canonicalize(ump);
}
+#ifdef DEBUGFACTOR
static void umodpoly_from_ex(umodpoly& ump, const ex& e, const ex& x, const cl_I& modulus)
{
umodpoly_from_ex(ump, e, x, find_modint_ring(modulus));
}
+#endif
static ex upoly_to_ex(const upoly& a, const ex& x)
{
upvec factors;
};
-static ex factor_univariate(const ex& poly, const ex& x)
+static ex factor_univariate(const ex& poly, const ex& x, unsigned int& prime)
{
ex unit, cont, prim_ex;
poly.unitcontprim(x, unit, cont, prim_ex);
upoly_from_ex(prim, prim_ex, x);
// determine proper prime and minimize number of modular factors
- unsigned int p = 3, lastp = 3;
+ prime = 3;
+ unsigned int lastp = prime;
cl_modint_ring R;
unsigned int trials = 0;
unsigned int minfactors = 0;
- cl_I lc = lcoeff(prim);
+ cl_I lc = lcoeff(prim) * the<cl_I>(ex_to<numeric>(cont).to_cl_N());
upvec factors;
while ( trials < 2 ) {
umodpoly modpoly;
while ( true ) {
- p = next_prime(p);
- if ( !zerop(rem(lc, p)) ) {
- R = find_modint_ring(p);
+ prime = next_prime(prime);
+ if ( !zerop(rem(lc, prime)) ) {
+ R = find_modint_ring(prime);
umodpoly_from_upoly(modpoly, prim, R);
if ( squarefree(modpoly) ) break;
}
if ( minfactors == 0 || trialfactors.size() < minfactors ) {
factors = trialfactors;
minfactors = trialfactors.size();
- lastp = p;
+ lastp = prime;
trials = 1;
}
else {
++trials;
}
}
- p = lastp;
- R = find_modint_ring(p);
+ prime = lastp;
+ R = find_modint_ring(prime);
// lift all factor combinations
stack<ModFactors> tocheck;
const size_t n = tocheck.top().factors.size();
factor_partition part(tocheck.top().factors);
while ( true ) {
- hensel_univar(tocheck.top().poly, p, part.left(), part.right(), f1, f2);
+ hensel_univar(tocheck.top().poly, prime, part.left(), part.right(), f1, f2);
if ( !f1.empty() ) {
if ( part.size_left() == 1 ) {
if ( part.size_right() == 1 ) {
return unit * cont * result;
}
+static inline ex factor_univariate(const ex& poly, const ex& x)
+{
+ unsigned int prime;
+ return factor_univariate(poly, x, prime);
+}
+
struct EvalPoint
{
ex x;
};
// forward declaration
-vector<ex> multivar_diophant(const vector<ex>& a_, const ex& x, const ex& c, const vector<EvalPoint>& I, unsigned int d, unsigned int p, unsigned int k);
+static vector<ex> multivar_diophant(const vector<ex>& a_, const ex& x, const ex& c, const vector<EvalPoint>& I, unsigned int d, unsigned int p, unsigned int k);
-upvec multiterm_eea_lift(const upvec& a, const ex& x, unsigned int p, unsigned int k)
+static upvec multiterm_eea_lift(const upvec& a, const ex& x, unsigned int p, unsigned int k)
{
const size_t r = a.size();
cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
/**
* Assert: a not empty.
*/
-void change_modulus(const cl_modint_ring& R, umodpoly& a)
+static void change_modulus(const cl_modint_ring& R, umodpoly& a)
{
if ( a.empty() ) return;
cl_modint_ring oldR = a[0].ring();
canonicalize(a);
}
-void eea_lift(const umodpoly& a, const umodpoly& b, const ex& x, unsigned int p, unsigned int k, umodpoly& s_, umodpoly& t_)
+static void eea_lift(const umodpoly& a, const umodpoly& b, const ex& x, unsigned int p, unsigned int k, umodpoly& s_, umodpoly& t_)
{
cl_modint_ring R = find_modint_ring(p);
umodpoly amod = a;
s_ = s; t_ = t;
}
-upvec univar_diophant(const upvec& a, const ex& x, unsigned int m, unsigned int p, unsigned int k)
+static upvec univar_diophant(const upvec& a, const ex& x, unsigned int m, unsigned int p, unsigned int k)
{
cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
return map(e.expand());
}
-vector<ex> multivar_diophant(const vector<ex>& a_, const ex& x, const ex& c, const vector<EvalPoint>& I, unsigned int d, unsigned int p, unsigned int k)
+static vector<ex> multivar_diophant(const vector<ex>& a_, const ex& x, const ex& c, const vector<EvalPoint>& I,
+ unsigned int d, unsigned int p, unsigned int k)
{
vector<ex> a = a_;
}
#endif // def DEBUGFACTOR
-ex hensel_multivar(const ex& a, const ex& x, const vector<EvalPoint>& I, unsigned int p, const cl_I& l, const upvec& u, const vector<ex>& lcU)
+static ex hensel_multivar(const ex& a, const ex& x, const vector<EvalPoint>& I, unsigned int p, const cl_I& l, const upvec& u, const vector<ex>& lcU)
{
const size_t nu = I.size() + 1;
const cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),l));
if ( is_a<power>(e) ) {
result.append(1);
result.append(e.op(0));
- result.append(e.op(1));
return result;
}
if ( is_a<symbol>(e) || is_a<add>(e) ) {
result.append(1);
result.append(e);
- result.append(1);
return result;
}
if ( is_a<mul>(e) ) {
}
if ( is_a<power>(op) ) {
result.append(op.op(0));
- result.append(op.op(1));
}
if ( is_a<symbol>(op) || is_a<add>(op) ) {
result.append(op);
- result.append(1);
}
}
result.prepend(nfac);
}
#endif // def DEBUGFACTOR
-static bool checkdivisors(const lst& f, vector<numeric>& d)
+/** Checks whether in a set of numbers each has a unique prime factor.
+ *
+ * @param[in] f list of numbers to check
+ * @return true: if number set is bad, false: otherwise
+ */
+static bool checkdivisors(const lst& f)
{
- const int k = f.nops()-2;
+ const int k = f.nops();
numeric q, r;
- d[0] = ex_to<numeric>(f.op(0) * f.op(f.nops()-1));
- if ( d[0] == 1 && k == 1 && abs(f.op(1)) != 1 ) {
- return false;
- }
- for ( int i=1; i<=k; ++i ) {
+ vector<numeric> d(k);
+ d[0] = ex_to<numeric>(abs(f.op(0)));
+ for ( int i=1; i<k; ++i ) {
q = ex_to<numeric>(abs(f.op(i)));
for ( int j=i-1; j>=0; --j ) {
r = d[j];
return false;
}
-static bool generate_set(const ex& u, const ex& vn, const exset& syms, const ex& f, const numeric& modulus, vector<numeric>& a, vector<numeric>& d)
+/** Generates a set of evaluation points for a multivariate polynomial.
+ * The set fulfills the following conditions:
+ * 1. lcoeff(evaluated_polynomial) does not vanish
+ * 2. factors of lcoeff(evaluated_polynomial) have each a unique prime factor
+ * 3. evaluated_polynomial is square free
+ * See [W1] for more details.
+ *
+ * @param[in] u multivariate polynomial to be factored
+ * @param[in] vn leading coefficient of u in x (x==first symbol in syms)
+ * @param[in] syms set of symbols that appear in u
+ * @param[in] f lst containing the factors of the leading coefficient vn
+ * @param[in,out] modulus integer modulus for random number generation (i.e. |a_i| < modulus)
+ * @param[out] u0 returns the evaluated (univariate) polynomial
+ * @param[out] a returns the valid evaluation points. must have initial size equal
+ * number of symbols-1 before calling generate_set
+ */
+static void generate_set(const ex& u, const ex& vn, const exset& syms, const lst& f,
+ numeric& modulus, ex& u0, vector<numeric>& a)
{
- // computation of d is actually not necessary
const ex& x = *syms.begin();
- bool trying = true;
- do {
- ex u0 = u;
+ while ( true ) {
+ ++modulus;
+ /* generate a set of integers ... */
+ u0 = u;
ex vna = vn;
ex vnatry;
exset::const_iterator s = syms.begin();
do {
a[i] = mod(numeric(rand()), 2*modulus) - modulus;
vnatry = vna.subs(*s == a[i]);
+ /* ... for which the leading coefficient doesn't vanish ... */
} while ( vnatry == 0 );
vna = vnatry;
u0 = u0.subs(*s == a[i]);
++s;
}
- if ( gcd(u0,u0.diff(ex_to<symbol>(x))) != 1 ) {
+ /* ... for which u0 is square free ... */
+ ex g = gcd(u0, u0.diff(ex_to<symbol>(x)));
+ if ( !is_a<numeric>(g) ) {
continue;
}
- if ( is_a<numeric>(vn) ) {
- trying = false;
- }
- else {
- lst fnum;
- lst::const_iterator i = ex_to<lst>(f).begin();
- fnum.append(*i++);
- bool problem = false;
- while ( i!=ex_to<lst>(f).end() ) {
- ex fs = *i;
- if ( !is_a<numeric>(fs) ) {
+ if ( !is_a<numeric>(vn) ) {
+ /* ... and for which the evaluated factors have each an unique prime factor */
+ lst fnum = f;
+ fnum.let_op(0) = fnum.op(0) * u0.content(x);
+ for ( size_t i=1; i<fnum.nops(); ++i ) {
+ if ( !is_a<numeric>(fnum.op(i)) ) {
s = syms.begin();
++s;
- for ( size_t j=0; j<a.size(); ++j ) {
- fs = fs.subs(*s == a[j]);
- ++s;
- }
- if ( abs(fs) == 1 ) {
- problem = true;
- break;
+ for ( size_t j=0; j<a.size(); ++j, ++s ) {
+ fnum.let_op(i) = fnum.op(i).subs(*s == a[j]);
}
}
- fnum.append(fs);
- ++i; ++i;
}
- if ( problem ) {
- return true;
+ if ( checkdivisors(fnum) ) {
+ continue;
}
- ex con = u0.content(x);
- fnum.append(con);
- trying = checkdivisors(fnum, d);
}
- } while ( trying );
- return false;
+ /* ok, we have a valid set now */
+ return;
+ }
}
+// forward declaration
+static ex factor_sqrfree(const ex& poly);
+
+/**
+ * ASSERT: poly is expanded
+ */
static ex factor_multivariate(const ex& poly, const exset& syms)
{
exset::const_iterator s;
const ex& x = *syms.begin();
/* make polynomial primitive */
- ex p = poly.expand().collect(x);
+ ex p = poly.collect(x);
ex cont = p.lcoeff(x);
- for ( numeric i=p.degree(x)-1; i>=p.ldegree(x); --i ) {
- cont = gcd(cont, p.coeff(x,ex_to<numeric>(i).to_int()));
+ for ( int i=p.degree(x)-1; i>=p.ldegree(x); --i ) {
+ cont = gcd(cont, p.coeff(x,i));
if ( cont == 1 ) break;
}
ex pp = expand(normal(p / cont));
if ( !is_a<numeric>(cont) ) {
- return factor(cont) * factor(pp);
+ return factor_sqrfree(cont) * factor_sqrfree(pp);
}
/* factor leading coefficient */
- pp = pp.collect(x);
- ex vn = pp.lcoeff(x);
- pp = pp.expand();
+ ex vn = pp.collect(x).lcoeff(x);
ex vnlst;
if ( is_a<numeric>(vn) ) {
vnlst = lst(vn);
vnlst = put_factors_into_lst(vnfactors);
}
- const numeric maxtrials = 3;
- numeric modulus = (vnlst.nops()-1 > 3) ? vnlst.nops()-1 : 3;
- numeric minimalr = -1;
+ const unsigned int maxtrials = 3;
+ numeric modulus = (vnlst.nops() > 3) ? vnlst.nops() : 3;
vector<numeric> a(syms.size()-1, 0);
- vector<numeric> d((vnlst.nops()-1)/2+1, 0);
+ /* try now to factorize until we are successful */
while ( true ) {
- numeric trialcount = 0;
+
+ unsigned int trialcount = 0;
+ unsigned int prime;
+ int factor_count = 0;
+ int min_factor_count = -1;
ex u, delta;
- unsigned int prime = 3;
- size_t factor_count = 0;
- ex ufac;
- ex ufaclst;
+ ex ufac, ufaclst;
+
+ /* try several evaluation points to reduce the number of modular factors */
while ( trialcount < maxtrials ) {
- bool problem = generate_set(pp, vn, syms, vnlst, modulus, a, d);
- if ( problem ) {
- ++modulus;
- continue;
- }
- u = pp;
- s = syms.begin();
- ++s;
- for ( size_t i=0; i<a.size(); ++i ) {
- u = u.subs(*s == a[i]);
- ++s;
- }
- delta = u.content(x);
-
- // determine proper prime
- prime = 3;
- cl_modint_ring R = find_modint_ring(prime);
- while ( true ) {
- if ( irem(ex_to<numeric>(u.lcoeff(x)), prime) != 0 ) {
- umodpoly modpoly;
- umodpoly_from_ex(modpoly, u, x, R);
- if ( squarefree(modpoly) ) break;
- }
- prime = next_prime(prime);
- R = find_modint_ring(prime);
- }
- ufac = factor(u);
+ /* generate a set of valid evaluation points */
+ generate_set(pp, vn, syms, ex_to<lst>(vnlst), modulus, u, a);
+
+ ufac = factor_univariate(u, x, prime);
ufaclst = put_factors_into_lst(ufac);
- factor_count = (ufaclst.nops()-1)/2;
-
- // veto factorization for which gcd(u_i, u_j) != 1 for all i,j
- upvec tryu;
- for ( size_t i=0; i<(ufaclst.nops()-1)/2; ++i ) {
- umodpoly newu;
- umodpoly_from_ex(newu, ufaclst.op(i*2+1), x, R);
- tryu.push_back(newu);
- }
- bool veto = false;
- for ( size_t i=0; i<tryu.size()-1; ++i ) {
- for ( size_t j=i+1; j<tryu.size(); ++j ) {
- umodpoly tryg;
- gcd(tryu[i], tryu[j], tryg);
- if ( unequal_one(tryg) ) {
- veto = true;
- goto escape_quickly;
- }
- }
- }
- escape_quickly: ;
- if ( veto ) {
- continue;
- }
+ factor_count = ufaclst.nops()-1;
+ delta = ufaclst.op(0);
if ( factor_count <= 1 ) {
+ /* irreducible */
return poly;
}
-
- if ( minimalr < 0 ) {
- minimalr = factor_count;
+ if ( min_factor_count < 0 ) {
+ /* first time here */
+ min_factor_count = factor_count;
}
- else if ( minimalr == factor_count ) {
+ else if ( min_factor_count == factor_count ) {
+ /* one less to try */
++trialcount;
- ++modulus;
}
- else if ( minimalr > factor_count ) {
- minimalr = factor_count;
+ else if ( min_factor_count > factor_count ) {
+ /* new minimum, reset trial counter */
+ min_factor_count = factor_count;
trialcount = 0;
}
- if ( minimalr <= 1 ) {
- return poly;
- }
}
- vector<numeric> ftilde((vnlst.nops()-1)/2+1);
- ftilde[0] = ex_to<numeric>(vnlst.op(0));
- for ( size_t i=1; i<ftilde.size(); ++i ) {
- ex ft = vnlst.op((i-1)*2+1);
- s = syms.begin();
- ++s;
- for ( size_t j=0; j<a.size(); ++j ) {
- ft = ft.subs(*s == a[j]);
- ++s;
+ // determine true leading coefficients for the Hensel lifting
+ vector<ex> C(factor_count);
+ if ( is_a<numeric>(vn) ) {
+ for ( size_t i=1; i<ufaclst.nops(); ++i ) {
+ C[i-1] = ufaclst.op(i).lcoeff(x);
}
- ftilde[i] = ex_to<numeric>(ft);
}
+ else {
+ vector<numeric> ftilde(vnlst.nops()-1);
+ for ( size_t i=0; i<ftilde.size(); ++i ) {
+ ex ft = vnlst.op(i+1);
+ s = syms.begin();
+ ++s;
+ for ( size_t j=0; j<a.size(); ++j ) {
+ ft = ft.subs(*s == a[j]);
+ ++s;
+ }
+ ftilde[i] = ex_to<numeric>(ft);
+ }
- vector<bool> used_flag((vnlst.nops()-1)/2+1, false);
- vector<ex> D(factor_count, 1);
- for ( size_t i=0; i<=factor_count; ++i ) {
- numeric prefac;
- if ( i == 0 ) {
- prefac = ex_to<numeric>(ufaclst.op(0));
- ftilde[0] = ftilde[0] / prefac;
- vnlst.let_op(0) = vnlst.op(0) / prefac;
- continue;
+ vector<bool> used_flag(ftilde.size(), false);
+ vector<ex> D(factor_count, 1);
+ if ( delta == 1 ) {
+ for ( int i=0; i<factor_count; ++i ) {
+ numeric prefac = ex_to<numeric>(ufaclst.op(i+1).lcoeff(x));
+ for ( int j=ftilde.size()-1; j>=0; --j ) {
+ int count = 0;
+ while ( irem(prefac, ftilde[j]) == 0 ) {
+ prefac = iquo(prefac, ftilde[j]);
+ ++count;
+ }
+ if ( count ) {
+ used_flag[j] = true;
+ D[i] = D[i] * pow(vnlst.op(j+1), count);
+ }
+ }
+ C[i] = D[i] * prefac;
+ }
}
else {
- prefac = ex_to<numeric>(ufaclst.op(2*(i-1)+1).lcoeff(x));
- }
- for ( size_t j=(vnlst.nops()-1)/2+1; j>0; --j ) {
- if ( abs(ftilde[j-1]) == 1 ) {
- used_flag[j-1] = true;
- continue;
- }
- numeric g = gcd(prefac, ftilde[j-1]);
- if ( g != 1 ) {
- prefac = prefac / g;
- numeric count = abs(iquo(g, ftilde[j-1]));
- used_flag[j-1] = true;
- if ( i > 0 ) {
- if ( j == 1 ) {
- D[i-1] = D[i-1] * pow(vnlst.op(0), count);
+ for ( int i=0; i<factor_count; ++i ) {
+ numeric prefac = ex_to<numeric>(ufaclst.op(i+1).lcoeff(x));
+ for ( int j=ftilde.size()-1; j>=0; --j ) {
+ int count = 0;
+ while ( irem(prefac, ftilde[j]) == 0 ) {
+ prefac = iquo(prefac, ftilde[j]);
+ ++count;
}
- else {
- D[i-1] = D[i-1] * pow(vnlst.op(2*(j-2)+1), count);
+ while ( irem(ex_to<numeric>(delta)*prefac, ftilde[j]) == 0 ) {
+ numeric g = gcd(prefac, ex_to<numeric>(ftilde[j]));
+ prefac = iquo(prefac, g);
+ delta = delta / (ftilde[j]/g);
+ ufaclst.let_op(i+1) = ufaclst.op(i+1) * (ftilde[j]/g);
+ ++count;
+ }
+ if ( count ) {
+ used_flag[j] = true;
+ D[i] = D[i] * pow(vnlst.op(j+1), count);
}
}
- else {
- ftilde[j-1] = ftilde[j-1] / prefac;
- break;
- }
- ++j;
+ C[i] = D[i] * prefac;
}
}
- }
-
- bool some_factor_unused = false;
- for ( size_t i=0; i<used_flag.size(); ++i ) {
- if ( !used_flag[i] ) {
- some_factor_unused = true;
- break;
- }
- }
- if ( some_factor_unused ) {
- continue;
- }
- vector<ex> C(factor_count);
- if ( delta == 1 ) {
- for ( size_t i=0; i<D.size(); ++i ) {
- ex Dtilde = D[i];
- s = syms.begin();
- ++s;
- for ( size_t j=0; j<a.size(); ++j ) {
- Dtilde = Dtilde.subs(*s == a[j]);
- ++s;
+ bool some_factor_unused = false;
+ for ( size_t i=0; i<used_flag.size(); ++i ) {
+ if ( !used_flag[i] ) {
+ some_factor_unused = true;
+ break;
}
- C[i] = D[i] * (ufaclst.op(2*i+1).lcoeff(x) / Dtilde);
}
- }
- else {
- for ( size_t i=0; i<D.size(); ++i ) {
- ex Dtilde = D[i];
- s = syms.begin();
- ++s;
- for ( size_t j=0; j<a.size(); ++j ) {
- Dtilde = Dtilde.subs(*s == a[j]);
- ++s;
- }
- ex ui;
- if ( i == 0 ) {
- ui = ufaclst.op(0);
- }
- else {
- ui = ufaclst.op(2*(i-1)+1);
- }
- while ( true ) {
- ex d = gcd(ui.lcoeff(x), Dtilde);
- C[i] = D[i] * ( ui.lcoeff(x) / d );
- ui = ui * ( Dtilde[i] / d );
- delta = delta / ( Dtilde[i] / d );
- if ( delta == 1 ) break;
- ui = delta * ui;
- C[i] = delta * C[i];
- pp = pp * pow(delta, D.size()-1);
- }
+ if ( some_factor_unused ) {
+ continue;
}
}
+ if ( delta != 1 ) {
+ C[0] = C[0] * delta;
+ ufaclst.let_op(1) = ufaclst.op(1) * delta;
+ }
+
EvalPoint ep;
vector<EvalPoint> epv;
s = syms.begin();
// calc bound p^l
int maxdeg = 0;
- for ( size_t i=0; i<factor_count; ++i ) {
- if ( ufaclst[2*i+1].degree(x) > maxdeg ) {
- maxdeg = ufaclst[2*i+1].degree(x);
+ for ( int i=1; i<=factor_count; ++i ) {
+ if ( ufaclst.op(i).degree(x) > maxdeg ) {
+ maxdeg = ufaclst[i].degree(x);
}
}
cl_I B = 2*calc_bound(u, x, maxdeg);
l = l + 1;
pl = pl * prime;
}
-
- upvec uvec;
cl_modint_ring R = find_modint_ring(expt_pos(cl_I(prime),l));
- for ( size_t i=0; i<(ufaclst.nops()-1)/2; ++i ) {
- umodpoly newu;
- umodpoly_from_ex(newu, ufaclst.op(i*2+1), x, R);
- uvec.push_back(newu);
+ upvec modfactors(ufaclst.nops()-1);
+ for ( size_t i=1; i<ufaclst.nops(); ++i ) {
+ umodpoly_from_ex(modfactors[i-1], ufaclst.op(i), x, R);
}
- ex res = hensel_multivar(ufaclst.op(0)*pp, x, epv, prime, l, uvec, C);
+ ex res = hensel_multivar(pp, x, epv, prime, l, modfactors, C);
if ( res != lst() ) {
- ex result = cont * ufaclst.op(0);
+ ex result = cont;
for ( size_t i=0; i<res.nops(); ++i ) {
result *= res.op(i).content(x) * res.op(i).unit(x);
result *= res.op(i).primpart(x);
}
// make poly square free
- ex sfpoly = sqrfree(poly, syms);
+ ex sfpoly = sqrfree(poly.expand(), syms);
// factorize the square free components
if ( is_a<power>(sfpoly) ) {
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