea_lift(a[1], a[0], x, p, k, s, t);
1810 umodpoly bmod = umodpoly_to_umodpoly(s, R, m);
1812 remdiv(bmod, a[0], buf, q);
1813 result.push_back(buf);
1814 umodpoly t1mod = umodpoly_to_umodpoly(t, R, m);
1815 buf = t1mod + q * a[1];
1816 result.push_back(buf);
1822 /** Map used by function make_modular().
1823 * Finds every coefficient in a polynomial and replaces it by is value in the
1824 * given modular ring R (symmetric representation).
1826 struct make_modular_map : public map_function {
1828 make_modular_map(const cl_modint_ring& R_) : R(R_) { }
1829 ex operator()(const ex& e)
1831 if ( is_a<add>(e) || is_a<mul>(e) ) {
1832 return e.map(*this);
1834 else if ( is_a<numeric>(e) ) {
1835 numeric mod(R->modulus);
1836 numeric halfmod = (mod-1)/2;
1837 cl_MI emod = R->canonhom(the<cl_I>(ex_to<numeric>(e).to_cl_N()));
1838 numeric n(R->retract(emod));
1839 if ( n > halfmod ) {
1850 /** Helps mimicking modular multivariate polynomial arithmetic.
1852 * @param e expression of which to make the coefficients equal to their value
1853 * in the modular ring R (symmetric representation)
1854 * @param R modular ring
1855 * @return resulting expression
1857 static ex make_modular(const ex& e, const cl_modint_ring& R)
1859 make_modular_map map(R);
1860 return map(e.expand());
1863 /** Utility function for multivariate Hensel lifting.
1865 * Returns the polynomials s_i that fulfill
1866 * s_1*b_1 + ... + s_r*b_r == c mod <I^(d+1),p^k>
1867 * with given b_1 = a_1 * ... * a_{i-1} * a_{i+1} * ... * a_r
1869 * The implementation follows the algorithm in chapter 6 of [GCL].
1871 * @param a_ vector of multivariate factors mod p^k
1872 * @param x symbol (equiv. x_1 in [GCL])
1873 * @param c polynomial mod p^k
1874 * @param I vector of evaluation points
1875 * @param d maximum total degree of result
1876 * @param p prime number
1877 * @param k p^k is modulus
1878 * @return vector of polynomials (s_i)
1880 static vector<ex> multivar_diophant(const vector<ex>& a_, const ex& x, const ex& c, const vector<EvalPoint>& I,
1881 unsigned int d, unsigned int p, unsigned int k)
1885 const cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
1886 const size_t r = a.size();
1887 const size_t nu = I.size() + 1;
1891 ex xnu = I.back().x;
1892 int alphanu = I.back().evalpoint;
1895 for ( size_t i=0; i<r; ++i ) {
1899 for ( size_t i=0; i<r; ++i ) {
1900 b[i] = normal(A / a[i]);
1903 vector<ex> anew = a;
1904 for ( size_t i=0; i<r; ++i ) {
1905 anew[i] = anew[i].subs(xnu == alphanu);
1907 ex cnew = c.subs(xnu == alphanu);
1908 vector<EvalPoint> Inew = I;
1910 sigma = multivar_diophant(anew, x, cnew, Inew, d, p, k);
1913 for ( size_t i=0; i<r; ++i ) {
1914 buf -= sigma[i] * b[i];
1916 ex e = make_modular(buf, R);
1919 for ( size_t m=1; !e.is_zero() && e.has(xnu) && m<=d; ++m ) {
1920 monomial *= (xnu - alphanu);
1921 monomial = expand(monomial);
1922 ex cm = e.diff(ex_to<symbol>(xnu), m).subs(xnu==alphanu) / factorial(m);
1923 cm = make_modular(cm, R);
1924 if ( !cm.is_zero() ) {
1925 vector<ex> delta_s = multivar_diophant(anew, x, cm, Inew, d, p, k);
1927 for ( size_t j=0; j<delta_s.size(); ++j ) {
1928 delta_s[j] *= monomial;
1929 sigma[j] += delta_s[j];
1930 buf -= delta_s[j] * b[j];
1932 e = make_modular(buf, R);
1938 for ( size_t i=0; i<a.size(); ++i ) {
1940 umodpoly_from_ex(up, a[i], x, R);
1944 sigma.insert(sigma.begin(), r, 0);
1947 if ( is_a<add>(c) ) {
1955 for ( size_t i=0; i<nterms; ++i ) {
1956 int m = z.degree(x);
1957 cl_I cm = the<cl_I>(ex_to<numeric>(z.lcoeff(x)).to_cl_N());
1958 upvec delta_s = univar_diophant(amod, x, m, p, k);
1961 while ( poscm < 0 ) {
1962 poscm = poscm + expt_pos(cl_I(p),k);
1964 modcm = cl_MI(R, poscm);
1965 for ( size_t j=0; j<delta_s.size(); ++j ) {
1966 delta_s[j] = delta_s[j] * modcm;
1967 sigma[j] = sigma[j] + umodpoly_to_ex(delta_s[j], x);
1975 for ( size_t i=0; i<sigma.size(); ++i ) {
1976 sigma[i] = make_modular(sigma[i], R);
1982 /** Multivariate Hensel lifting.
1983 * The implementation follows the algorithm in chapter 6 of [GCL].
1984 * Since we don't have a data type for modular multivariate polynomials, the
1985 * respective operations are done in a GiNaC::ex and the function
1986 * make_modular() is then called to make the coefficient modular p^l.
1988 * @param a multivariate polynomial primitive in x
1989 * @param x symbol (equiv. x_1 in [GCL])
1990 * @param I vector of evaluation points (x_2==a_2,x_3==a_3,...)
1991 * @param p prime number (should not divide lcoeff(a mod I))
1992 * @param l p^l is the modulus of the lifted univariate field
1993 * @param u vector of modular (mod p^l) factors of a mod I
1994 * @param lcU correct leading coefficient of the univariate factors of a mod I
1995 * @return list GiNaC::lst with lifted factors (multivariate factors of a),
1996 * empty if Hensel lifting did not succeed
1998 static ex hensel_multivar(const ex& a, const ex& x, const vector<EvalPoint>& I,
1999 unsigned int p, const cl_I& l, const upvec& u, const vector<ex>& lcU)
2001 const size_t nu = I.size() + 1;
2002 const cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),l));
2007 for ( size_t j=nu; j>=2; --j ) {
2009 int alpha = I[j-2].evalpoint;
2010 A[j-2] = A[j-1].subs(x==alpha);
2011 A[j-2] = make_modular(A[j-2], R);
2014 int maxdeg = a.degree(I.front().x);
2015 for ( size_t i=1; i<I.size(); ++i ) {
2016 int maxdeg2 = a.degree(I[i].x);
2017 if ( maxdeg2 > maxdeg ) maxdeg = maxdeg2;
2020 const size_t n = u.size();
2022 for ( size_t i=0; i<n; ++i ) {
2023 U[i] = umodpoly_to_ex(u[i], x);
2026 for ( size_t j=2; j<=nu; ++j ) {
2029 for ( size_t m=0; m<n; ++m) {
2030 if ( lcU[m] != 1 ) {
2032 for ( size_t i=j-1; i<nu-1; ++i ) {
2033 coef = coef.subs(I[i].x == I[i].evalpoint);
2035 coef = make_modular(coef, R);
2036 int deg = U[m].degree(x);
2037 U[m] = U[m] - U[m].lcoeff(x) * pow(x,deg) + coef * pow(x,deg);
2041 for ( size_t i=0; i<n; ++i ) {
2044 ex e = expand(A[j-1] - Uprod);
2046 vector<EvalPoint> newI;
2047 for ( size_t i=1; i<=j-2; ++i ) {
2048 newI.push_back(I[i-1]);
2052 int alphaj = I[j-2].evalpoint;
2053 size_t deg = A[j-1].degree(xj);
2054 for ( size_t k=1; k<=deg; ++k ) {
2055 if ( !e.is_zero() ) {
2056 monomial *= (xj - alphaj);
2057 monomial = expand(monomial);
2058 ex dif = e.diff(ex_to<symbol>(xj), k);
2059 ex c = dif.subs(xj==alphaj) / factorial(k);
2060 if ( !c.is_zero() ) {
2061 vector<ex> deltaU = multivar_diophant(U1, x, c, newI, maxdeg, p, cl_I_to_uint(l));
2062 for ( size_t i=0; i<n; ++i ) {
2063 deltaU[i] *= monomial;
2065 U[i] = make_modular(U[i], R);
2068 for ( size_t i=0; i<n; ++i ) {
2072 e = make_modular(e, R);
2079 for ( size_t i=0; i<U.size(); ++i ) {
2082 if ( expand(a-acand).is_zero() ) {
2084 for ( size_t i=0; i<U.size(); ++i ) {
2095 /** Takes a factorized expression and puts the factors in a lst. The exponents
2096 * of the factors are discarded, e.g. 7*x^2*(y+1)^4 --> {7,x,y+1}. The first
2097 * element of the list is always the numeric coefficient.
2099 static ex put_factors_into_lst(const ex& e)
2102 if ( is_a<numeric>(e) ) {
2106 if ( is_a<power>(e) ) {
2108 result.append(e.op(0));
2111 if ( is_a<symbol>(e) || is_a<add>(e) ) {
2116 if ( is_a<mul>(e) ) {
2118 for ( size_t i=0; i<e.nops(); ++i ) {
2120 if ( is_a<numeric>(op) ) {
2123 if ( is_a<power>(op) ) {
2124 result.append(op.op(0));
2126 if ( is_a<symbol>(op) || is_a<add>(op) ) {
2130 result.prepend(nfac);
2133 throw runtime_error("put_factors_into_lst: bad term.");
2136 /** Checks a set of numbers for whether each number has a unique prime factor.
2138 * @param[in] f list of numbers to check
2139 * @return true: if number set is bad, false: if set is okay (has unique
2142 static bool checkdivisors(const lst& f)
2144 const int k = f.nops();
2146 vector<numeric> d(k);
2147 d[0] = ex_to<numeric>(abs(f.op(0)));
2148 for ( int i=1; i<k; ++i ) {
2149 q = ex_to<numeric>(abs(f.op(i)));
2150 for ( int j=i-1; j>=0; --j ) {
2165 /** Generates a set of evaluation points for a multivariate polynomial.
2166 * The set fulfills the following conditions:
2167 * 1. lcoeff(evaluated_polynomial) does not vanish
2168 * 2. factors of lcoeff(evaluated_polynomial) have each a unique prime factor
2169 * 3. evaluated_polynomial is square free
2170 * See [Wan] for more details.
2172 * @param[in] u multivariate polynomial to be factored
2173 * @param[in] vn leading coefficient of u in x (x==first symbol in syms)
2174 * @param[in] syms set of symbols that appear in u
2175 * @param[in] f lst containing the factors of the leading coefficient vn
2176 * @param[in,out] modulus integer modulus for random number generation (i.e. |a_i| < modulus)
2177 * @param[out] u0 returns the evaluated (univariate) polynomial
2178 * @param[out] a returns the valid evaluation points. must have initial size equal
2179 * number of symbols-1 before calling generate_set
2181 static void generate_set(const ex& u, const ex& vn, const exset& syms, const lst& f,
2182 numeric& modulus, ex& u0, vector<numeric>& a)
2184 const ex& x = *syms.begin();
2187 // generate a set of integers ...
2191 exset::const_iterator s = syms.begin();
2193 for ( size_t i=0; i<a.size(); ++i ) {
2195 a[i] = mod(numeric(rand()), 2*modulus) - modulus;
2196 vnatry = vna.subs(*s == a[i]);
2197 // ... for which the leading coefficient doesn't vanish ...
2198 } while ( vnatry == 0 );
2200 u0 = u0.subs(*s == a[i]);
2203 // ... for which u0 is square free ...
2204 ex g = gcd(u0, u0.diff(ex_to<symbol>(x)));
2205 if ( !is_a<numeric>(g) ) {
2208 if ( !is_a<numeric>(vn) ) {
2209 // ... and for which the evaluated factors have each an unique prime factor
2211 fnum.let_op(0) = fnum.op(0) * u0.content(x);
2212 for ( size_t i=1; i<fnum.nops(); ++i ) {
2213 if ( !is_a<numeric>(fnum.op(i)) ) {
2216 for ( size_t j=0; j<a.size(); ++j, ++s ) {
2217 fnum.let_op(i) = fnum.op(i).subs(*s == a[j]);
2221 if ( checkdivisors(fnum) ) {
2225 // ok, we have a valid set now
2230 // forward declaration
2231 static ex factor_sqrfree(const ex& poly);
2233 /** Multivariate factorization.
2235 * The implementation is based on the algorithm described in [Wan].
2236 * An evaluation homomorphism (a set of integers) is determined that fulfills
2237 * certain criteria. The evaluated polynomial is univariate and is factorized
2238 * by factor_univariate(). The main work then is to find the correct leading
2239 * coefficients of the univariate factors. They have to correspond to the
2240 * factors of the (multivariate) leading coefficient of the input polynomial
2241 * (as defined for a specific variable x). After that the Hensel lifting can be
2244 * @param[in] poly expanded, square free polynomial
2245 * @param[in] syms contains the symbols in the polynomial
2246 * @return factorized polynomial
2248 static ex factor_multivariate(const ex& poly, const exset& syms)
2250 exset::const_iterator s;
2251 const ex& x = *syms.begin();
2253 // make polynomial primitive
2255 poly.unitcontprim(x, unit, cont, pp);
2256 if ( !is_a<numeric>(cont) ) {
2257 return factor_sqrfree(cont) * factor_sqrfree(pp);
2260 // factor leading coefficient
2261 ex vn = pp.collect(x).lcoeff(x);
2263 if ( is_a<numeric>(vn) ) {
2267 ex vnfactors = factor(vn);
2268 vnlst = put_factors_into_lst(vnfactors);
2271 const unsigned int maxtrials = 3;
2272 numeric modulus = (vnlst.nops() > 3) ? vnlst.nops() : 3;
2273 vector<numeric> a(syms.size()-1, 0);
2275 // try now to factorize until we are successful
2278 unsigned int trialcount = 0;
2280 int factor_count = 0;
2281 int min_factor_count = -1;
2285 // try several evaluation points to reduce the number of factors
2286 while ( trialcount < maxtrials ) {
2288 // generate a set of valid evaluation points
2289 generate_set(pp, vn, syms, ex_to<lst>(vnlst), modulus, u, a);
2291 ufac = factor_univariate(u, x, prime);
2292 ufaclst = put_factors_into_lst(ufac);
2293 factor_count = ufaclst.nops()-1;
2294 delta = ufaclst.op(0);
2296 if ( factor_count <= 1 ) {
2300 if ( min_factor_count < 0 ) {
2302 min_factor_count = factor_count;
2304 else if ( min_factor_count == factor_count ) {
2308 else if ( min_factor_count > factor_count ) {
2309 // new minimum, reset trial counter
2310 min_factor_count = factor_count;
2315 // determine true leading coefficients for the Hensel lifting
2316 vector<ex> C(factor_count);
2317 if ( is_a<numeric>(vn) ) {
2319 for ( size_t i=1; i<ufaclst.nops(); ++i ) {
2320 C[i-1] = ufaclst.op(i).lcoeff(x);
2325 // we use the property of the ftilde having a unique prime factor.
2326 // details can be found in [Wan].
2328 vector<numeric> ftilde(vnlst.nops()-1);
2329 for ( size_t i=0; i<ftilde.size(); ++i ) {
2330 ex ft = vnlst.op(i+1);
2333 for ( size_t j=0; j<a.size(); ++j ) {
2334 ft = ft.subs(*s == a[j]);
2337 ftilde[i] = ex_to<numeric>(ft);
2339 // calculate D and C
2340 vector<bool> used_flag(ftilde.size(), false);
2341 vector<ex> D(factor_count, 1);
2343 for ( int i=0; i<factor_count; ++i ) {
2344 numeric prefac = ex_to<numeric>(ufaclst.op(i+1).lcoeff(x));
2345 for ( int j=ftilde.size()-1; j>=0; --j ) {
2347 while ( irem(prefac, ftilde[j]) == 0 ) {
2348 prefac = iquo(prefac, ftilde[j]);
2352 used_flag[j] = true;
2353 D[i] = D[i] * pow(vnlst.op(j+1), count);
2356 C[i] = D[i] * prefac;
2360 for ( int i=0; i<factor_count; ++i ) {
2361 numeric prefac = ex_to<numeric>(ufaclst.op(i+1).lcoeff(x));
2362 for ( int j=ftilde.size()-1; j>=0; --j ) {
2364 while ( irem(prefac, ftilde[j]) == 0 ) {
2365 prefac = iquo(prefac, ftilde[j]);
2368 while ( irem(ex_to<numeric>(delta)*prefac, ftilde[j]) == 0 ) {
2369 numeric g = gcd(prefac, ex_to<numeric>(ftilde[j]));
2370 prefac = iquo(prefac, g);
2371 delta = delta / (ftilde[j]/g);
2372 ufaclst.let_op(i+1) = ufaclst.op(i+1) * (ftilde[j]/g);
2376 used_flag[j] = true;
2377 D[i] = D[i] * pow(vnlst.op(j+1), count);
2380 C[i] = D[i] * prefac;
2383 // check if something went wrong
2384 bool some_factor_unused = false;
2385 for ( size_t i=0; i<used_flag.size(); ++i ) {
2386 if ( !used_flag[i] ) {
2387 some_factor_unused = true;
2391 if ( some_factor_unused ) {
2396 // multiply the remaining content of the univariate polynomial into the
2399 C[0] = C[0] * delta;
2400 ufaclst.let_op(1) = ufaclst.op(1) * delta;
2403 // set up evaluation points
2405 vector<EvalPoint> epv;
2408 for ( size_t i=0; i<a.size(); ++i ) {
2410 ep.evalpoint = a[i].to_int();
2416 for ( int i=1; i<=factor_count; ++i ) {
2417 if ( ufaclst.op(i).degree(x) > maxdeg ) {
2418 maxdeg = ufaclst[i].degree(x);
2421 cl_I B = 2*calc_bound(u, x, maxdeg);
2429 // set up modular factors (mod p^l)
2430 cl_modint_ring R = find_modint_ring(expt_pos(cl_I(prime),l));
2431 upvec modfactors(ufaclst.nops()-1);
2432 for ( size_t i=1; i<ufaclst.nops(); ++i ) {
2433 umodpoly_from_ex(modfactors[i-1], ufaclst.op(i), x, R);
2436 // try Hensel lifting
2437 ex res = hensel_multivar(pp, x, epv, prime, l, modfactors, C);
2438 if ( res != lst() ) {
2439 ex result = cont * unit;
2440 for ( size_t i=0; i<res.nops(); ++i ) {
2441 result *= res.op(i).content(x) * res.op(i).unit(x);
2442 result *= res.op(i).primpart(x);
2449 /** Finds all symbols in an expression. Used by factor_sqrfree() and factor().
2451 struct find_symbols_map : public map_function {
2453 ex operator()(const ex& e)
2455 if ( is_a<symbol>(e) ) {
2459 return e.map(*this);
2463 /** Factorizes a polynomial that is square free. It calls either the univariate
2464 * or the multivariate factorization functions.
2466 static ex factor_sqrfree(const ex& poly)
2468 // determine all symbols in poly
2469 find_symbols_map findsymbols;
2471 if ( findsymbols.syms.size() == 0 ) {
2475 if ( findsymbols.syms.size() == 1 ) {
2477 const ex& x = *(findsymbols.syms.begin());
2478 if ( poly.ldegree(x) > 0 ) {
2479 // pull out direct factors
2480 int ld = poly.ldegree(x);
2481 ex res = factor_univariate(expand(poly/pow(x, ld)), x);
2482 return res * pow(x,ld);
2485 ex res = factor_univariate(poly, x);
2490 // multivariate case
2491 ex res = factor_multivariate(poly, findsymbols.syms);
2495 /** Map used by factor() when factor_options::all is given to access all
2496 * subexpressions and to call factor() on them.
2498 struct apply_factor_map : public map_function {
2500 apply_factor_map(unsigned options_) : options(options_) { }
2501 ex operator()(const ex& e)
2503 if ( e.info(info_flags::polynomial) ) {
2504 return factor(e, options);
2506 if ( is_a<add>(e) ) {
2508 for ( size_t i=0; i<e.nops(); ++i ) {
2509 if ( e.op(i).info(info_flags::polynomial) ) {
2518 return factor(s1, options) + s2.map(*this);
2520 return e.map(*this);
2524 } // anonymous namespace
2526 /** Interface function to the outside world. It checks the arguments, tries a
2527 * square free factorization, and then calls factor_sqrfree to do the hard
2530 ex factor(const ex& poly, unsigned options)
2533 if ( !poly.info(info_flags::polynomial) ) {
2534 if ( options & factor_options::all ) {
2535 options &= ~factor_options::all;
2536 apply_factor_map factor_map(options);
2537 return factor_map(poly);
2542 // determine all symbols in poly
2543 find_symbols_map findsymbols;
2545 if ( findsymbols.syms.size() == 0 ) {
2549 exset::const_iterator i=findsymbols.syms.begin(), end=findsymbols.syms.end();
2550 for ( ; i!=end; ++i ) {
2554 // make poly square free
2555 ex sfpoly = sqrfree(poly.expand(), syms);
2557 // factorize the square free components
2558 if ( is_a<power>(sfpoly) ) {
2559 // case: (polynomial)^exponent
2560 const ex& base = sfpoly.op(0);
2561 if ( !is_a<add>(base) ) {
2562 // simple case: (monomial)^exponent
2565 ex f = factor_sqrfree(base);
2566 return pow(f, sfpoly.op(1));
2568 if ( is_a<mul>(sfpoly) ) {
2569 // case: multiple factors
2571 for ( size_t i=0; i<sfpoly.nops(); ++i ) {
2572 const ex& t = sfpoly.op(i);
2573 if ( is_a<power>(t) ) {
2574 const ex& base = t.op(0);
2575 if ( !is_a<add>(base) ) {
2579 ex f = factor_sqrfree(base);
2580 res *= pow(f, t.op(1));
2583 else if ( is_a<add>(t) ) {
2584 ex f = factor_sqrfree(t);
2593 if ( is_a<symbol>(sfpoly) ) {
2596 // case: (polynomial)
2597 ex f = factor_sqrfree(sfpoly);
2601 } // namespace GiNaC