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) 1999-2018 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 auto i = v.begin(), end = v.end();
96 static ostream& operator<<(ostream& o, const vector<cl_I>& v)
98 auto i = v.begin(), end = v.end();
100 o << *i << "[" << i-v.begin() << "]" << " ";
105 static ostream& operator<<(ostream& o, const vector<cl_MI>& v)
107 auto i = v.begin(), end = v.end();
109 o << *i << "[" << i-v.begin() << "]" << " ";
114 ostream& operator<<(ostream& o, const vector<numeric>& v)
116 for ( size_t i=0; i<v.size(); ++i ) {
121 ostream& operator<<(ostream& o, const vector<vector<cl_MI>>& v)
123 auto i = v.begin(), end = v.end();
125 o << i-v.begin() << ": " << *i << endl;
132 #define DCOUTVAR(var)
133 #define DCOUT2(str,var)
134 #endif // def DEBUGFACTOR
136 // anonymous namespace to hide all utility functions
139 ////////////////////////////////////////////////////////////////////////////////
140 // modular univariate polynomial code
142 typedef std::vector<cln::cl_MI> umodpoly;
143 typedef std::vector<cln::cl_I> upoly;
144 typedef vector<umodpoly> upvec;
146 // COPY FROM UPOLY.HPP
148 // CHANGED size_t -> int !!!
149 template<typename T> static int degree(const T& p)
154 template<typename T> static typename T::value_type lcoeff(const T& p)
156 return p[p.size() - 1];
159 static bool normalize_in_field(umodpoly& a)
163 if ( lcoeff(a) == a[0].ring()->one() ) {
167 const cln::cl_MI lc_1 = recip(lcoeff(a));
168 for (std::size_t k = a.size(); k-- != 0; )
173 template<typename T> static void
174 canonicalize(T& p, const typename T::size_type hint = std::numeric_limits<typename T::size_type>::max())
179 std::size_t i = p.size() - 1;
180 // Be fast if the polynomial is already canonicalized
187 bool is_zero = false;
205 p.erase(p.begin() + i, p.end());
208 // END COPY FROM UPOLY.HPP
210 static void expt_pos(umodpoly& a, unsigned int q)
212 if ( a.empty() ) return;
213 cl_MI zero = a[0].ring()->zero();
215 a.resize(degree(a)*q+1, zero);
216 for ( int i=deg; i>0; --i ) {
222 template<bool COND, typename T = void> struct enable_if
227 template<typename T> struct enable_if<false, T> { /* empty */ };
229 template<typename T> struct uvar_poly_p
231 static const bool value = false;
234 template<> struct uvar_poly_p<upoly>
236 static const bool value = true;
239 template<> struct uvar_poly_p<umodpoly>
241 static const bool value = true;
245 // Don't define this for anything but univariate polynomials.
246 static typename enable_if<uvar_poly_p<T>::value, T>::type
247 operator+(const T& a, const T& b)
254 for ( ; i<sb; ++i ) {
257 for ( ; i<sa; ++i ) {
266 for ( ; i<sa; ++i ) {
269 for ( ; i<sb; ++i ) {
278 // Don't define this for anything but univariate polynomials. Otherwise
279 // overload resolution might fail (this actually happens when compiling
280 // GiNaC with g++ 3.4).
281 static typename enable_if<uvar_poly_p<T>::value, T>::type
282 operator-(const T& a, const T& b)
289 for ( ; i<sb; ++i ) {
292 for ( ; i<sa; ++i ) {
301 for ( ; i<sa; ++i ) {
304 for ( ; i<sb; ++i ) {
312 static upoly operator*(const upoly& a, const upoly& b)
315 if ( a.empty() || b.empty() ) return c;
317 int n = degree(a) + degree(b);
319 for ( int i=0 ; i<=n; ++i ) {
320 for ( int j=0 ; j<=i; ++j ) {
321 if ( j > degree(a) || (i-j) > degree(b) ) continue;
322 c[i] = c[i] + a[j] * b[i-j];
329 static umodpoly operator*(const umodpoly& a, const umodpoly& b)
332 if ( a.empty() || b.empty() ) return c;
334 int n = degree(a) + degree(b);
335 c.resize(n+1, a[0].ring()->zero());
336 for ( int i=0 ; i<=n; ++i ) {
337 for ( int j=0 ; j<=i; ++j ) {
338 if ( j > degree(a) || (i-j) > degree(b) ) continue;
339 c[i] = c[i] + a[j] * b[i-j];
346 static upoly operator*(const upoly& a, const cl_I& x)
353 for ( size_t i=0; i<a.size(); ++i ) {
359 static upoly operator/(const upoly& a, const cl_I& x)
366 for ( size_t i=0; i<a.size(); ++i ) {
367 r[i] = exquo(a[i],x);
372 static umodpoly operator*(const umodpoly& a, const cl_MI& x)
374 umodpoly r(a.size());
375 for ( size_t i=0; i<a.size(); ++i ) {
382 static void upoly_from_ex(upoly& up, const ex& e, const ex& x)
384 // assert: e is in Z[x]
385 int deg = e.degree(x);
387 int ldeg = e.ldegree(x);
388 for ( ; deg>=ldeg; --deg ) {
389 up[deg] = the<cl_I>(ex_to<numeric>(e.coeff(x, deg)).to_cl_N());
391 for ( ; deg>=0; --deg ) {
397 static void umodpoly_from_upoly(umodpoly& ump, const upoly& e, const cl_modint_ring& R)
401 for ( ; deg>=0; --deg ) {
402 ump[deg] = R->canonhom(e[deg]);
407 static void umodpoly_from_ex(umodpoly& ump, const ex& e, const ex& x, const cl_modint_ring& R)
409 // assert: e is in Z[x]
410 int deg = e.degree(x);
412 int ldeg = e.ldegree(x);
413 for ( ; deg>=ldeg; --deg ) {
414 cl_I coeff = the<cl_I>(ex_to<numeric>(e.coeff(x, deg)).to_cl_N());
415 ump[deg] = R->canonhom(coeff);
417 for ( ; deg>=0; --deg ) {
418 ump[deg] = R->zero();
424 static void umodpoly_from_ex(umodpoly& ump, const ex& e, const ex& x, const cl_I& modulus)
426 umodpoly_from_ex(ump, e, x, find_modint_ring(modulus));
430 static ex upoly_to_ex(const upoly& a, const ex& x)
432 if ( a.empty() ) return 0;
434 for ( int i=degree(a); i>=0; --i ) {
435 e += numeric(a[i]) * pow(x, i);
440 static ex umodpoly_to_ex(const umodpoly& a, const ex& x)
442 if ( a.empty() ) return 0;
443 cl_modint_ring R = a[0].ring();
444 cl_I mod = R->modulus;
445 cl_I halfmod = (mod-1) >> 1;
447 for ( int i=degree(a); i>=0; --i ) {
448 cl_I n = R->retract(a[i]);
450 e += numeric(n-mod) * pow(x, i);
452 e += numeric(n) * pow(x, i);
458 static upoly umodpoly_to_upoly(const umodpoly& a)
461 if ( a.empty() ) return e;
462 cl_modint_ring R = a[0].ring();
463 cl_I mod = R->modulus;
464 cl_I halfmod = (mod-1) >> 1;
465 for ( int i=degree(a); i>=0; --i ) {
466 cl_I n = R->retract(a[i]);
476 static umodpoly umodpoly_to_umodpoly(const umodpoly& a, const cl_modint_ring& R, unsigned int m)
479 if ( a.empty() ) return e;
480 cl_modint_ring oldR = a[0].ring();
481 size_t sa = a.size();
482 e.resize(sa+m, R->zero());
483 for ( size_t i=0; i<sa; ++i ) {
484 e[i+m] = R->canonhom(oldR->retract(a[i]));
490 /** Divides all coefficients of the polynomial a by the integer x.
491 * All coefficients are supposed to be divisible by x. If they are not, the
492 * the<cl_I> cast will raise an exception.
494 * @param[in,out] a polynomial of which the coefficients will be reduced by x
495 * @param[in] x integer that divides the coefficients
497 static void reduce_coeff(umodpoly& a, const cl_I& x)
499 if ( a.empty() ) return;
501 cl_modint_ring R = a[0].ring();
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;
683 friend ostream& operator<<(ostream& o, const modular_matrix& m);
686 modular_matrix(size_t r_, size_t c_, const cl_MI& init) : r(r_), c(c_)
690 size_t rowsize() const { return r; }
691 size_t colsize() const { return c; }
692 cl_MI& operator()(size_t row, size_t col) { return m[row*c + col]; }
693 cl_MI operator()(size_t row, size_t col) const { return m[row*c + col]; }
694 void mul_col(size_t col, const cl_MI x)
696 for ( size_t rc=0; rc<r; ++rc ) {
697 std::size_t i = c*rc + col;
701 void sub_col(size_t col1, size_t col2, const cl_MI fac)
703 for ( size_t rc=0; rc<r; ++rc ) {
704 std::size_t i1 = col1 + c*rc;
705 std::size_t i2 = col2 + c*rc;
706 m[i1] = m[i1] - m[i2]*fac;
709 void switch_col(size_t col1, size_t col2)
711 for ( size_t rc=0; rc<r; ++rc ) {
712 std::size_t i1 = col1 + rc*c;
713 std::size_t i2 = col2 + rc*c;
714 std::swap(m[i1], m[i2]);
717 void mul_row(size_t row, const cl_MI x)
719 for ( size_t cc=0; cc<c; ++cc ) {
720 std::size_t i = row*c + cc;
724 void sub_row(size_t row1, size_t row2, const cl_MI fac)
726 for ( size_t cc=0; cc<c; ++cc ) {
727 std::size_t i1 = row1*c + cc;
728 std::size_t i2 = row2*c + cc;
729 m[i1] = m[i1] - m[i2]*fac;
732 void switch_row(size_t row1, size_t row2)
734 for ( size_t cc=0; cc<c; ++cc ) {
735 std::size_t i1 = row1*c + cc;
736 std::size_t i2 = row2*c + cc;
737 std::swap(m[i1], m[i2]);
740 bool is_col_zero(size_t col) const
742 for ( size_t rr=0; rr<r; ++rr ) {
743 std::size_t i = col + rr*c;
744 if ( !zerop(m[i]) ) {
750 bool is_row_zero(size_t row) const
752 for ( size_t cc=0; cc<c; ++cc ) {
753 std::size_t i = row*c + cc;
754 if ( !zerop(m[i]) ) {
760 void set_row(size_t row, const vector<cl_MI>& newrow)
762 for (std::size_t i2 = 0; i2 < newrow.size(); ++i2) {
763 std::size_t i1 = row*c + i2;
767 mvec::const_iterator row_begin(size_t row) const { return m.begin()+row*c; }
768 mvec::const_iterator row_end(size_t row) const { return m.begin()+row*c+r; }
775 modular_matrix operator*(const modular_matrix& m1, const modular_matrix& m2)
777 const unsigned int r = m1.rowsize();
778 const unsigned int c = m2.colsize();
779 modular_matrix o(r,c,m1(0,0));
781 for ( size_t i=0; i<r; ++i ) {
782 for ( size_t j=0; j<c; ++j ) {
784 buf = m1(i,0) * m2(0,j);
785 for ( size_t k=1; k<c; ++k ) {
786 buf = buf + m1(i,k)*m2(k,j);
794 ostream& operator<<(ostream& o, const modular_matrix& m)
796 cl_modint_ring R = m(0,0).ring();
798 for ( size_t i=0; i<m.rowsize(); ++i ) {
800 for ( size_t j=0; j<m.colsize()-1; ++j ) {
801 o << R->retract(m(i,j)) << ",";
803 o << R->retract(m(i,m.colsize()-1)) << "}";
804 if ( i != m.rowsize()-1 ) {
811 #endif // def DEBUGFACTOR
813 // END modular matrix
814 ////////////////////////////////////////////////////////////////////////////////
816 /** Calculates the Q matrix for a polynomial. Used by Berlekamp's algorithm.
818 * @param[in] a_ modular polynomial
819 * @param[out] Q Q matrix
821 static void q_matrix(const umodpoly& a_, modular_matrix& Q)
824 normalize_in_field(a);
827 unsigned int q = cl_I_to_uint(a[0].ring()->modulus);
828 umodpoly r(n, a[0].ring()->zero());
829 r[0] = a[0].ring()->one();
831 unsigned int max = (n-1) * q;
832 for ( size_t m=1; m<=max; ++m ) {
833 cl_MI rn_1 = r.back();
834 for ( size_t i=n-1; i>0; --i ) {
835 r[i] = r[i-1] - (rn_1 * a[i]);
838 if ( (m % q) == 0 ) {
844 /** Determine the nullspace of a matrix M-1.
846 * @param[in,out] M matrix, will be modified
847 * @param[out] basis calculated nullspace of M-1
849 static void nullspace(modular_matrix& M, vector<mvec>& basis)
851 const size_t n = M.rowsize();
852 const cl_MI one = M(0,0).ring()->one();
853 for ( size_t i=0; i<n; ++i ) {
854 M(i,i) = M(i,i) - one;
856 for ( size_t r=0; r<n; ++r ) {
858 for ( ; cc<n; ++cc ) {
859 if ( !zerop(M(r,cc)) ) {
861 if ( !zerop(M(cc,cc)) ) {
873 M.mul_col(r, recip(M(r,r)));
874 for ( cc=0; cc<n; ++cc ) {
876 M.sub_col(cc, r, M(r,cc));
882 for ( size_t i=0; i<n; ++i ) {
883 M(i,i) = M(i,i) - one;
885 for ( size_t i=0; i<n; ++i ) {
886 if ( !M.is_row_zero(i) ) {
887 mvec nu(M.row_begin(i), M.row_end(i));
893 /** Berlekamp's modular factorization.
895 * The implementation follows the algorithm in chapter 8 of [GCL].
897 * @param[in] a modular polynomial
898 * @param[out] upv vector containing modular factors. if upv was not empty the
899 * new elements are added at the end
901 static void berlekamp(const umodpoly& a, upvec& upv)
903 cl_modint_ring R = a[0].ring();
904 umodpoly one(1, R->one());
906 // find nullspace of Q matrix
907 modular_matrix Q(degree(a), degree(a), R->zero());
912 const unsigned int k = nu.size();
918 list<umodpoly> factors = {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.");
941 factors.push_back(g);
943 for (auto & i : factors) {
948 for (auto & i : factors) {
962 // modular square free factorization is not used at the moment so we deactivate
966 /** Calculates a^(1/prime).
968 * @param[in] a polynomial
969 * @param[in] prime prime number -> exponent 1/prime
970 * @param[in] ap resulting polynomial
972 static void expt_1_over_p(const umodpoly& a, unsigned int prime, umodpoly& ap)
974 size_t newdeg = degree(a)/prime;
977 for ( size_t i=1; i<=newdeg; ++i ) {
982 /** Modular square free factorization.
984 * @param[in] a polynomial
985 * @param[out] factors modular factors
986 * @param[out] mult corresponding multiplicities (exponents)
988 static void modsqrfree(const umodpoly& a, upvec& factors, vector<int>& mult)
990 const unsigned int prime = cl_I_to_uint(a[0].ring()->modulus);
999 while ( unequal_one(w) ) {
1004 factors.push_back(z);
1012 if ( unequal_one(c) ) {
1014 expt_1_over_p(c, prime, cp);
1015 size_t previ = mult.size();
1016 modsqrfree(cp, factors, mult);
1017 for ( size_t i=previ; i<mult.size(); ++i ) {
1023 expt_1_over_p(a, prime, ap);
1024 size_t previ = mult.size();
1025 modsqrfree(ap, factors, mult);
1026 for ( size_t i=previ; i<mult.size(); ++i ) {
1032 #endif // deactivation of square free factorization
1034 /** Distinct degree factorization (DDF).
1036 * The implementation follows the algorithm in chapter 8 of [GCL].
1038 * @param[in] a_ modular polynomial
1039 * @param[out] degrees vector containing the degrees of the factors of the
1040 * corresponding polynomials in ddfactors.
1041 * @param[out] ddfactors vector containing polynomials which factors have the
1042 * degree given in degrees.
1044 static void distinct_degree_factor(const umodpoly& a_, vector<int>& degrees, upvec& ddfactors)
1048 cl_modint_ring R = a[0].ring();
1049 int q = cl_I_to_int(R->modulus);
1050 int nhalf = degree(a)/2;
1058 while ( i <= nhalf ) {
1063 umodpoly wx = w - x;
1065 if ( unequal_one(buf) ) {
1066 degrees.push_back(i);
1067 ddfactors.push_back(buf);
1069 if ( unequal_one(buf) ) {
1073 nhalf = degree(a)/2;
1079 if ( unequal_one(a) ) {
1080 degrees.push_back(degree(a));
1081 ddfactors.push_back(a);
1085 /** Modular same degree factorization.
1086 * Same degree factorization is a kind of misnomer. It performs distinct degree
1087 * factorization, but instead of using the Cantor-Zassenhaus algorithm it
1088 * (sub-optimally) uses Berlekamp's algorithm for the factors of the same
1091 * @param[in] a modular polynomial
1092 * @param[out] upv vector containing modular factors. if upv was not empty the
1093 * new elements are added at the end
1095 static void same_degree_factor(const umodpoly& a, upvec& upv)
1097 cl_modint_ring R = a[0].ring();
1099 vector<int> degrees;
1101 distinct_degree_factor(a, degrees, ddfactors);
1103 for ( size_t i=0; i<degrees.size(); ++i ) {
1104 if ( degrees[i] == degree(ddfactors[i]) ) {
1105 upv.push_back(ddfactors[i]);
1107 berlekamp(ddfactors[i], upv);
1112 // Yes, we can (choose).
1113 #define USE_SAME_DEGREE_FACTOR
1115 /** Modular univariate factorization.
1117 * In principle, we have two algorithms at our disposal: Berlekamp's algorithm
1118 * and same degree factorization (SDF). SDF seems to be slightly faster in
1119 * almost all cases so it is activated as default.
1121 * @param[in] p modular polynomial
1122 * @param[out] upv vector containing modular factors. if upv was not empty the
1123 * new elements are added at the end
1125 static void factor_modular(const umodpoly& p, upvec& upv)
1127 #ifdef USE_SAME_DEGREE_FACTOR
1128 same_degree_factor(p, upv);
1134 /** Calculates modular polynomials s and t such that a*s+b*t==1.
1135 * Assertion: a and b are relatively prime and not zero.
1137 * @param[in] a polynomial
1138 * @param[in] b polynomial
1139 * @param[out] s polynomial
1140 * @param[out] t polynomial
1142 static void exteuclid(const umodpoly& a, const umodpoly& b, umodpoly& s, umodpoly& t)
1144 if ( degree(a) < degree(b) ) {
1145 exteuclid(b, a, t, s);
1149 umodpoly one(1, a[0].ring()->one());
1150 umodpoly c = a; normalize_in_field(c);
1151 umodpoly d = b; normalize_in_field(d);
1159 umodpoly r = c - q * d;
1160 umodpoly r1 = s - q * d1;
1161 umodpoly r2 = t - q * d2;
1165 if ( r.empty() ) break;
1170 cl_MI fac = recip(lcoeff(a) * lcoeff(c));
1171 for (auto & i : s) {
1175 fac = recip(lcoeff(b) * lcoeff(c));
1176 for (auto & i : t) {
1182 /** Replaces the leading coefficient in a polynomial by a given number.
1184 * @param[in] poly polynomial to change
1185 * @param[in] lc new leading coefficient
1186 * @return changed polynomial
1188 static upoly replace_lc(const upoly& poly, const cl_I& lc)
1190 if ( poly.empty() ) return poly;
1196 /** Calculates the bound for the modulus.
1199 static inline cl_I calc_bound(const ex& a, const ex& x, int maxdeg)
1203 for ( int i=a.degree(x); i>=a.ldegree(x); --i ) {
1204 cl_I aa = abs(the<cl_I>(ex_to<numeric>(a.coeff(x, i)).to_cl_N()));
1205 if ( aa > maxcoeff ) maxcoeff = aa;
1206 coeff = coeff + square(aa);
1208 cl_I coeffnorm = ceiling1(the<cl_R>(cln::sqrt(coeff)));
1209 cl_I B = coeffnorm * expt_pos(cl_I(2), cl_I(maxdeg));
1210 return ( B > maxcoeff ) ? B : maxcoeff;
1213 /** Calculates the bound for the modulus.
1216 static inline cl_I calc_bound(const upoly& a, int maxdeg)
1220 for ( int i=degree(a); i>=0; --i ) {
1221 cl_I aa = abs(a[i]);
1222 if ( aa > maxcoeff ) maxcoeff = aa;
1223 coeff = coeff + square(aa);
1225 cl_I coeffnorm = ceiling1(the<cl_R>(cln::sqrt(coeff)));
1226 cl_I B = coeffnorm * expt_pos(cl_I(2), cl_I(maxdeg));
1227 return ( B > maxcoeff ) ? B : maxcoeff;
1230 /** Hensel lifting as used by factor_univariate().
1232 * The implementation follows the algorithm in chapter 6 of [GCL].
1234 * @param[in] a_ primitive univariate polynomials
1235 * @param[in] p prime number that does not divide lcoeff(a)
1236 * @param[in] u1_ modular factor of a (mod p)
1237 * @param[in] w1_ modular factor of a (mod p), relatively prime to u1_,
1238 * fulfilling u1_*w1_ == a mod p
1239 * @param[out] u lifted factor
1240 * @param[out] w lifted factor, u*w = a
1242 static void hensel_univar(const upoly& a_, unsigned int p, const umodpoly& u1_, const umodpoly& w1_, upoly& u, upoly& w)
1245 const cl_modint_ring& R = u1_[0].ring();
1248 int maxdeg = (degree(u1_) > degree(w1_)) ? degree(u1_) : degree(w1_);
1249 cl_I maxmodulus = 2*calc_bound(a, maxdeg);
1252 cl_I alpha = lcoeff(a);
1255 normalize_in_field(nu1);
1257 normalize_in_field(nw1);
1259 phi = umodpoly_to_upoly(nu1) * alpha;
1261 umodpoly_from_upoly(u1, phi, R);
1262 phi = umodpoly_to_upoly(nw1) * alpha;
1264 umodpoly_from_upoly(w1, phi, R);
1269 exteuclid(u1, w1, s, t);
1272 u = replace_lc(umodpoly_to_upoly(u1), alpha);
1273 w = replace_lc(umodpoly_to_upoly(w1), alpha);
1274 upoly e = a - u * w;
1278 while ( !e.empty() && modulus < maxmodulus ) {
1279 upoly c = e / modulus;
1280 phi = umodpoly_to_upoly(s) * c;
1281 umodpoly sigmatilde;
1282 umodpoly_from_upoly(sigmatilde, phi, R);
1283 phi = umodpoly_to_upoly(t) * c;
1285 umodpoly_from_upoly(tautilde, phi, R);
1287 remdiv(sigmatilde, w1, r, q);
1289 phi = umodpoly_to_upoly(tautilde) + umodpoly_to_upoly(q) * umodpoly_to_upoly(u1);
1291 umodpoly_from_upoly(tau, phi, R);
1292 u = u + umodpoly_to_upoly(tau) * modulus;
1293 w = w + umodpoly_to_upoly(sigma) * modulus;
1295 modulus = modulus * p;
1301 for ( size_t i=1; i<u.size(); ++i ) {
1303 if ( g == 1 ) break;
1317 /** Returns a new prime number.
1319 * @param[in] p prime number
1320 * @return next prime number after p
1322 static unsigned int next_prime(unsigned int p)
1324 static vector<unsigned int> primes;
1325 if (primes.empty()) {
1328 if ( p >= primes.back() ) {
1329 unsigned int candidate = primes.back() + 2;
1331 size_t n = primes.size()/2;
1332 for ( size_t i=0; i<n; ++i ) {
1333 if (candidate % primes[i])
1338 primes.push_back(candidate);
1344 for (auto & it : primes) {
1349 throw logic_error("next_prime: should not reach this point!");
1352 /** Manages the splitting a vector of of modular factors into two partitions.
1354 class factor_partition
1357 /** Takes the vector of modular factors and initializes the first partition */
1358 factor_partition(const upvec& factors_) : factors(factors_)
1364 one.resize(1, factors.front()[0].ring()->one());
1369 int operator[](size_t i) const { return k[i]; }
1370 size_t size() const { return n; }
1371 size_t size_left() const { return n-len; }
1372 size_t size_right() const { return len; }
1373 /** Initializes the next partition.
1374 Returns true, if there is one, false otherwise. */
1377 if ( last == n-1 ) {
1387 while ( k[last] == 0 ) { --last; }
1388 if ( last == 0 && n == 2*len ) return false;
1390 for ( size_t i=0; i<=len-rem; ++i ) {
1394 fill(k.begin()+last, k.end(), 0);
1401 if ( len > n/2 ) return false;
1402 fill(k.begin(), k.begin()+len, 1);
1403 fill(k.begin()+len+1, k.end(), 0);
1411 /** Get first partition */
1412 umodpoly& left() { return lr[0]; }
1413 /** Get second partition */
1414 umodpoly& right() { return lr[1]; }
1423 while ( i < n && k[i] == group ) { ++d; ++i; }
1425 if ( cache[pos].size() >= d ) {
1426 lr[group] = lr[group] * cache[pos][d-1];
1428 if ( cache[pos].size() == 0 ) {
1429 cache[pos].push_back(factors[pos] * factors[pos+1]);
1431 size_t j = pos + cache[pos].size() + 1;
1432 d -= cache[pos].size();
1434 umodpoly buf = cache[pos].back() * factors[j];
1435 cache[pos].push_back(buf);
1439 lr[group] = lr[group] * cache[pos].back();
1442 lr[group] = lr[group] * factors[pos];
1453 for ( size_t i=0; i<n; ++i ) {
1454 lr[k[i]] = lr[k[i]] * factors[i];
1460 vector<vector<umodpoly>> cache;
1469 /** Contains a pair of univariate polynomial and its modular factors.
1470 * Used by factor_univariate().
1478 /** Univariate polynomial factorization.
1480 * Modular factorization is tried for several primes to minimize the number of
1481 * modular factors. Then, Hensel lifting is performed.
1483 * @param[in] poly expanded square free univariate polynomial
1484 * @param[in] x symbol
1485 * @param[in,out] prime prime number to start trying modular factorization with,
1486 * output value is the prime number actually used
1488 static ex factor_univariate(const ex& poly, const ex& x, unsigned int& prime)
1490 ex unit, cont, prim_ex;
1491 poly.unitcontprim(x, unit, cont, prim_ex);
1493 upoly_from_ex(prim, prim_ex, x);
1495 // determine proper prime and minimize number of modular factors
1497 unsigned int lastp = prime;
1499 unsigned int trials = 0;
1500 unsigned int minfactors = 0;
1502 const numeric& cont_n = ex_to<numeric>(cont);
1504 if (cont_n.is_integer()) {
1505 i_cont = the<cl_I>(cont_n.to_cl_N());
1507 // poly \in Q[x] => poly = q ipoly, ipoly \in Z[x], q \in Q
1508 // factor(poly) \equiv q factor(ipoly)
1511 cl_I lc = lcoeff(prim)*i_cont;
1513 while ( trials < 2 ) {
1516 prime = next_prime(prime);
1517 if ( !zerop(rem(lc, prime)) ) {
1518 R = find_modint_ring(prime);
1519 umodpoly_from_upoly(modpoly, prim, R);
1520 if ( squarefree(modpoly) ) break;
1524 // do modular factorization
1526 factor_modular(modpoly, trialfactors);
1527 if ( trialfactors.size() <= 1 ) {
1528 // irreducible for sure
1532 if ( minfactors == 0 || trialfactors.size() < minfactors ) {
1533 factors = trialfactors;
1534 minfactors = trialfactors.size();
1542 R = find_modint_ring(prime);
1544 // lift all factor combinations
1545 stack<ModFactors> tocheck;
1548 mf.factors = factors;
1552 while ( tocheck.size() ) {
1553 const size_t n = tocheck.top().factors.size();
1554 factor_partition part(tocheck.top().factors);
1556 // call Hensel lifting
1557 hensel_univar(tocheck.top().poly, prime, part.left(), part.right(), f1, f2);
1558 if ( !f1.empty() ) {
1559 // successful, update the stack and the result
1560 if ( part.size_left() == 1 ) {
1561 if ( part.size_right() == 1 ) {
1562 result *= upoly_to_ex(f1, x) * upoly_to_ex(f2, x);
1566 result *= upoly_to_ex(f1, x);
1567 tocheck.top().poly = f2;
1568 for ( size_t i=0; i<n; ++i ) {
1569 if ( part[i] == 0 ) {
1570 tocheck.top().factors.erase(tocheck.top().factors.begin()+i);
1576 else if ( part.size_right() == 1 ) {
1577 if ( part.size_left() == 1 ) {
1578 result *= upoly_to_ex(f1, x) * upoly_to_ex(f2, x);
1582 result *= upoly_to_ex(f2, x);
1583 tocheck.top().poly = f1;
1584 for ( size_t i=0; i<n; ++i ) {
1585 if ( part[i] == 1 ) {
1586 tocheck.top().factors.erase(tocheck.top().factors.begin()+i);
1592 upvec newfactors1(part.size_left()), newfactors2(part.size_right());
1593 auto i1 = newfactors1.begin(), i2 = newfactors2.begin();
1594 for ( size_t i=0; i<n; ++i ) {
1596 *i2++ = tocheck.top().factors[i];
1598 *i1++ = tocheck.top().factors[i];
1601 tocheck.top().factors = newfactors1;
1602 tocheck.top().poly = f1;
1604 mf.factors = newfactors2;
1611 if ( !part.next() ) {
1612 // if no more combinations left, return polynomial as
1614 result *= upoly_to_ex(tocheck.top().poly, x);
1622 return unit * cont * result;
1625 /** Second interface to factor_univariate() to be used if the information about
1626 * the prime is not needed.
1628 static inline ex factor_univariate(const ex& poly, const ex& x)
1631 return factor_univariate(poly, x, prime);
1634 /** Represents an evaluation point (<symbol>==<integer>).
1643 ostream& operator<<(ostream& o, const vector<EvalPoint>& v)
1645 for ( size_t i=0; i<v.size(); ++i ) {
1646 o << "(" << v[i].x << "==" << v[i].evalpoint << ") ";
1650 #endif // def DEBUGFACTOR
1652 // forward declaration
1653 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);
1655 /** Utility function for multivariate Hensel lifting.
1657 * Solves the equation
1658 * s_1*b_1 + ... + s_r*b_r == 1 mod p^k
1659 * with deg(s_i) < deg(a_i)
1660 * and with given b_1 = a_1 * ... * a_{i-1} * a_{i+1} * ... * a_r
1662 * The implementation follows the algorithm in chapter 6 of [GCL].
1664 * @param[in] a vector of modular univariate polynomials
1665 * @param[in] x symbol
1666 * @param[in] p prime number
1667 * @param[in] k p^k is modulus
1668 * @return vector of polynomials (s_i)
1670 static upvec multiterm_eea_lift(const upvec& a, const ex& x, unsigned int p, unsigned int k)
1672 const size_t r = a.size();
1673 cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
1676 for ( size_t j=r-2; j>=1; --j ) {
1677 q[j-1] = a[j] * q[j];
1679 umodpoly beta(1, R->one());
1681 for ( size_t j=1; j<r; ++j ) {
1682 vector<ex> mdarg(2);
1683 mdarg[0] = umodpoly_to_ex(q[j-1], x);
1684 mdarg[1] = umodpoly_to_ex(a[j-1], x);
1685 vector<EvalPoint> empty;
1686 vector<ex> exsigma = multivar_diophant(mdarg, x, umodpoly_to_ex(beta, x), empty, 0, p, k);
1688 umodpoly_from_ex(sigma1, exsigma[0], x, R);
1690 umodpoly_from_ex(sigma2, exsigma[1], x, R);
1692 s.push_back(sigma2);
1698 /** Changes the modulus of a modular polynomial. Used by eea_lift().
1700 * @param[in] R new modular ring
1701 * @param[in,out] a polynomial to change (in situ)
1703 static void change_modulus(const cl_modint_ring& R, umodpoly& a)
1705 if ( a.empty() ) return;
1706 cl_modint_ring oldR = a[0].ring();
1707 for (auto & i : a) {
1708 i = R->canonhom(oldR->retract(i));
1713 /** Utility function for multivariate Hensel lifting.
1715 * Solves s*a + t*b == 1 mod p^k given a,b.
1717 * The implementation follows the algorithm in chapter 6 of [GCL].
1719 * @param[in] a polynomial
1720 * @param[in] b polynomial
1721 * @param[in] x symbol
1722 * @param[in] p prime number
1723 * @param[in] k p^k is modulus
1724 * @param[out] s_ output polynomial
1725 * @param[out] t_ output polynomial
1727 static void eea_lift(const umodpoly& a, const umodpoly& b, const ex& x, unsigned int p, unsigned int k, umodpoly& s_, umodpoly& t_)
1729 cl_modint_ring R = find_modint_ring(p);
1731 change_modulus(R, amod);
1733 change_modulus(R, bmod);
1737 exteuclid(amod, bmod, smod, tmod);
1739 cl_modint_ring Rpk = find_modint_ring(expt_pos(cl_I(p),k));
1741 change_modulus(Rpk, s);
1743 change_modulus(Rpk, t);
1746 umodpoly one(1, Rpk->one());
1747 for ( size_t j=1; j<k; ++j ) {
1748 umodpoly e = one - a * s - b * t;
1749 reduce_coeff(e, modulus);
1751 change_modulus(R, c);
1752 umodpoly sigmabar = smod * c;
1753 umodpoly taubar = tmod * c;
1755 remdiv(sigmabar, bmod, sigma, q);
1756 umodpoly tau = taubar + q * amod;
1757 umodpoly sadd = sigma;
1758 change_modulus(Rpk, sadd);
1759 cl_MI modmodulus(Rpk, modulus);
1760 s = s + sadd * modmodulus;
1761 umodpoly tadd = tau;
1762 change_modulus(Rpk, tadd);
1763 t = t + tadd * modmodulus;
1764 modulus = modulus * p;
1770 /** Utility function for multivariate Hensel lifting.
1772 * Solves the equation
1773 * s_1*b_1 + ... + s_r*b_r == x^m mod p^k
1774 * with given b_1 = a_1 * ... * a_{i-1} * a_{i+1} * ... * a_r
1776 * The implementation follows the algorithm in chapter 6 of [GCL].
1778 * @param a vector with univariate polynomials mod p^k
1780 * @param m exponent of x^m in the equation to solve
1781 * @param p prime number
1782 * @param k p^k is modulus
1783 * @return vector of polynomials (s_i)
1785 static upvec univar_diophant(const upvec& a, const ex& x, unsigned int m, unsigned int p, unsigned int k)
1787 cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
1789 const size_t r = a.size();
1792 upvec s = multiterm_eea_lift(a, x, p, k);
1793 for ( size_t j=0; j<r; ++j ) {
1794 umodpoly bmod = umodpoly_to_umodpoly(s[j], R, m);
1796 rem(bmod, a[j], buf);
1797 result.push_back(buf);
1801 eea_lift(a[1], a[0], x, p, k, s, t);
1802 umodpoly bmod = umodpoly_to_umodpoly(s, R, m);
1804 remdiv(bmod, a[0], buf, q);
1805 result.push_back(buf);
1806 umodpoly t1mod = umodpoly_to_umodpoly(t, R, m);
1807 buf = t1mod + q * a[1];
1808 result.push_back(buf);
1814 /** Map used by function make_modular().
1815 * Finds every coefficient in a polynomial and replaces it by is value in the
1816 * given modular ring R (symmetric representation).
1818 struct make_modular_map : public map_function {
1820 make_modular_map(const cl_modint_ring& R_) : R(R_) { }
1821 ex operator()(const ex& e) override
1823 if ( is_a<add>(e) || is_a<mul>(e) ) {
1824 return e.map(*this);
1826 else if ( is_a<numeric>(e) ) {
1827 numeric mod(R->modulus);
1828 numeric halfmod = (mod-1)/2;
1829 cl_MI emod = R->canonhom(the<cl_I>(ex_to<numeric>(e).to_cl_N()));
1830 numeric n(R->retract(emod));
1831 if ( n > halfmod ) {
1841 /** Helps mimicking modular multivariate polynomial arithmetic.
1843 * @param e expression of which to make the coefficients equal to their value
1844 * in the modular ring R (symmetric representation)
1845 * @param R modular ring
1846 * @return resulting expression
1848 static ex make_modular(const ex& e, const cl_modint_ring& R)
1850 make_modular_map map(R);
1851 return map(e.expand());
1854 /** Utility function for multivariate Hensel lifting.
1856 * Returns the polynomials s_i that fulfill
1857 * s_1*b_1 + ... + s_r*b_r == c mod <I^(d+1),p^k>
1858 * with given b_1 = a_1 * ... * a_{i-1} * a_{i+1} * ... * a_r
1860 * The implementation follows the algorithm in chapter 6 of [GCL].
1862 * @param a_ vector of multivariate factors mod p^k
1863 * @param x symbol (equiv. x_1 in [GCL])
1864 * @param c polynomial mod p^k
1865 * @param I vector of evaluation points
1866 * @param d maximum total degree of result
1867 * @param p prime number
1868 * @param k p^k is modulus
1869 * @return vector of polynomials (s_i)
1871 static vector<ex> multivar_diophant(const vector<ex>& a_, const ex& x, const ex& c, const vector<EvalPoint>& I,
1872 unsigned int d, unsigned int p, unsigned int k)
1876 const cl_I modulus = expt_pos(cl_I(p),k);
1877 const cl_modint_ring R = find_modint_ring(modulus);
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);
1950 cl_I poscm = plusp(cm) ? cm : mod(cm, modulus);
1951 modcm = cl_MI(R, poscm);
1952 for ( size_t j=0; j<delta_s.size(); ++j ) {
1953 delta_s[j] = delta_s[j] * modcm;
1954 sigma[j] = sigma[j] + umodpoly_to_ex(delta_s[j], x);
1962 for ( size_t i=0; i<sigma.size(); ++i ) {
1963 sigma[i] = make_modular(sigma[i], R);
1969 /** Multivariate Hensel lifting.
1970 * The implementation follows the algorithm in chapter 6 of [GCL].
1971 * Since we don't have a data type for modular multivariate polynomials, the
1972 * respective operations are done in a GiNaC::ex and the function
1973 * make_modular() is then called to make the coefficient modular p^l.
1975 * @param a multivariate polynomial primitive in x
1976 * @param x symbol (equiv. x_1 in [GCL])
1977 * @param I vector of evaluation points (x_2==a_2,x_3==a_3,...)
1978 * @param p prime number (should not divide lcoeff(a mod I))
1979 * @param l p^l is the modulus of the lifted univariate field
1980 * @param u vector of modular (mod p^l) factors of a mod I
1981 * @param lcU correct leading coefficient of the univariate factors of a mod I
1982 * @return list GiNaC::lst with lifted factors (multivariate factors of a),
1983 * empty if Hensel lifting did not succeed
1985 static ex hensel_multivar(const ex& a, const ex& x, const vector<EvalPoint>& I,
1986 unsigned int p, const cl_I& l, const upvec& u, const vector<ex>& lcU)
1988 const size_t nu = I.size() + 1;
1989 const cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),l));
1994 for ( size_t j=nu; j>=2; --j ) {
1996 int alpha = I[j-2].evalpoint;
1997 A[j-2] = A[j-1].subs(x==alpha);
1998 A[j-2] = make_modular(A[j-2], R);
2001 int maxdeg = a.degree(I.front().x);
2002 for ( size_t i=1; i<I.size(); ++i ) {
2003 int maxdeg2 = a.degree(I[i].x);
2004 if ( maxdeg2 > maxdeg ) maxdeg = maxdeg2;
2007 const size_t n = u.size();
2009 for ( size_t i=0; i<n; ++i ) {
2010 U[i] = umodpoly_to_ex(u[i], x);
2013 for ( size_t j=2; j<=nu; ++j ) {
2016 for ( size_t m=0; m<n; ++m) {
2017 if ( lcU[m] != 1 ) {
2019 for ( size_t i=j-1; i<nu-1; ++i ) {
2020 coef = coef.subs(I[i].x == I[i].evalpoint);
2022 coef = make_modular(coef, R);
2023 int deg = U[m].degree(x);
2024 U[m] = U[m] - U[m].lcoeff(x) * pow(x,deg) + coef * pow(x,deg);
2028 for ( size_t i=0; i<n; ++i ) {
2031 ex e = expand(A[j-1] - Uprod);
2033 vector<EvalPoint> newI;
2034 for ( size_t i=1; i<=j-2; ++i ) {
2035 newI.push_back(I[i-1]);
2039 int alphaj = I[j-2].evalpoint;
2040 size_t deg = A[j-1].degree(xj);
2041 for ( size_t k=1; k<=deg; ++k ) {
2042 if ( !e.is_zero() ) {
2043 monomial *= (xj - alphaj);
2044 monomial = expand(monomial);
2045 ex dif = e.diff(ex_to<symbol>(xj), k);
2046 ex c = dif.subs(xj==alphaj) / factorial(k);
2047 if ( !c.is_zero() ) {
2048 vector<ex> deltaU = multivar_diophant(U1, x, c, newI, maxdeg, p, cl_I_to_uint(l));
2049 for ( size_t i=0; i<n; ++i ) {
2050 deltaU[i] *= monomial;
2052 U[i] = make_modular(U[i], R);
2055 for ( size_t i=0; i<n; ++i ) {
2059 e = make_modular(e, R);
2066 for ( size_t i=0; i<U.size(); ++i ) {
2069 if ( expand(a-acand).is_zero() ) {
2070 return lst(U.begin(), U.end());
2076 /** Takes a factorized expression and puts the factors in a lst. The exponents
2077 * of the factors are discarded, e.g. 7*x^2*(y+1)^4 --> {7,x,y+1}. The first
2078 * element of the list is always the numeric coefficient.
2080 static ex put_factors_into_lst(const ex& e)
2083 if ( is_a<numeric>(e) ) {
2087 if ( is_a<power>(e) ) {
2089 result.append(e.op(0));
2092 if ( is_a<symbol>(e) || is_a<add>(e) ) {
2093 ex icont(e.integer_content());
2094 result.append(icont);
2095 result.append(e/icont);
2098 if ( is_a<mul>(e) ) {
2100 for ( size_t i=0; i<e.nops(); ++i ) {
2102 if ( is_a<numeric>(op) ) {
2105 if ( is_a<power>(op) ) {
2106 result.append(op.op(0));
2108 if ( is_a<symbol>(op) || is_a<add>(op) ) {
2112 result.prepend(nfac);
2115 throw runtime_error("put_factors_into_lst: bad term.");
2118 /** Checks a set of numbers for whether each number has a unique prime factor.
2120 * @param[in] f list of numbers to check
2121 * @return true: if number set is bad, false: if set is okay (has unique
2124 static bool checkdivisors(const lst& f)
2126 const int k = f.nops();
2128 vector<numeric> d(k);
2129 d[0] = ex_to<numeric>(abs(f.op(0)));
2130 for ( int i=1; i<k; ++i ) {
2131 q = ex_to<numeric>(abs(f.op(i)));
2132 for ( int j=i-1; j>=0; --j ) {
2147 /** Generates a set of evaluation points for a multivariate polynomial.
2148 * The set fulfills the following conditions:
2149 * 1. lcoeff(evaluated_polynomial) does not vanish
2150 * 2. factors of lcoeff(evaluated_polynomial) have each a unique prime factor
2151 * 3. evaluated_polynomial is square free
2152 * See [Wan] for more details.
2154 * @param[in] u multivariate polynomial to be factored
2155 * @param[in] vn leading coefficient of u in x (x==first symbol in syms)
2156 * @param[in] syms set of symbols that appear in u
2157 * @param[in] f lst containing the factors of the leading coefficient vn
2158 * @param[in,out] modulus integer modulus for random number generation (i.e. |a_i| < modulus)
2159 * @param[out] u0 returns the evaluated (univariate) polynomial
2160 * @param[out] a returns the valid evaluation points. must have initial size equal
2161 * number of symbols-1 before calling generate_set
2163 static void generate_set(const ex& u, const ex& vn, const exset& syms, const lst& f,
2164 numeric& modulus, ex& u0, vector<numeric>& a)
2166 const ex& x = *syms.begin();
2169 // generate a set of integers ...
2173 exset::const_iterator s = syms.begin();
2175 for ( size_t i=0; i<a.size(); ++i ) {
2177 a[i] = mod(numeric(rand()), 2*modulus) - modulus;
2178 vnatry = vna.subs(*s == a[i]);
2179 // ... for which the leading coefficient doesn't vanish ...
2180 } while ( vnatry == 0 );
2182 u0 = u0.subs(*s == a[i]);
2185 // ... for which u0 is square free ...
2186 ex g = gcd(u0, u0.diff(ex_to<symbol>(x)));
2187 if ( !is_a<numeric>(g) ) {
2190 if ( !is_a<numeric>(vn) ) {
2191 // ... and for which the evaluated factors have each an unique prime factor
2193 fnum.let_op(0) = fnum.op(0) * u0.content(x);
2194 for ( size_t i=1; i<fnum.nops(); ++i ) {
2195 if ( !is_a<numeric>(fnum.op(i)) ) {
2198 for ( size_t j=0; j<a.size(); ++j, ++s ) {
2199 fnum.let_op(i) = fnum.op(i).subs(*s == a[j]);
2203 if ( checkdivisors(fnum) ) {
2207 // ok, we have a valid set now
2212 // forward declaration
2213 static ex factor_sqrfree(const ex& poly);
2215 /** Multivariate factorization.
2217 * The implementation is based on the algorithm described in [Wan].
2218 * An evaluation homomorphism (a set of integers) is determined that fulfills
2219 * certain criteria. The evaluated polynomial is univariate and is factorized
2220 * by factor_univariate(). The main work then is to find the correct leading
2221 * coefficients of the univariate factors. They have to correspond to the
2222 * factors of the (multivariate) leading coefficient of the input polynomial
2223 * (as defined for a specific variable x). After that the Hensel lifting can be
2226 * @param[in] poly expanded, square free polynomial
2227 * @param[in] syms contains the symbols in the polynomial
2228 * @return factorized polynomial
2230 static ex factor_multivariate(const ex& poly, const exset& syms)
2232 exset::const_iterator s;
2233 const ex& x = *syms.begin();
2235 // make polynomial primitive
2237 poly.unitcontprim(x, unit, cont, pp);
2238 if ( !is_a<numeric>(cont) ) {
2239 return factor_sqrfree(cont) * factor_sqrfree(pp);
2242 // factor leading coefficient
2243 ex vn = pp.collect(x).lcoeff(x);
2245 if ( is_a<numeric>(vn) ) {
2249 ex vnfactors = factor(vn);
2250 vnlst = put_factors_into_lst(vnfactors);
2253 const unsigned int maxtrials = 3;
2254 numeric modulus = (vnlst.nops() > 3) ? vnlst.nops() : 3;
2255 vector<numeric> a(syms.size()-1, 0);
2257 // try now to factorize until we are successful
2260 unsigned int trialcount = 0;
2262 int factor_count = 0;
2263 int min_factor_count = -1;
2267 // try several evaluation points to reduce the number of factors
2268 while ( trialcount < maxtrials ) {
2270 // generate a set of valid evaluation points
2271 generate_set(pp, vn, syms, ex_to<lst>(vnlst), modulus, u, a);
2273 ufac = factor_univariate(u, x, prime);
2274 ufaclst = put_factors_into_lst(ufac);
2275 factor_count = ufaclst.nops()-1;
2276 delta = ufaclst.op(0);
2278 if ( factor_count <= 1 ) {
2282 if ( min_factor_count < 0 ) {
2284 min_factor_count = factor_count;
2286 else if ( min_factor_count == factor_count ) {
2290 else if ( min_factor_count > factor_count ) {
2291 // new minimum, reset trial counter
2292 min_factor_count = factor_count;
2297 // determine true leading coefficients for the Hensel lifting
2298 vector<ex> C(factor_count);
2299 if ( is_a<numeric>(vn) ) {
2301 for ( size_t i=1; i<ufaclst.nops(); ++i ) {
2302 C[i-1] = ufaclst.op(i).lcoeff(x);
2306 // we use the property of the ftilde having a unique prime factor.
2307 // details can be found in [Wan].
2309 vector<numeric> ftilde(vnlst.nops()-1);
2310 for ( size_t i=0; i<ftilde.size(); ++i ) {
2311 ex ft = vnlst.op(i+1);
2314 for ( size_t j=0; j<a.size(); ++j ) {
2315 ft = ft.subs(*s == a[j]);
2318 ftilde[i] = ex_to<numeric>(ft);
2320 // calculate D and C
2321 vector<bool> used_flag(ftilde.size(), false);
2322 vector<ex> D(factor_count, 1);
2324 for ( int i=0; i<factor_count; ++i ) {
2325 numeric prefac = ex_to<numeric>(ufaclst.op(i+1).lcoeff(x));
2326 for ( int j=ftilde.size()-1; j>=0; --j ) {
2328 while ( irem(prefac, ftilde[j]) == 0 ) {
2329 prefac = iquo(prefac, ftilde[j]);
2333 used_flag[j] = true;
2334 D[i] = D[i] * pow(vnlst.op(j+1), count);
2337 C[i] = D[i] * prefac;
2340 for ( int i=0; i<factor_count; ++i ) {
2341 numeric prefac = ex_to<numeric>(ufaclst.op(i+1).lcoeff(x));
2342 for ( int j=ftilde.size()-1; j>=0; --j ) {
2344 while ( irem(prefac, ftilde[j]) == 0 ) {
2345 prefac = iquo(prefac, ftilde[j]);
2348 while ( irem(ex_to<numeric>(delta)*prefac, ftilde[j]) == 0 ) {
2349 numeric g = gcd(prefac, ex_to<numeric>(ftilde[j]));
2350 prefac = iquo(prefac, g);
2351 delta = delta / (ftilde[j]/g);
2352 ufaclst.let_op(i+1) = ufaclst.op(i+1) * (ftilde[j]/g);
2356 used_flag[j] = true;
2357 D[i] = D[i] * pow(vnlst.op(j+1), count);
2360 C[i] = D[i] * prefac;
2363 // check if something went wrong
2364 bool some_factor_unused = false;
2365 for ( size_t i=0; i<used_flag.size(); ++i ) {
2366 if ( !used_flag[i] ) {
2367 some_factor_unused = true;
2371 if ( some_factor_unused ) {
2376 // multiply the remaining content of the univariate polynomial into the
2379 C[0] = C[0] * delta;
2380 ufaclst.let_op(1) = ufaclst.op(1) * delta;
2383 // set up evaluation points
2385 vector<EvalPoint> epv;
2388 for ( size_t i=0; i<a.size(); ++i ) {
2390 ep.evalpoint = a[i].to_int();
2396 for ( int i=1; i<=factor_count; ++i ) {
2397 if ( ufaclst.op(i).degree(x) > maxdeg ) {
2398 maxdeg = ufaclst[i].degree(x);
2401 cl_I B = 2*calc_bound(u, x, maxdeg);
2409 // set up modular factors (mod p^l)
2410 cl_modint_ring R = find_modint_ring(expt_pos(cl_I(prime),l));
2411 upvec modfactors(ufaclst.nops()-1);
2412 for ( size_t i=1; i<ufaclst.nops(); ++i ) {
2413 umodpoly_from_ex(modfactors[i-1], ufaclst.op(i), x, R);
2416 // try Hensel lifting
2417 ex res = hensel_multivar(pp, x, epv, prime, l, modfactors, C);
2418 if ( res != lst{} ) {
2419 ex result = cont * unit;
2420 for ( size_t i=0; i<res.nops(); ++i ) {
2421 result *= res.op(i).content(x) * res.op(i).unit(x);
2422 result *= res.op(i).primpart(x);
2429 /** Finds all symbols in an expression. Used by factor_sqrfree() and factor().
2431 struct find_symbols_map : public map_function {
2433 ex operator()(const ex& e) override
2435 if ( is_a<symbol>(e) ) {
2439 return e.map(*this);
2443 /** Factorizes a polynomial that is square free. It calls either the univariate
2444 * or the multivariate factorization functions.
2446 static ex factor_sqrfree(const ex& poly)
2448 // determine all symbols in poly
2449 find_symbols_map findsymbols;
2451 if ( findsymbols.syms.size() == 0 ) {
2455 if ( findsymbols.syms.size() == 1 ) {
2457 const ex& x = *(findsymbols.syms.begin());
2458 if ( poly.ldegree(x) > 0 ) {
2459 // pull out direct factors
2460 int ld = poly.ldegree(x);
2461 ex res = factor_univariate(expand(poly/pow(x, ld)), x);
2462 return res * pow(x,ld);
2464 ex res = factor_univariate(poly, x);
2469 // multivariate case
2470 ex res = factor_multivariate(poly, findsymbols.syms);
2474 /** Map used by factor() when factor_options::all is given to access all
2475 * subexpressions and to call factor() on them.
2477 struct apply_factor_map : public map_function {
2479 apply_factor_map(unsigned options_) : options(options_) { }
2480 ex operator()(const ex& e) override
2482 if ( e.info(info_flags::polynomial) ) {
2483 return factor(e, options);
2485 if ( is_a<add>(e) ) {
2487 for ( size_t i=0; i<e.nops(); ++i ) {
2488 if ( e.op(i).info(info_flags::polynomial) ) {
2494 return factor(s1, options) + s2.map(*this);
2496 return e.map(*this);
2500 /** Iterate through explicit factors of e, call yield(f, k) for
2501 * each factor of the form f^k.
2503 * Note that this function doesn't factor e itself, it only
2504 * iterates through the factors already explicitly present.
2506 template <typename F> void
2507 factor_iter(const ex &e, F yield)
2510 for (const auto &f : e) {
2511 if (is_a<power>(f)) {
2512 yield(f.op(0), f.op(1));
2518 if (is_a<power>(e)) {
2519 yield(e.op(0), e.op(1));
2526 /** This function factorizes a polynomial. It checks the arguments,
2527 * tries a square free factorization, and then calls factor_sqrfree
2528 * to do the hard work.
2530 * This function expands its argument, so for polynomials with
2531 * explicit factors it's better to call it on each one separately
2532 * (or use factor() which does just that).
2534 static ex factor1(const ex& poly, unsigned options)
2537 if ( !poly.info(info_flags::polynomial) ) {
2538 if ( options & factor_options::all ) {
2539 options &= ~factor_options::all;
2540 apply_factor_map factor_map(options);
2541 return factor_map(poly);
2546 // determine all symbols in poly
2547 find_symbols_map findsymbols;
2549 if ( findsymbols.syms.size() == 0 ) {
2553 for (auto & i : findsymbols.syms ) {
2557 // make poly square free
2558 ex sfpoly = sqrfree(poly.expand(), syms);
2560 // factorize the square free components
2563 [&](const ex &f, const ex &k) {
2564 if ( is_a<add>(f) ) {
2565 res *= pow(factor_sqrfree(f), k);
2567 // simple case: (monomial)^exponent
2574 } // anonymous namespace
2576 /** Interface function to the outside world. It uses factor1()
2577 * on each of the explicitly present factors of poly.
2579 ex factor(const ex& poly, unsigned options)
2583 [&](const ex &f1, const ex &k1) {
2584 factor_iter(factor1(f1, options),
2585 [&](const ex &f2, const ex &k2) {
2586 result *= pow(f2, k1*k2);
2592 } // namespace GiNaC