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) 1999-2017 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"
85 #define DCOUT(str) cout << #str << endl
86 #define DCOUTVAR(var) cout << #var << ": " << var << endl
87 #define DCOUT2(str,var) cout << #str << ": " << var << endl
88 ostream& operator<<(ostream& o, const vector<int>& v)
90 auto i = v.begin(), end = v.end();
97 static ostream& operator<<(ostream& o, const vector<cl_I>& v)
99 auto i = v.begin(), end = v.end();
101 o << *i << "[" << i-v.begin() << "]" << " ";
106 static ostream& operator<<(ostream& o, const vector<cl_MI>& v)
108 auto i = v.begin(), end = v.end();
110 o << *i << "[" << i-v.begin() << "]" << " ";
115 ostream& operator<<(ostream& o, const vector<numeric>& v)
117 for ( size_t i=0; i<v.size(); ++i ) {
122 ostream& operator<<(ostream& o, const vector<vector<cl_MI>>& v)
124 auto i = v.begin(), end = v.end();
126 o << i-v.begin() << ": " << *i << endl;
133 #define DCOUTVAR(var)
134 #define DCOUT2(str,var)
135 #endif // def DEBUGFACTOR
137 // anonymous namespace to hide all utility functions
140 ////////////////////////////////////////////////////////////////////////////////
141 // modular univariate polynomial code
143 typedef std::vector<cln::cl_MI> umodpoly;
144 typedef std::vector<cln::cl_I> upoly;
145 typedef vector<umodpoly> upvec;
147 // COPY FROM UPOLY.HPP
149 // CHANGED size_t -> int !!!
150 template<typename T> static int degree(const T& p)
155 template<typename T> static typename T::value_type lcoeff(const T& p)
157 return p[p.size() - 1];
160 static bool normalize_in_field(umodpoly& a)
164 if ( lcoeff(a) == a[0].ring()->one() ) {
168 const cln::cl_MI lc_1 = recip(lcoeff(a));
169 for (std::size_t k = a.size(); k-- != 0; )
174 template<typename T> static void
175 canonicalize(T& p, const typename T::size_type hint = std::numeric_limits<typename T::size_type>::max())
180 std::size_t i = p.size() - 1;
181 // Be fast if the polynomial is already canonicalized
188 bool is_zero = false;
206 p.erase(p.begin() + i, p.end());
209 // END COPY FROM UPOLY.HPP
211 static void expt_pos(umodpoly& a, unsigned int q)
213 if ( a.empty() ) return;
214 cl_MI zero = a[0].ring()->zero();
216 a.resize(degree(a)*q+1, zero);
217 for ( int i=deg; i>0; --i ) {
223 template<bool COND, typename T = void> struct enable_if
228 template<typename T> struct enable_if<false, T> { /* empty */ };
230 template<typename T> struct uvar_poly_p
232 static const bool value = false;
235 template<> struct uvar_poly_p<upoly>
237 static const bool value = true;
240 template<> struct uvar_poly_p<umodpoly>
242 static const bool value = true;
246 // Don't define this for anything but univariate polynomials.
247 static typename enable_if<uvar_poly_p<T>::value, T>::type
248 operator+(const T& a, const T& b)
255 for ( ; i<sb; ++i ) {
258 for ( ; i<sa; ++i ) {
267 for ( ; i<sa; ++i ) {
270 for ( ; i<sb; ++i ) {
279 // Don't define this for anything but univariate polynomials. Otherwise
280 // overload resolution might fail (this actually happens when compiling
281 // GiNaC with g++ 3.4).
282 static typename enable_if<uvar_poly_p<T>::value, T>::type
283 operator-(const T& a, const T& b)
290 for ( ; i<sb; ++i ) {
293 for ( ; i<sa; ++i ) {
302 for ( ; i<sa; ++i ) {
305 for ( ; i<sb; ++i ) {
313 static upoly operator*(const upoly& a, const upoly& b)
316 if ( a.empty() || b.empty() ) return c;
318 int n = degree(a) + degree(b);
320 for ( int i=0 ; i<=n; ++i ) {
321 for ( int j=0 ; j<=i; ++j ) {
322 if ( j > degree(a) || (i-j) > degree(b) ) continue;
323 c[i] = c[i] + a[j] * b[i-j];
330 static umodpoly operator*(const umodpoly& a, const umodpoly& b)
333 if ( a.empty() || b.empty() ) return c;
335 int n = degree(a) + degree(b);
336 c.resize(n+1, a[0].ring()->zero());
337 for ( int i=0 ; i<=n; ++i ) {
338 for ( int j=0 ; j<=i; ++j ) {
339 if ( j > degree(a) || (i-j) > degree(b) ) continue;
340 c[i] = c[i] + a[j] * b[i-j];
347 static upoly operator*(const upoly& a, const cl_I& x)
354 for ( size_t i=0; i<a.size(); ++i ) {
360 static upoly operator/(const upoly& a, const cl_I& x)
367 for ( size_t i=0; i<a.size(); ++i ) {
368 r[i] = exquo(a[i],x);
373 static umodpoly operator*(const umodpoly& a, const cl_MI& x)
375 umodpoly r(a.size());
376 for ( size_t i=0; i<a.size(); ++i ) {
383 static void upoly_from_ex(upoly& up, const ex& e, const ex& x)
385 // assert: e is in Z[x]
386 int deg = e.degree(x);
388 int ldeg = e.ldegree(x);
389 for ( ; deg>=ldeg; --deg ) {
390 up[deg] = the<cl_I>(ex_to<numeric>(e.coeff(x, deg)).to_cl_N());
392 for ( ; deg>=0; --deg ) {
398 static void umodpoly_from_upoly(umodpoly& ump, const upoly& e, const cl_modint_ring& R)
402 for ( ; deg>=0; --deg ) {
403 ump[deg] = R->canonhom(e[deg]);
408 static void umodpoly_from_ex(umodpoly& ump, const ex& e, const ex& x, const cl_modint_ring& R)
410 // assert: e is in Z[x]
411 int deg = e.degree(x);
413 int ldeg = e.ldegree(x);
414 for ( ; deg>=ldeg; --deg ) {
415 cl_I coeff = the<cl_I>(ex_to<numeric>(e.coeff(x, deg)).to_cl_N());
416 ump[deg] = R->canonhom(coeff);
418 for ( ; deg>=0; --deg ) {
419 ump[deg] = R->zero();
425 static void umodpoly_from_ex(umodpoly& ump, const ex& e, const ex& x, const cl_I& modulus)
427 umodpoly_from_ex(ump, e, x, find_modint_ring(modulus));
431 static ex upoly_to_ex(const upoly& a, const ex& x)
433 if ( a.empty() ) return 0;
435 for ( int i=degree(a); i>=0; --i ) {
436 e += numeric(a[i]) * pow(x, i);
441 static ex umodpoly_to_ex(const umodpoly& a, const ex& x)
443 if ( a.empty() ) return 0;
444 cl_modint_ring R = a[0].ring();
445 cl_I mod = R->modulus;
446 cl_I halfmod = (mod-1) >> 1;
448 for ( int i=degree(a); i>=0; --i ) {
449 cl_I n = R->retract(a[i]);
451 e += numeric(n-mod) * pow(x, i);
453 e += numeric(n) * pow(x, i);
459 static upoly umodpoly_to_upoly(const umodpoly& a)
462 if ( a.empty() ) return e;
463 cl_modint_ring R = a[0].ring();
464 cl_I mod = R->modulus;
465 cl_I halfmod = (mod-1) >> 1;
466 for ( int i=degree(a); i>=0; --i ) {
467 cl_I n = R->retract(a[i]);
477 static umodpoly umodpoly_to_umodpoly(const umodpoly& a, const cl_modint_ring& R, unsigned int m)
480 if ( a.empty() ) return e;
481 cl_modint_ring oldR = a[0].ring();
482 size_t sa = a.size();
483 e.resize(sa+m, R->zero());
484 for ( size_t i=0; i<sa; ++i ) {
485 e[i+m] = R->canonhom(oldR->retract(a[i]));
491 /** Divides all coefficients of the polynomial a by the integer x.
492 * All coefficients are supposed to be divisible by x. If they are not, the
493 * the<cl_I> cast will raise an exception.
495 * @param[in,out] a polynomial of which the coefficients will be reduced by x
496 * @param[in] x integer that divides the coefficients
498 static void reduce_coeff(umodpoly& a, const cl_I& x)
500 if ( a.empty() ) return;
502 cl_modint_ring R = a[0].ring();
504 // cln cannot perform this division in the modular field
505 cl_I c = R->retract(i);
506 i = cl_MI(R, the<cl_I>(c / x));
510 /** Calculates remainder of a/b.
511 * Assertion: a and b not empty.
513 * @param[in] a polynomial dividend
514 * @param[in] b polynomial divisor
515 * @param[out] r polynomial remainder
517 static void rem(const umodpoly& a, const umodpoly& b, umodpoly& r)
526 cl_MI qk = div(r[n+k], b[n]);
528 for ( int i=0; i<n; ++i ) {
529 unsigned int j = n + k - 1 - i;
530 r[j] = r[j] - qk * b[j-k];
535 fill(r.begin()+n, r.end(), a[0].ring()->zero());
539 /** Calculates quotient of a/b.
540 * Assertion: a and b not empty.
542 * @param[in] a polynomial dividend
543 * @param[in] b polynomial divisor
544 * @param[out] q polynomial quotient
546 static void div(const umodpoly& a, const umodpoly& b, umodpoly& q)
555 q.resize(k+1, a[0].ring()->zero());
557 cl_MI qk = div(r[n+k], b[n]);
560 for ( int i=0; i<n; ++i ) {
561 unsigned int j = n + k - 1 - i;
562 r[j] = r[j] - qk * b[j-k];
570 /** Calculates quotient and remainder of a/b.
571 * Assertion: a and b not empty.
573 * @param[in] a polynomial dividend
574 * @param[in] b polynomial divisor
575 * @param[out] r polynomial remainder
576 * @param[out] q polynomial quotient
578 static void remdiv(const umodpoly& a, const umodpoly& b, umodpoly& r, umodpoly& q)
587 q.resize(k+1, a[0].ring()->zero());
589 cl_MI qk = div(r[n+k], b[n]);
592 for ( int i=0; i<n; ++i ) {
593 unsigned int j = n + k - 1 - i;
594 r[j] = r[j] - qk * b[j-k];
599 fill(r.begin()+n, r.end(), a[0].ring()->zero());
604 /** Calculates the GCD of polynomial a and b.
606 * @param[in] a polynomial
607 * @param[in] b polynomial
610 static void gcd(const umodpoly& a, const umodpoly& b, umodpoly& c)
612 if ( degree(a) < degree(b) ) return gcd(b, a, c);
615 normalize_in_field(c);
617 normalize_in_field(d);
619 while ( !d.empty() ) {
624 normalize_in_field(c);
627 /** Calculates the derivative of the polynomial a.
629 * @param[in] a polynomial of which to take the derivative
630 * @param[out] d result/derivative
632 static void deriv(const umodpoly& a, umodpoly& d)
635 if ( a.size() <= 1 ) return;
637 d.insert(d.begin(), a.begin()+1, a.end());
639 for ( int i=1; i<max; ++i ) {
645 static bool unequal_one(const umodpoly& a)
647 if ( a.empty() ) return true;
648 return ( a.size() != 1 || a[0] != a[0].ring()->one() );
651 static bool equal_one(const umodpoly& a)
653 return ( a.size() == 1 && a[0] == a[0].ring()->one() );
656 /** Returns true if polynomial a is square free.
658 * @param[in] a polynomial to check
659 * @return true if polynomial is square free, false otherwise
661 static bool squarefree(const umodpoly& a)
673 // END modular univariate polynomial code
674 ////////////////////////////////////////////////////////////////////////////////
676 ////////////////////////////////////////////////////////////////////////////////
679 typedef vector<cl_MI> mvec;
684 friend ostream& operator<<(ostream& o, const modular_matrix& m);
687 modular_matrix(size_t r_, size_t c_, const cl_MI& init) : r(r_), c(c_)
691 size_t rowsize() const { return r; }
692 size_t colsize() const { return c; }
693 cl_MI& operator()(size_t row, size_t col) { return m[row*c + col]; }
694 cl_MI operator()(size_t row, size_t col) const { return m[row*c + col]; }
695 void mul_col(size_t col, const cl_MI x)
697 for ( size_t rc=0; rc<r; ++rc ) {
698 std::size_t i = c*rc + col;
702 void sub_col(size_t col1, size_t col2, const cl_MI fac)
704 for ( size_t rc=0; rc<r; ++rc ) {
705 std::size_t i1 = col1 + c*rc;
706 std::size_t i2 = col2 + c*rc;
707 m[i1] = m[i1] - m[i2]*fac;
710 void switch_col(size_t col1, size_t col2)
712 for ( size_t rc=0; rc<r; ++rc ) {
713 std::size_t i1 = col1 + rc*c;
714 std::size_t i2 = col2 + rc*c;
715 std::swap(m[i1], m[i2]);
718 void mul_row(size_t row, const cl_MI x)
720 for ( size_t cc=0; cc<c; ++cc ) {
721 std::size_t i = row*c + cc;
725 void sub_row(size_t row1, size_t row2, const cl_MI fac)
727 for ( size_t cc=0; cc<c; ++cc ) {
728 std::size_t i1 = row1*c + cc;
729 std::size_t i2 = row2*c + cc;
730 m[i1] = m[i1] - m[i2]*fac;
733 void switch_row(size_t row1, size_t row2)
735 for ( size_t cc=0; cc<c; ++cc ) {
736 std::size_t i1 = row1*c + cc;
737 std::size_t i2 = row2*c + cc;
738 std::swap(m[i1], m[i2]);
741 bool is_col_zero(size_t col) const
743 for ( size_t rr=0; rr<r; ++rr ) {
744 std::size_t i = col + rr*c;
745 if ( !zerop(m[i]) ) {
751 bool is_row_zero(size_t row) const
753 for ( size_t cc=0; cc<c; ++cc ) {
754 std::size_t i = row*c + cc;
755 if ( !zerop(m[i]) ) {
761 void set_row(size_t row, const vector<cl_MI>& newrow)
763 for (std::size_t i2 = 0; i2 < newrow.size(); ++i2) {
764 std::size_t i1 = row*c + i2;
768 mvec::const_iterator row_begin(size_t row) const { return m.begin()+row*c; }
769 mvec::const_iterator row_end(size_t row) const { return m.begin()+row*c+r; }
776 modular_matrix operator*(const modular_matrix& m1, const modular_matrix& m2)
778 const unsigned int r = m1.rowsize();
779 const unsigned int c = m2.colsize();
780 modular_matrix o(r,c,m1(0,0));
782 for ( size_t i=0; i<r; ++i ) {
783 for ( size_t j=0; j<c; ++j ) {
785 buf = m1(i,0) * m2(0,j);
786 for ( size_t k=1; k<c; ++k ) {
787 buf = buf + m1(i,k)*m2(k,j);
795 ostream& operator<<(ostream& o, const modular_matrix& m)
797 cl_modint_ring R = m(0,0).ring();
799 for ( size_t i=0; i<m.rowsize(); ++i ) {
801 for ( size_t j=0; j<m.colsize()-1; ++j ) {
802 o << R->retract(m(i,j)) << ",";
804 o << R->retract(m(i,m.colsize()-1)) << "}";
805 if ( i != m.rowsize()-1 ) {
812 #endif // def DEBUGFACTOR
814 // END modular matrix
815 ////////////////////////////////////////////////////////////////////////////////
817 /** Calculates the Q matrix for a polynomial. Used by Berlekamp's algorithm.
819 * @param[in] a_ modular polynomial
820 * @param[out] Q Q matrix
822 static void q_matrix(const umodpoly& a_, modular_matrix& Q)
825 normalize_in_field(a);
828 unsigned int q = cl_I_to_uint(a[0].ring()->modulus);
829 umodpoly r(n, a[0].ring()->zero());
830 r[0] = a[0].ring()->one();
832 unsigned int max = (n-1) * q;
833 for ( size_t m=1; m<=max; ++m ) {
834 cl_MI rn_1 = r.back();
835 for ( size_t i=n-1; i>0; --i ) {
836 r[i] = r[i-1] - (rn_1 * a[i]);
839 if ( (m % q) == 0 ) {
845 /** Determine the nullspace of a matrix M-1.
847 * @param[in,out] M matrix, will be modified
848 * @param[out] basis calculated nullspace of M-1
850 static void nullspace(modular_matrix& M, vector<mvec>& basis)
852 const size_t n = M.rowsize();
853 const cl_MI one = M(0,0).ring()->one();
854 for ( size_t i=0; i<n; ++i ) {
855 M(i,i) = M(i,i) - one;
857 for ( size_t r=0; r<n; ++r ) {
859 for ( ; cc<n; ++cc ) {
860 if ( !zerop(M(r,cc)) ) {
862 if ( !zerop(M(cc,cc)) ) {
874 M.mul_col(r, recip(M(r,r)));
875 for ( cc=0; cc<n; ++cc ) {
877 M.sub_col(cc, r, M(r,cc));
883 for ( size_t i=0; i<n; ++i ) {
884 M(i,i) = M(i,i) - one;
886 for ( size_t i=0; i<n; ++i ) {
887 if ( !M.is_row_zero(i) ) {
888 mvec nu(M.row_begin(i), M.row_end(i));
894 /** Berlekamp's modular factorization.
896 * The implementation follows the algorithm in chapter 8 of [GCL].
898 * @param[in] a modular polynomial
899 * @param[out] upv vector containing modular factors. if upv was not empty the
900 * new elements are added at the end
902 static void berlekamp(const umodpoly& a, upvec& upv)
904 cl_modint_ring R = a[0].ring();
905 umodpoly one(1, R->one());
907 // find nullspace of Q matrix
908 modular_matrix Q(degree(a), degree(a), R->zero());
913 const unsigned int k = nu.size();
919 list<umodpoly> factors = {a};
920 unsigned int size = 1;
922 unsigned int q = cl_I_to_uint(R->modulus);
924 list<umodpoly>::iterator u = factors.begin();
926 // calculate all gcd's
928 for ( unsigned int s=0; s<q; ++s ) {
929 umodpoly nur = nu[r];
930 nur[0] = nur[0] - cl_MI(R, s);
934 if ( unequal_one(g) && g != *u ) {
937 if ( equal_one(uo) ) {
938 throw logic_error("berlekamp: unexpected divisor.");
942 factors.push_back(g);
944 for (auto & i : factors) {
949 for (auto & i : factors) {
963 // modular square free factorization is not used at the moment so we deactivate
967 /** Calculates a^(1/prime).
969 * @param[in] a polynomial
970 * @param[in] prime prime number -> exponent 1/prime
971 * @param[in] ap resulting polynomial
973 static void expt_1_over_p(const umodpoly& a, unsigned int prime, umodpoly& ap)
975 size_t newdeg = degree(a)/prime;
978 for ( size_t i=1; i<=newdeg; ++i ) {
983 /** Modular square free factorization.
985 * @param[in] a polynomial
986 * @param[out] factors modular factors
987 * @param[out] mult corresponding multiplicities (exponents)
989 static void modsqrfree(const umodpoly& a, upvec& factors, vector<int>& mult)
991 const unsigned int prime = cl_I_to_uint(a[0].ring()->modulus);
1000 while ( unequal_one(w) ) {
1005 factors.push_back(z);
1013 if ( unequal_one(c) ) {
1015 expt_1_over_p(c, prime, cp);
1016 size_t previ = mult.size();
1017 modsqrfree(cp, factors, mult);
1018 for ( size_t i=previ; i<mult.size(); ++i ) {
1024 expt_1_over_p(a, prime, ap);
1025 size_t previ = mult.size();
1026 modsqrfree(ap, factors, mult);
1027 for ( size_t i=previ; i<mult.size(); ++i ) {
1033 #endif // deactivation of square free factorization
1035 /** Distinct degree factorization (DDF).
1037 * The implementation follows the algorithm in chapter 8 of [GCL].
1039 * @param[in] a_ modular polynomial
1040 * @param[out] degrees vector containing the degrees of the factors of the
1041 * corresponding polynomials in ddfactors.
1042 * @param[out] ddfactors vector containing polynomials which factors have the
1043 * degree given in degrees.
1045 static void distinct_degree_factor(const umodpoly& a_, vector<int>& degrees, upvec& ddfactors)
1049 cl_modint_ring R = a[0].ring();
1050 int q = cl_I_to_int(R->modulus);
1051 int nhalf = degree(a)/2;
1059 while ( i <= nhalf ) {
1064 umodpoly wx = w - x;
1066 if ( unequal_one(buf) ) {
1067 degrees.push_back(i);
1068 ddfactors.push_back(buf);
1070 if ( unequal_one(buf) ) {
1074 nhalf = degree(a)/2;
1080 if ( unequal_one(a) ) {
1081 degrees.push_back(degree(a));
1082 ddfactors.push_back(a);
1086 /** Modular same degree factorization.
1087 * Same degree factorization is a kind of misnomer. It performs distinct degree
1088 * factorization, but instead of using the Cantor-Zassenhaus algorithm it
1089 * (sub-optimally) uses Berlekamp's algorithm for the factors of the same
1092 * @param[in] a modular polynomial
1093 * @param[out] upv vector containing modular factors. if upv was not empty the
1094 * new elements are added at the end
1096 static void same_degree_factor(const umodpoly& a, upvec& upv)
1098 cl_modint_ring R = a[0].ring();
1100 vector<int> degrees;
1102 distinct_degree_factor(a, degrees, ddfactors);
1104 for ( size_t i=0; i<degrees.size(); ++i ) {
1105 if ( degrees[i] == degree(ddfactors[i]) ) {
1106 upv.push_back(ddfactors[i]);
1108 berlekamp(ddfactors[i], upv);
1113 // Yes, we can (choose).
1114 #define USE_SAME_DEGREE_FACTOR
1116 /** Modular univariate factorization.
1118 * In principle, we have two algorithms at our disposal: Berlekamp's algorithm
1119 * and same degree factorization (SDF). SDF seems to be slightly faster in
1120 * almost all cases so it is activated as default.
1122 * @param[in] p modular polynomial
1123 * @param[out] upv vector containing modular factors. if upv was not empty the
1124 * new elements are added at the end
1126 static void factor_modular(const umodpoly& p, upvec& upv)
1128 #ifdef USE_SAME_DEGREE_FACTOR
1129 same_degree_factor(p, upv);
1135 /** Calculates modular polynomials s and t such that a*s+b*t==1.
1136 * Assertion: a and b are relatively prime and not zero.
1138 * @param[in] a polynomial
1139 * @param[in] b polynomial
1140 * @param[out] s polynomial
1141 * @param[out] t polynomial
1143 static void exteuclid(const umodpoly& a, const umodpoly& b, umodpoly& s, umodpoly& t)
1145 if ( degree(a) < degree(b) ) {
1146 exteuclid(b, a, t, s);
1150 umodpoly one(1, a[0].ring()->one());
1151 umodpoly c = a; normalize_in_field(c);
1152 umodpoly d = b; normalize_in_field(d);
1160 umodpoly r = c - q * d;
1161 umodpoly r1 = s - q * d1;
1162 umodpoly r2 = t - q * d2;
1166 if ( r.empty() ) break;
1171 cl_MI fac = recip(lcoeff(a) * lcoeff(c));
1172 for (auto & i : s) {
1176 fac = recip(lcoeff(b) * lcoeff(c));
1177 for (auto & i : t) {
1183 /** Replaces the leading coefficient in a polynomial by a given number.
1185 * @param[in] poly polynomial to change
1186 * @param[in] lc new leading coefficient
1187 * @return changed polynomial
1189 static upoly replace_lc(const upoly& poly, const cl_I& lc)
1191 if ( poly.empty() ) return poly;
1197 /** Calculates the bound for the modulus.
1200 static inline cl_I calc_bound(const ex& a, const ex& x, int maxdeg)
1204 for ( int i=a.degree(x); i>=a.ldegree(x); --i ) {
1205 cl_I aa = abs(the<cl_I>(ex_to<numeric>(a.coeff(x, i)).to_cl_N()));
1206 if ( aa > maxcoeff ) maxcoeff = aa;
1207 coeff = coeff + square(aa);
1209 cl_I coeffnorm = ceiling1(the<cl_R>(cln::sqrt(coeff)));
1210 cl_I B = coeffnorm * expt_pos(cl_I(2), cl_I(maxdeg));
1211 return ( B > maxcoeff ) ? B : maxcoeff;
1214 /** Calculates the bound for the modulus.
1217 static inline cl_I calc_bound(const upoly& a, int maxdeg)
1221 for ( int i=degree(a); i>=0; --i ) {
1222 cl_I aa = abs(a[i]);
1223 if ( aa > maxcoeff ) maxcoeff = aa;
1224 coeff = coeff + square(aa);
1226 cl_I coeffnorm = ceiling1(the<cl_R>(cln::sqrt(coeff)));
1227 cl_I B = coeffnorm * expt_pos(cl_I(2), cl_I(maxdeg));
1228 return ( B > maxcoeff ) ? B : maxcoeff;
1231 /** Hensel lifting as used by factor_univariate().
1233 * The implementation follows the algorithm in chapter 6 of [GCL].
1235 * @param[in] a_ primitive univariate polynomials
1236 * @param[in] p prime number that does not divide lcoeff(a)
1237 * @param[in] u1_ modular factor of a (mod p)
1238 * @param[in] w1_ modular factor of a (mod p), relatively prime to u1_,
1239 * fulfilling u1_*w1_ == a mod p
1240 * @param[out] u lifted factor
1241 * @param[out] w lifted factor, u*w = a
1243 static void hensel_univar(const upoly& a_, unsigned int p, const umodpoly& u1_, const umodpoly& w1_, upoly& u, upoly& w)
1246 const cl_modint_ring& R = u1_[0].ring();
1249 int maxdeg = (degree(u1_) > degree(w1_)) ? degree(u1_) : degree(w1_);
1250 cl_I maxmodulus = 2*calc_bound(a, maxdeg);
1253 cl_I alpha = lcoeff(a);
1256 normalize_in_field(nu1);
1258 normalize_in_field(nw1);
1260 phi = umodpoly_to_upoly(nu1) * alpha;
1262 umodpoly_from_upoly(u1, phi, R);
1263 phi = umodpoly_to_upoly(nw1) * alpha;
1265 umodpoly_from_upoly(w1, phi, R);
1270 exteuclid(u1, w1, s, t);
1273 u = replace_lc(umodpoly_to_upoly(u1), alpha);
1274 w = replace_lc(umodpoly_to_upoly(w1), alpha);
1275 upoly e = a - u * w;
1279 while ( !e.empty() && modulus < maxmodulus ) {
1280 upoly c = e / modulus;
1281 phi = umodpoly_to_upoly(s) * c;
1282 umodpoly sigmatilde;
1283 umodpoly_from_upoly(sigmatilde, phi, R);
1284 phi = umodpoly_to_upoly(t) * c;
1286 umodpoly_from_upoly(tautilde, phi, R);
1288 remdiv(sigmatilde, w1, r, q);
1290 phi = umodpoly_to_upoly(tautilde) + umodpoly_to_upoly(q) * umodpoly_to_upoly(u1);
1292 umodpoly_from_upoly(tau, phi, R);
1293 u = u + umodpoly_to_upoly(tau) * modulus;
1294 w = w + umodpoly_to_upoly(sigma) * modulus;
1296 modulus = modulus * p;
1302 for ( size_t i=1; i<u.size(); ++i ) {
1304 if ( g == 1 ) break;
1318 /** Returns a new prime number.
1320 * @param[in] p prime number
1321 * @return next prime number after p
1323 static unsigned int next_prime(unsigned int p)
1325 static vector<unsigned int> primes;
1326 if (primes.empty()) {
1329 if ( p >= primes.back() ) {
1330 unsigned int candidate = primes.back() + 2;
1332 size_t n = primes.size()/2;
1333 for ( size_t i=0; i<n; ++i ) {
1334 if (candidate % primes[i])
1339 primes.push_back(candidate);
1345 for (auto & it : primes) {
1350 throw logic_error("next_prime: should not reach this point!");
1353 /** Manages the splitting a vector of of modular factors into two partitions.
1355 class factor_partition
1358 /** Takes the vector of modular factors and initializes the first partition */
1359 factor_partition(const upvec& factors_) : factors(factors_)
1365 one.resize(1, factors.front()[0].ring()->one());
1370 int operator[](size_t i) const { return k[i]; }
1371 size_t size() const { return n; }
1372 size_t size_left() const { return n-len; }
1373 size_t size_right() const { return len; }
1374 /** Initializes the next partition.
1375 Returns true, if there is one, false otherwise. */
1378 if ( last == n-1 ) {
1388 while ( k[last] == 0 ) { --last; }
1389 if ( last == 0 && n == 2*len ) return false;
1391 for ( size_t i=0; i<=len-rem; ++i ) {
1395 fill(k.begin()+last, k.end(), 0);
1402 if ( len > n/2 ) return false;
1403 fill(k.begin(), k.begin()+len, 1);
1404 fill(k.begin()+len+1, k.end(), 0);
1412 /** Get first partition */
1413 umodpoly& left() { return lr[0]; }
1414 /** Get second partition */
1415 umodpoly& right() { return lr[1]; }
1424 while ( i < n && k[i] == group ) { ++d; ++i; }
1426 if ( cache[pos].size() >= d ) {
1427 lr[group] = lr[group] * cache[pos][d-1];
1429 if ( cache[pos].size() == 0 ) {
1430 cache[pos].push_back(factors[pos] * factors[pos+1]);
1432 size_t j = pos + cache[pos].size() + 1;
1433 d -= cache[pos].size();
1435 umodpoly buf = cache[pos].back() * factors[j];
1436 cache[pos].push_back(buf);
1440 lr[group] = lr[group] * cache[pos].back();
1443 lr[group] = lr[group] * factors[pos];
1454 for ( size_t i=0; i<n; ++i ) {
1455 lr[k[i]] = lr[k[i]] * factors[i];
1461 vector<vector<umodpoly>> cache;
1470 /** Contains a pair of univariate polynomial and its modular factors.
1471 * Used by factor_univariate().
1479 /** Univariate polynomial factorization.
1481 * Modular factorization is tried for several primes to minimize the number of
1482 * modular factors. Then, Hensel lifting is performed.
1484 * @param[in] poly expanded square free univariate polynomial
1485 * @param[in] x symbol
1486 * @param[in,out] prime prime number to start trying modular factorization with,
1487 * output value is the prime number actually used
1489 static ex factor_univariate(const ex& poly, const ex& x, unsigned int& prime)
1491 ex unit, cont, prim_ex;
1492 poly.unitcontprim(x, unit, cont, prim_ex);
1494 upoly_from_ex(prim, prim_ex, x);
1496 // determine proper prime and minimize number of modular factors
1498 unsigned int lastp = prime;
1500 unsigned int trials = 0;
1501 unsigned int minfactors = 0;
1503 const numeric& cont_n = ex_to<numeric>(cont);
1505 if (cont_n.is_integer()) {
1506 i_cont = the<cl_I>(cont_n.to_cl_N());
1508 // poly \in Q[x] => poly = q ipoly, ipoly \in Z[x], q \in Q
1509 // factor(poly) \equiv q factor(ipoly)
1512 cl_I lc = lcoeff(prim)*i_cont;
1514 while ( trials < 2 ) {
1517 prime = next_prime(prime);
1518 if ( !zerop(rem(lc, prime)) ) {
1519 R = find_modint_ring(prime);
1520 umodpoly_from_upoly(modpoly, prim, R);
1521 if ( squarefree(modpoly) ) break;
1525 // do modular factorization
1527 factor_modular(modpoly, trialfactors);
1528 if ( trialfactors.size() <= 1 ) {
1529 // irreducible for sure
1533 if ( minfactors == 0 || trialfactors.size() < minfactors ) {
1534 factors = trialfactors;
1535 minfactors = trialfactors.size();
1543 R = find_modint_ring(prime);
1545 // lift all factor combinations
1546 stack<ModFactors> tocheck;
1549 mf.factors = factors;
1553 while ( tocheck.size() ) {
1554 const size_t n = tocheck.top().factors.size();
1555 factor_partition part(tocheck.top().factors);
1557 // call Hensel lifting
1558 hensel_univar(tocheck.top().poly, prime, part.left(), part.right(), f1, f2);
1559 if ( !f1.empty() ) {
1560 // successful, update the stack and the result
1561 if ( part.size_left() == 1 ) {
1562 if ( part.size_right() == 1 ) {
1563 result *= upoly_to_ex(f1, x) * upoly_to_ex(f2, x);
1567 result *= upoly_to_ex(f1, x);
1568 tocheck.top().poly = f2;
1569 for ( size_t i=0; i<n; ++i ) {
1570 if ( part[i] == 0 ) {
1571 tocheck.top().factors.erase(tocheck.top().factors.begin()+i);
1577 else if ( part.size_right() == 1 ) {
1578 if ( part.size_left() == 1 ) {
1579 result *= upoly_to_ex(f1, x) * upoly_to_ex(f2, x);
1583 result *= upoly_to_ex(f2, x);
1584 tocheck.top().poly = f1;
1585 for ( size_t i=0; i<n; ++i ) {
1586 if ( part[i] == 1 ) {
1587 tocheck.top().factors.erase(tocheck.top().factors.begin()+i);
1593 upvec newfactors1(part.size_left()), newfactors2(part.size_right());
1594 auto i1 = newfactors1.begin(), i2 = newfactors2.begin();
1595 for ( size_t i=0; i<n; ++i ) {
1597 *i2++ = tocheck.top().factors[i];
1599 *i1++ = tocheck.top().factors[i];
1602 tocheck.top().factors = newfactors1;
1603 tocheck.top().poly = f1;
1605 mf.factors = newfactors2;
1612 if ( !part.next() ) {
1613 // if no more combinations left, return polynomial as
1615 result *= upoly_to_ex(tocheck.top().poly, x);
1623 return unit * cont * result;
1626 /** Second interface to factor_univariate() to be used if the information about
1627 * the prime is not needed.
1629 static inline ex factor_univariate(const ex& poly, const ex& x)
1632 return factor_univariate(poly, x, prime);
1635 /** Represents an evaluation point (<symbol>==<integer>).
1644 ostream& operator<<(ostream& o, const vector<EvalPoint>& v)
1646 for ( size_t i=0; i<v.size(); ++i ) {
1647 o << "(" << v[i].x << "==" << v[i].evalpoint << ") ";
1651 #endif // def DEBUGFACTOR
1653 // forward declaration
1654 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);
1656 /** Utility function for multivariate Hensel lifting.
1658 * Solves the equation
1659 * s_1*b_1 + ... + s_r*b_r == 1 mod p^k
1660 * with deg(s_i) < deg(a_i)
1661 * and with given b_1 = a_1 * ... * a_{i-1} * a_{i+1} * ... * a_r
1663 * The implementation follows the algorithm in chapter 6 of [GCL].
1665 * @param[in] a vector of modular univariate polynomials
1666 * @param[in] x symbol
1667 * @param[in] p prime number
1668 * @param[in] k p^k is modulus
1669 * @return vector of polynomials (s_i)
1671 static upvec multiterm_eea_lift(const upvec& a, const ex& x, unsigned int p, unsigned int k)
1673 const size_t r = a.size();
1674 cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
1677 for ( size_t j=r-2; j>=1; --j ) {
1678 q[j-1] = a[j] * q[j];
1680 umodpoly beta(1, R->one());
1682 for ( size_t j=1; j<r; ++j ) {
1683 vector<ex> mdarg(2);
1684 mdarg[0] = umodpoly_to_ex(q[j-1], x);
1685 mdarg[1] = umodpoly_to_ex(a[j-1], x);
1686 vector<EvalPoint> empty;
1687 vector<ex> exsigma = multivar_diophant(mdarg, x, umodpoly_to_ex(beta, x), empty, 0, p, k);
1689 umodpoly_from_ex(sigma1, exsigma[0], x, R);
1691 umodpoly_from_ex(sigma2, exsigma[1], x, R);
1693 s.push_back(sigma2);
1699 /** Changes the modulus of a modular polynomial. Used by eea_lift().
1701 * @param[in] R new modular ring
1702 * @param[in,out] a polynomial to change (in situ)
1704 static void change_modulus(const cl_modint_ring& R, umodpoly& a)
1706 if ( a.empty() ) return;
1707 cl_modint_ring oldR = a[0].ring();
1708 for (auto & i : a) {
1709 i = R->canonhom(oldR->retract(i));
1714 /** Utility function for multivariate Hensel lifting.
1716 * Solves s*a + t*b == 1 mod p^k given a,b.
1718 * The implementation follows the algorithm in chapter 6 of [GCL].
1720 * @param[in] a polynomial
1721 * @param[in] b polynomial
1722 * @param[in] x symbol
1723 * @param[in] p prime number
1724 * @param[in] k p^k is modulus
1725 * @param[out] s_ output polynomial
1726 * @param[out] t_ output polynomial
1728 static void eea_lift(const umodpoly& a, const umodpoly& b, const ex& x, unsigned int p, unsigned int k, umodpoly& s_, umodpoly& t_)
1730 cl_modint_ring R = find_modint_ring(p);
1732 change_modulus(R, amod);
1734 change_modulus(R, bmod);
1738 exteuclid(amod, bmod, smod, tmod);
1740 cl_modint_ring Rpk = find_modint_ring(expt_pos(cl_I(p),k));
1742 change_modulus(Rpk, s);
1744 change_modulus(Rpk, t);
1747 umodpoly one(1, Rpk->one());
1748 for ( size_t j=1; j<k; ++j ) {
1749 umodpoly e = one - a * s - b * t;
1750 reduce_coeff(e, modulus);
1752 change_modulus(R, c);
1753 umodpoly sigmabar = smod * c;
1754 umodpoly taubar = tmod * c;
1756 remdiv(sigmabar, bmod, sigma, q);
1757 umodpoly tau = taubar + q * amod;
1758 umodpoly sadd = sigma;
1759 change_modulus(Rpk, sadd);
1760 cl_MI modmodulus(Rpk, modulus);
1761 s = s + sadd * modmodulus;
1762 umodpoly tadd = tau;
1763 change_modulus(Rpk, tadd);
1764 t = t + tadd * modmodulus;
1765 modulus = modulus * p;
1771 /** Utility function for multivariate Hensel lifting.
1773 * Solves the equation
1774 * s_1*b_1 + ... + s_r*b_r == x^m mod p^k
1775 * with given b_1 = a_1 * ... * a_{i-1} * a_{i+1} * ... * a_r
1777 * The implementation follows the algorithm in chapter 6 of [GCL].
1779 * @param a vector with univariate polynomials mod p^k
1781 * @param m exponent of x^m in the equation to solve
1782 * @param p prime number
1783 * @param k p^k is modulus
1784 * @return vector of polynomials (s_i)
1786 static upvec univar_diophant(const upvec& a, const ex& x, unsigned int m, unsigned int p, unsigned int k)
1788 cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
1790 const size_t r = a.size();
1793 upvec s = multiterm_eea_lift(a, x, p, k);
1794 for ( size_t j=0; j<r; ++j ) {
1795 umodpoly bmod = umodpoly_to_umodpoly(s[j], R, m);
1797 rem(bmod, a[j], buf);
1798 result.push_back(buf);
1802 eea_lift(a[1], a[0], x, p, k, s, t);
1803 umodpoly bmod = umodpoly_to_umodpoly(s, R, m);
1805 remdiv(bmod, a[0], buf, q);
1806 result.push_back(buf);
1807 umodpoly t1mod = umodpoly_to_umodpoly(t, R, m);
1808 buf = t1mod + q * a[1];
1809 result.push_back(buf);
1815 /** Map used by function make_modular().
1816 * Finds every coefficient in a polynomial and replaces it by is value in the
1817 * given modular ring R (symmetric representation).
1819 struct make_modular_map : public map_function {
1821 make_modular_map(const cl_modint_ring& R_) : R(R_) { }
1822 ex operator()(const ex& e) override
1824 if ( is_a<add>(e) || is_a<mul>(e) ) {
1825 return e.map(*this);
1827 else if ( is_a<numeric>(e) ) {
1828 numeric mod(R->modulus);
1829 numeric halfmod = (mod-1)/2;
1830 cl_MI emod = R->canonhom(the<cl_I>(ex_to<numeric>(e).to_cl_N()));
1831 numeric n(R->retract(emod));
1832 if ( n > halfmod ) {
1842 /** Helps mimicking modular multivariate polynomial arithmetic.
1844 * @param e expression of which to make the coefficients equal to their value
1845 * in the modular ring R (symmetric representation)
1846 * @param R modular ring
1847 * @return resulting expression
1849 static ex make_modular(const ex& e, const cl_modint_ring& R)
1851 make_modular_map map(R);
1852 return map(e.expand());
1855 /** Utility function for multivariate Hensel lifting.
1857 * Returns the polynomials s_i that fulfill
1858 * s_1*b_1 + ... + s_r*b_r == c mod <I^(d+1),p^k>
1859 * with given b_1 = a_1 * ... * a_{i-1} * a_{i+1} * ... * a_r
1861 * The implementation follows the algorithm in chapter 6 of [GCL].
1863 * @param a_ vector of multivariate factors mod p^k
1864 * @param x symbol (equiv. x_1 in [GCL])
1865 * @param c polynomial mod p^k
1866 * @param I vector of evaluation points
1867 * @param d maximum total degree of result
1868 * @param p prime number
1869 * @param k p^k is modulus
1870 * @return vector of polynomials (s_i)
1872 static vector<ex> multivar_diophant(const vector<ex>& a_, const ex& x, const ex& c, const vector<EvalPoint>& I,
1873 unsigned int d, unsigned int p, unsigned int k)
1877 const cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
1878 const size_t r = a.size();
1879 const size_t nu = I.size() + 1;
1883 ex xnu = I.back().x;
1884 int alphanu = I.back().evalpoint;
1887 for ( size_t i=0; i<r; ++i ) {
1891 for ( size_t i=0; i<r; ++i ) {
1892 b[i] = normal(A / a[i]);
1895 vector<ex> anew = a;
1896 for ( size_t i=0; i<r; ++i ) {
1897 anew[i] = anew[i].subs(xnu == alphanu);
1899 ex cnew = c.subs(xnu == alphanu);
1900 vector<EvalPoint> Inew = I;
1902 sigma = multivar_diophant(anew, x, cnew, Inew, d, p, k);
1905 for ( size_t i=0; i<r; ++i ) {
1906 buf -= sigma[i] * b[i];
1908 ex e = make_modular(buf, R);
1911 for ( size_t m=1; !e.is_zero() && e.has(xnu) && m<=d; ++m ) {
1912 monomial *= (xnu - alphanu);
1913 monomial = expand(monomial);
1914 ex cm = e.diff(ex_to<symbol>(xnu), m).subs(xnu==alphanu) / factorial(m);
1915 cm = make_modular(cm, R);
1916 if ( !cm.is_zero() ) {
1917 vector<ex> delta_s = multivar_diophant(anew, x, cm, Inew, d, p, k);
1919 for ( size_t j=0; j<delta_s.size(); ++j ) {
1920 delta_s[j] *= monomial;
1921 sigma[j] += delta_s[j];
1922 buf -= delta_s[j] * b[j];
1924 e = make_modular(buf, R);
1929 for ( size_t i=0; i<a.size(); ++i ) {
1931 umodpoly_from_ex(up, a[i], x, R);
1935 sigma.insert(sigma.begin(), r, 0);
1938 if ( is_a<add>(c) ) {
1945 for ( size_t i=0; i<nterms; ++i ) {
1946 int m = z.degree(x);
1947 cl_I cm = the<cl_I>(ex_to<numeric>(z.lcoeff(x)).to_cl_N());
1948 upvec delta_s = univar_diophant(amod, x, m, p, k);
1951 while ( poscm < 0 ) {
1952 poscm = poscm + expt_pos(cl_I(p),k);
1954 modcm = cl_MI(R, poscm);
1955 for ( size_t j=0; j<delta_s.size(); ++j ) {
1956 delta_s[j] = delta_s[j] * modcm;
1957 sigma[j] = sigma[j] + umodpoly_to_ex(delta_s[j], x);
1965 for ( size_t i=0; i<sigma.size(); ++i ) {
1966 sigma[i] = make_modular(sigma[i], R);
1972 /** Multivariate Hensel lifting.
1973 * The implementation follows the algorithm in chapter 6 of [GCL].
1974 * Since we don't have a data type for modular multivariate polynomials, the
1975 * respective operations are done in a GiNaC::ex and the function
1976 * make_modular() is then called to make the coefficient modular p^l.
1978 * @param a multivariate polynomial primitive in x
1979 * @param x symbol (equiv. x_1 in [GCL])
1980 * @param I vector of evaluation points (x_2==a_2,x_3==a_3,...)
1981 * @param p prime number (should not divide lcoeff(a mod I))
1982 * @param l p^l is the modulus of the lifted univariate field
1983 * @param u vector of modular (mod p^l) factors of a mod I
1984 * @param lcU correct leading coefficient of the univariate factors of a mod I
1985 * @return list GiNaC::lst with lifted factors (multivariate factors of a),
1986 * empty if Hensel lifting did not succeed
1988 static ex hensel_multivar(const ex& a, const ex& x, const vector<EvalPoint>& I,
1989 unsigned int p, const cl_I& l, const upvec& u, const vector<ex>& lcU)
1991 const size_t nu = I.size() + 1;
1992 const cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),l));
1997 for ( size_t j=nu; j>=2; --j ) {
1999 int alpha = I[j-2].evalpoint;
2000 A[j-2] = A[j-1].subs(x==alpha);
2001 A[j-2] = make_modular(A[j-2], R);
2004 int maxdeg = a.degree(I.front().x);
2005 for ( size_t i=1; i<I.size(); ++i ) {
2006 int maxdeg2 = a.degree(I[i].x);
2007 if ( maxdeg2 > maxdeg ) maxdeg = maxdeg2;
2010 const size_t n = u.size();
2012 for ( size_t i=0; i<n; ++i ) {
2013 U[i] = umodpoly_to_ex(u[i], x);
2016 for ( size_t j=2; j<=nu; ++j ) {
2019 for ( size_t m=0; m<n; ++m) {
2020 if ( lcU[m] != 1 ) {
2022 for ( size_t i=j-1; i<nu-1; ++i ) {
2023 coef = coef.subs(I[i].x == I[i].evalpoint);
2025 coef = make_modular(coef, R);
2026 int deg = U[m].degree(x);
2027 U[m] = U[m] - U[m].lcoeff(x) * pow(x,deg) + coef * pow(x,deg);
2031 for ( size_t i=0; i<n; ++i ) {
2034 ex e = expand(A[j-1] - Uprod);
2036 vector<EvalPoint> newI;
2037 for ( size_t i=1; i<=j-2; ++i ) {
2038 newI.push_back(I[i-1]);
2042 int alphaj = I[j-2].evalpoint;
2043 size_t deg = A[j-1].degree(xj);
2044 for ( size_t k=1; k<=deg; ++k ) {
2045 if ( !e.is_zero() ) {
2046 monomial *= (xj - alphaj);
2047 monomial = expand(monomial);
2048 ex dif = e.diff(ex_to<symbol>(xj), k);
2049 ex c = dif.subs(xj==alphaj) / factorial(k);
2050 if ( !c.is_zero() ) {
2051 vector<ex> deltaU = multivar_diophant(U1, x, c, newI, maxdeg, p, cl_I_to_uint(l));
2052 for ( size_t i=0; i<n; ++i ) {
2053 deltaU[i] *= monomial;
2055 U[i] = make_modular(U[i], R);
2058 for ( size_t i=0; i<n; ++i ) {
2062 e = make_modular(e, R);
2069 for ( size_t i=0; i<U.size(); ++i ) {
2072 if ( expand(a-acand).is_zero() ) {
2073 return lst(U.begin(), U.end());
2079 /** Takes a factorized expression and puts the factors in a lst. The exponents
2080 * of the factors are discarded, e.g. 7*x^2*(y+1)^4 --> {7,x,y+1}. The first
2081 * element of the list is always the numeric coefficient.
2083 static ex put_factors_into_lst(const ex& e)
2086 if ( is_a<numeric>(e) ) {
2090 if ( is_a<power>(e) ) {
2092 result.append(e.op(0));
2095 if ( is_a<symbol>(e) || is_a<add>(e) ) {
2096 ex icont(e.integer_content());
2097 result.append(icont);
2098 result.append(e/icont);
2101 if ( is_a<mul>(e) ) {
2103 for ( size_t i=0; i<e.nops(); ++i ) {
2105 if ( is_a<numeric>(op) ) {
2108 if ( is_a<power>(op) ) {
2109 result.append(op.op(0));
2111 if ( is_a<symbol>(op) || is_a<add>(op) ) {
2115 result.prepend(nfac);
2118 throw runtime_error("put_factors_into_lst: bad term.");
2121 /** Checks a set of numbers for whether each number has a unique prime factor.
2123 * @param[in] f list of numbers to check
2124 * @return true: if number set is bad, false: if set is okay (has unique
2127 static bool checkdivisors(const lst& f)
2129 const int k = f.nops();
2131 vector<numeric> d(k);
2132 d[0] = ex_to<numeric>(abs(f.op(0)));
2133 for ( int i=1; i<k; ++i ) {
2134 q = ex_to<numeric>(abs(f.op(i)));
2135 for ( int j=i-1; j>=0; --j ) {
2150 /** Generates a set of evaluation points for a multivariate polynomial.
2151 * The set fulfills the following conditions:
2152 * 1. lcoeff(evaluated_polynomial) does not vanish
2153 * 2. factors of lcoeff(evaluated_polynomial) have each a unique prime factor
2154 * 3. evaluated_polynomial is square free
2155 * See [Wan] for more details.
2157 * @param[in] u multivariate polynomial to be factored
2158 * @param[in] vn leading coefficient of u in x (x==first symbol in syms)
2159 * @param[in] syms set of symbols that appear in u
2160 * @param[in] f lst containing the factors of the leading coefficient vn
2161 * @param[in,out] modulus integer modulus for random number generation (i.e. |a_i| < modulus)
2162 * @param[out] u0 returns the evaluated (univariate) polynomial
2163 * @param[out] a returns the valid evaluation points. must have initial size equal
2164 * number of symbols-1 before calling generate_set
2166 static void generate_set(const ex& u, const ex& vn, const exset& syms, const lst& f,
2167 numeric& modulus, ex& u0, vector<numeric>& a)
2169 const ex& x = *syms.begin();
2172 // generate a set of integers ...
2176 exset::const_iterator s = syms.begin();
2178 for ( size_t i=0; i<a.size(); ++i ) {
2180 a[i] = mod(numeric(rand()), 2*modulus) - modulus;
2181 vnatry = vna.subs(*s == a[i]);
2182 // ... for which the leading coefficient doesn't vanish ...
2183 } while ( vnatry == 0 );
2185 u0 = u0.subs(*s == a[i]);
2188 // ... for which u0 is square free ...
2189 ex g = gcd(u0, u0.diff(ex_to<symbol>(x)));
2190 if ( !is_a<numeric>(g) ) {
2193 if ( !is_a<numeric>(vn) ) {
2194 // ... and for which the evaluated factors have each an unique prime factor
2196 fnum.let_op(0) = fnum.op(0) * u0.content(x);
2197 for ( size_t i=1; i<fnum.nops(); ++i ) {
2198 if ( !is_a<numeric>(fnum.op(i)) ) {
2201 for ( size_t j=0; j<a.size(); ++j, ++s ) {
2202 fnum.let_op(i) = fnum.op(i).subs(*s == a[j]);
2206 if ( checkdivisors(fnum) ) {
2210 // ok, we have a valid set now
2215 // forward declaration
2216 static ex factor_sqrfree(const ex& poly);
2218 /** Multivariate factorization.
2220 * The implementation is based on the algorithm described in [Wan].
2221 * An evaluation homomorphism (a set of integers) is determined that fulfills
2222 * certain criteria. The evaluated polynomial is univariate and is factorized
2223 * by factor_univariate(). The main work then is to find the correct leading
2224 * coefficients of the univariate factors. They have to correspond to the
2225 * factors of the (multivariate) leading coefficient of the input polynomial
2226 * (as defined for a specific variable x). After that the Hensel lifting can be
2229 * @param[in] poly expanded, square free polynomial
2230 * @param[in] syms contains the symbols in the polynomial
2231 * @return factorized polynomial
2233 static ex factor_multivariate(const ex& poly, const exset& syms)
2235 exset::const_iterator s;
2236 const ex& x = *syms.begin();
2238 // make polynomial primitive
2240 poly.unitcontprim(x, unit, cont, pp);
2241 if ( !is_a<numeric>(cont) ) {
2242 return factor_sqrfree(cont) * factor_sqrfree(pp);
2245 // factor leading coefficient
2246 ex vn = pp.collect(x).lcoeff(x);
2248 if ( is_a<numeric>(vn) ) {
2252 ex vnfactors = factor(vn);
2253 vnlst = put_factors_into_lst(vnfactors);
2256 const unsigned int maxtrials = 3;
2257 numeric modulus = (vnlst.nops() > 3) ? vnlst.nops() : 3;
2258 vector<numeric> a(syms.size()-1, 0);
2260 // try now to factorize until we are successful
2263 unsigned int trialcount = 0;
2265 int factor_count = 0;
2266 int min_factor_count = -1;
2270 // try several evaluation points to reduce the number of factors
2271 while ( trialcount < maxtrials ) {
2273 // generate a set of valid evaluation points
2274 generate_set(pp, vn, syms, ex_to<lst>(vnlst), modulus, u, a);
2276 ufac = factor_univariate(u, x, prime);
2277 ufaclst = put_factors_into_lst(ufac);
2278 factor_count = ufaclst.nops()-1;
2279 delta = ufaclst.op(0);
2281 if ( factor_count <= 1 ) {
2285 if ( min_factor_count < 0 ) {
2287 min_factor_count = factor_count;
2289 else if ( min_factor_count == factor_count ) {
2293 else if ( min_factor_count > factor_count ) {
2294 // new minimum, reset trial counter
2295 min_factor_count = factor_count;
2300 // determine true leading coefficients for the Hensel lifting
2301 vector<ex> C(factor_count);
2302 if ( is_a<numeric>(vn) ) {
2304 for ( size_t i=1; i<ufaclst.nops(); ++i ) {
2305 C[i-1] = ufaclst.op(i).lcoeff(x);
2309 // we use the property of the ftilde having a unique prime factor.
2310 // details can be found in [Wan].
2312 vector<numeric> ftilde(vnlst.nops()-1);
2313 for ( size_t i=0; i<ftilde.size(); ++i ) {
2314 ex ft = vnlst.op(i+1);
2317 for ( size_t j=0; j<a.size(); ++j ) {
2318 ft = ft.subs(*s == a[j]);
2321 ftilde[i] = ex_to<numeric>(ft);
2323 // calculate D and C
2324 vector<bool> used_flag(ftilde.size(), false);
2325 vector<ex> D(factor_count, 1);
2327 for ( int i=0; i<factor_count; ++i ) {
2328 numeric prefac = ex_to<numeric>(ufaclst.op(i+1).lcoeff(x));
2329 for ( int j=ftilde.size()-1; j>=0; --j ) {
2331 while ( irem(prefac, ftilde[j]) == 0 ) {
2332 prefac = iquo(prefac, ftilde[j]);
2336 used_flag[j] = true;
2337 D[i] = D[i] * pow(vnlst.op(j+1), count);
2340 C[i] = D[i] * prefac;
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]);
2351 while ( irem(ex_to<numeric>(delta)*prefac, ftilde[j]) == 0 ) {
2352 numeric g = gcd(prefac, ex_to<numeric>(ftilde[j]));
2353 prefac = iquo(prefac, g);
2354 delta = delta / (ftilde[j]/g);
2355 ufaclst.let_op(i+1) = ufaclst.op(i+1) * (ftilde[j]/g);
2359 used_flag[j] = true;
2360 D[i] = D[i] * pow(vnlst.op(j+1), count);
2363 C[i] = D[i] * prefac;
2366 // check if something went wrong
2367 bool some_factor_unused = false;
2368 for ( size_t i=0; i<used_flag.size(); ++i ) {
2369 if ( !used_flag[i] ) {
2370 some_factor_unused = true;
2374 if ( some_factor_unused ) {
2379 // multiply the remaining content of the univariate polynomial into the
2382 C[0] = C[0] * delta;
2383 ufaclst.let_op(1) = ufaclst.op(1) * delta;
2386 // set up evaluation points
2388 vector<EvalPoint> epv;
2391 for ( size_t i=0; i<a.size(); ++i ) {
2393 ep.evalpoint = a[i].to_int();
2399 for ( int i=1; i<=factor_count; ++i ) {
2400 if ( ufaclst.op(i).degree(x) > maxdeg ) {
2401 maxdeg = ufaclst[i].degree(x);
2404 cl_I B = 2*calc_bound(u, x, maxdeg);
2412 // set up modular factors (mod p^l)
2413 cl_modint_ring R = find_modint_ring(expt_pos(cl_I(prime),l));
2414 upvec modfactors(ufaclst.nops()-1);
2415 for ( size_t i=1; i<ufaclst.nops(); ++i ) {
2416 umodpoly_from_ex(modfactors[i-1], ufaclst.op(i), x, R);
2419 // try Hensel lifting
2420 ex res = hensel_multivar(pp, x, epv, prime, l, modfactors, C);
2421 if ( res != lst{} ) {
2422 ex result = cont * unit;
2423 for ( size_t i=0; i<res.nops(); ++i ) {
2424 result *= res.op(i).content(x) * res.op(i).unit(x);
2425 result *= res.op(i).primpart(x);
2432 /** Finds all symbols in an expression. Used by factor_sqrfree() and factor().
2434 struct find_symbols_map : public map_function {
2436 ex operator()(const ex& e) override
2438 if ( is_a<symbol>(e) ) {
2442 return e.map(*this);
2446 /** Factorizes a polynomial that is square free. It calls either the univariate
2447 * or the multivariate factorization functions.
2449 static ex factor_sqrfree(const ex& poly)
2451 // determine all symbols in poly
2452 find_symbols_map findsymbols;
2454 if ( findsymbols.syms.size() == 0 ) {
2458 if ( findsymbols.syms.size() == 1 ) {
2460 const ex& x = *(findsymbols.syms.begin());
2461 if ( poly.ldegree(x) > 0 ) {
2462 // pull out direct factors
2463 int ld = poly.ldegree(x);
2464 ex res = factor_univariate(expand(poly/pow(x, ld)), x);
2465 return res * pow(x,ld);
2467 ex res = factor_univariate(poly, x);
2472 // multivariate case
2473 ex res = factor_multivariate(poly, findsymbols.syms);
2477 /** Map used by factor() when factor_options::all is given to access all
2478 * subexpressions and to call factor() on them.
2480 struct apply_factor_map : public map_function {
2482 apply_factor_map(unsigned options_) : options(options_) { }
2483 ex operator()(const ex& e) override
2485 if ( e.info(info_flags::polynomial) ) {
2486 return factor(e, options);
2488 if ( is_a<add>(e) ) {
2490 for ( size_t i=0; i<e.nops(); ++i ) {
2491 if ( e.op(i).info(info_flags::polynomial) ) {
2497 return factor(s1, options) + s2.map(*this);
2499 return e.map(*this);
2503 /** Iterate through explicit factors of e, call yield(f, k) for
2504 * each factor of the form f^k.
2506 * Note that this function doesn't factor e itself, it only
2507 * iterates through the factors already explicitly present.
2509 template <typename F> void
2510 factor_iter(const ex &e, F yield)
2513 for (const auto &f : e) {
2514 if (is_a<power>(f)) {
2515 yield(f.op(0), f.op(1));
2521 if (is_a<power>(e)) {
2522 yield(e.op(0), e.op(1));
2529 /** This function factorizes a polynomial. It checks the arguments,
2530 * tries a square free factorization, and then calls factor_sqrfree
2531 * to do the hard work.
2533 * This function expands its argument, so for polynomials with
2534 * explicit factors it's better to call it on each one separately
2535 * (or use factor() which does just that).
2537 static ex factor1(const ex& poly, unsigned options)
2540 if ( !poly.info(info_flags::polynomial) ) {
2541 if ( options & factor_options::all ) {
2542 options &= ~factor_options::all;
2543 apply_factor_map factor_map(options);
2544 return factor_map(poly);
2549 // determine all symbols in poly
2550 find_symbols_map findsymbols;
2552 if ( findsymbols.syms.size() == 0 ) {
2556 for (auto & i : findsymbols.syms ) {
2560 // make poly square free
2561 ex sfpoly = sqrfree(poly.expand(), syms);
2563 // factorize the square free components
2566 [&](const ex &f, const ex &k) {
2567 if ( is_a<add>(f) ) {
2568 res *= pow(factor_sqrfree(f), k);
2570 // simple case: (monomial)^exponent
2577 } // anonymous namespace
2579 /** Interface function to the outside world. It uses factor1()
2580 * on each of the explicitly present factors of poly.
2582 ex factor(const ex& poly, unsigned options)
2586 [&](const ex &f1, const ex &k1) {
2587 factor_iter(factor1(f1, options),
2588 [&](const ex &f2, const ex &k2) {
2589 result *= pow(f2, k1*k2);
2595 } // namespace GiNaC