+struct EvalPoint
+{
+ ex x;
+ int evalpoint;
+};
+
+// 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);
+
+UniPolyVec multiterm_eea_lift(const UniPolyVec& a, const ex& x, unsigned int p, unsigned int k)
+{
+ DCOUT(multiterm_eea_lift);
+ DCOUTVAR(a);
+ DCOUTVAR(p);
+ DCOUTVAR(k);
+
+ const size_t r = a.size();
+ DCOUTVAR(r);
+ cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
+ UniPoly fill(R);
+ UniPolyVec q(r-1, fill);
+ q[r-2] = a[r-1];
+ for ( size_t j=r-2; j>=1; --j ) {
+ q[j-1] = a[j] * q[j];
+ }
+ DCOUTVAR(q);
+ UniPoly beta(R);
+ beta.set(0, R->one());
+ UniPolyVec s;
+ for ( size_t j=1; j<r; ++j ) {
+ DCOUTVAR(j);
+ DCOUTVAR(beta);
+ vector<ex> mdarg(2);
+ mdarg[0] = q[j-1].to_ex(x);
+ mdarg[1] = a[j-1].to_ex(x);
+ vector<EvalPoint> empty;
+ vector<ex> exsigma = multivar_diophant(mdarg, x, beta.to_ex(x), empty, 0, p, k);
+ UniPoly sigma1(R, exsigma[0], x);
+ UniPoly sigma2(R, exsigma[1], x);
+ beta = sigma1;
+ s.push_back(sigma2);
+ }
+ s.push_back(beta);
+
+ DCOUTVAR(s);
+ DCOUT(END multiterm_eea_lift);
+ return s;
+}
+
+void eea_lift(const UniPoly& a, const UniPoly& b, const ex& x, unsigned int p, unsigned int k, UniPoly& s_, UniPoly& t_)
+{
+ DCOUT(eea_lift);
+ DCOUTVAR(a);
+ DCOUTVAR(b);
+ DCOUTVAR(x);
+ DCOUTVAR(p);
+ DCOUTVAR(k);
+
+ cl_modint_ring R = find_modint_ring(p);
+ UniPoly amod(R, a);
+ UniPoly bmod(R, b);
+ DCOUTVAR(amod);
+ DCOUTVAR(bmod);
+
+ UniPoly smod(R), tmod(R), g(R);
+ exteuclid(amod, bmod, g, smod, tmod);
+
+ DCOUTVAR(smod);
+ DCOUTVAR(tmod);
+ DCOUTVAR(g);
+
+ cl_modint_ring Rpk = find_modint_ring(expt_pos(cl_I(p),k));
+ UniPoly s(Rpk, smod);
+ UniPoly t(Rpk, tmod);
+ DCOUTVAR(s);
+ DCOUTVAR(t);
+
+ cl_I modulus(p);
+
+ UniPoly one(Rpk);
+ one.set(0, Rpk->one());
+ for ( size_t j=1; j<k; ++j ) {
+ UniPoly e = one - a * s - b * t;
+ e.divide(modulus);
+ UniPoly c(R, e);
+ UniPoly sigmabar(R);
+ sigmabar = smod * c;
+ UniPoly taubar(R);
+ taubar = tmod * c;
+ UniPoly q(R);
+ div(sigmabar, bmod, q);
+ UniPoly sigma(R);
+ rem(sigmabar, bmod, sigma);
+ UniPoly tau(R);
+ tau = taubar + q * amod;
+ UniPoly sadd(Rpk, sigma);
+ cl_MI modmodulus(Rpk, modulus);
+ s = s + sadd * modmodulus;
+ UniPoly tadd(Rpk, tau);
+ t = t + tadd * modmodulus;
+ modulus = modulus * p;
+ }
+
+ s_ = s; t_ = t;
+
+ DCOUTVAR(s);
+ DCOUTVAR(t);
+ DCOUT2(check, a*s + b*t);
+ DCOUT(END eea_lift);
+}
+
+UniPolyVec univar_diophant(const UniPolyVec& a, const ex& x, unsigned int m, unsigned int p, unsigned int k)
+{
+ DCOUT(univar_diophant);
+ DCOUTVAR(a);
+ DCOUTVAR(x);
+ DCOUTVAR(m);
+ DCOUTVAR(p);
+ DCOUTVAR(k);
+
+ cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
+
+ const size_t r = a.size();
+ UniPolyVec result;
+ if ( r > 2 ) {
+ UniPolyVec s = multiterm_eea_lift(a, x, p, k);
+ for ( size_t j=0; j<r; ++j ) {
+ ex phi = expand(pow(x,m)*s[j].to_ex(x));
+ UniPoly bmod(R, phi, x);
+ UniPoly buf(R);
+ rem(bmod, a[j], buf);
+ result.push_back(buf);
+ }
+ }
+ else {
+ UniPoly s(R), t(R);
+ eea_lift(a[1], a[0], x, p, k, s, t);
+ ex phi = expand(pow(x,m)*s.to_ex(x));
+ UniPoly bmod(R, phi, x);
+ UniPoly buf(R);
+ rem(bmod, a[0], buf);
+ result.push_back(buf);
+ UniPoly q(R);
+ div(bmod, a[0], q);
+ phi = expand(pow(x,m)*t.to_ex(x));
+ UniPoly t1mod(R, phi, x);
+ buf = t1mod + q * a[1];
+ result.push_back(buf);
+ }
+
+ DCOUTVAR(result);
+ DCOUT(END univar_diophant);
+ return result;
+}
+
+struct make_modular_map : public map_function {
+ cl_modint_ring R;
+ make_modular_map(const cl_modint_ring& R_) : R(R_) { }
+ ex operator()(const ex& e)
+ {
+ if ( is_a<add>(e) || is_a<mul>(e) ) {
+ return e.map(*this);
+ }
+ else if ( is_a<numeric>(e) ) {
+ numeric mod(R->modulus);
+ numeric halfmod = (mod-1)/2;
+ cl_MI emod = R->canonhom(the<cl_I>(ex_to<numeric>(e).to_cl_N()));
+ numeric n(R->retract(emod));
+ if ( n > halfmod ) {
+ return n-mod;
+ }
+ else {
+ return n;
+ }
+ }
+ return e;
+ }
+};
+
+static ex make_modular(const ex& e, const cl_modint_ring& R)
+{
+ make_modular_map map(R);
+ return map(e);
+}
+
+vector<ex> multivar_diophant(const vector<ex>& a_, const ex& x, const ex& c, const vector<EvalPoint>& I, unsigned int d, unsigned int p, unsigned int k)
+{
+ vector<ex> a = a_;
+
+ DCOUT(multivar_diophant);
+#ifdef DEBUGFACTOR
+ cout << "a ";
+ for ( size_t i=0; i<a.size(); ++i ) {
+ cout << a[i] << " ";
+ }
+ cout << endl;
+#endif
+ DCOUTVAR(x);
+ DCOUTVAR(c);
+#ifdef DEBUGFACTOR
+ cout << "I ";
+ for ( size_t i=0; i<I.size(); ++i ) {
+ cout << I[i].x << "=" << I[i].evalpoint << " ";
+ }
+ cout << endl;
+#endif
+ DCOUTVAR(d);
+ DCOUTVAR(p);
+ DCOUTVAR(k);
+
+ const cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
+ const size_t r = a.size();
+ const size_t nu = I.size() + 1;
+ DCOUTVAR(r);
+ DCOUTVAR(nu);
+
+ vector<ex> sigma;
+ if ( nu > 1 ) {
+ ex xnu = I.back().x;
+ int alphanu = I.back().evalpoint;
+
+ ex A = 1;
+ for ( size_t i=0; i<r; ++i ) {
+ A *= a[i];
+ }
+ vector<ex> b(r);
+ for ( size_t i=0; i<r; ++i ) {
+ b[i] = normal(A / a[i]);
+ }
+
+ vector<ex> anew = a;
+ for ( size_t i=0; i<r; ++i ) {
+ a[i] = a[i].subs(xnu == alphanu);
+ }
+ ex cnew = c.subs(xnu == alphanu);
+ vector<EvalPoint> Inew = I;
+ Inew.pop_back();
+ vector<ex> sigma = multivar_diophant(anew, x, cnew, Inew, d, p, k);
+
+ ex buf = c;
+ for ( size_t i=0; i<r; ++i ) {
+ buf -= sigma[i] * b[i];
+ }
+ ex e = buf;
+ e = make_modular(e, R);
+
+ ex monomial = 1;
+ for ( size_t m=1; m<=d; ++m ) {
+ while ( !e.is_zero() ) {
+ monomial *= (xnu - alphanu);
+ monomial = expand(monomial);
+ ex cm = e.diff(ex_to<symbol>(xnu), m).subs(xnu==alphanu) / factorial(m);
+ if ( !cm.is_zero() ) {
+ vector<ex> delta_s = multivar_diophant(anew, x, cm, Inew, d, p, k);
+ ex buf = e;
+ for ( size_t j=0; j<delta_s.size(); ++j ) {
+ delta_s[j] *= monomial;
+ sigma[j] += delta_s[j];
+ buf -= delta_s[j] * b[j];
+ }
+ e = buf;
+ e = make_modular(e, R);
+ }
+ }
+ }
+ }
+ else {
+ UniPolyVec amod;
+ for ( size_t i=0; i<a.size(); ++i ) {
+ UniPoly up(R, a[i], x);
+ amod.push_back(up);
+ }
+
+ sigma.insert(sigma.begin(), r, 0);
+ size_t nterms;
+ ex z;
+ if ( is_a<add>(c) ) {
+ nterms = c.nops();
+ z = c.op(0);
+ }
+ else {
+ nterms = 1;
+ z = c;
+ }
+ DCOUTVAR(nterms);
+ for ( size_t i=0; i<nterms; ++i ) {
+ DCOUTVAR(z);
+ int m = z.degree(x);
+ DCOUTVAR(m);
+ cl_I cm = the<cl_I>(ex_to<numeric>(z.lcoeff(x)).to_cl_N());
+ DCOUTVAR(cm);
+ UniPolyVec delta_s = univar_diophant(amod, x, m, p, k);
+ cl_MI modcm;
+ cl_I poscm = cm;
+ while ( poscm < 0 ) {
+ poscm = poscm + expt_pos(cl_I(p),k);
+ }
+ modcm = cl_MI(R, poscm);
+ DCOUTVAR(modcm);
+ for ( size_t j=0; j<delta_s.size(); ++j ) {
+ delta_s[j] = delta_s[j] * modcm;
+ sigma[j] = sigma[j] + delta_s[j].to_ex(x);
+ }
+ DCOUTVAR(delta_s);
+#ifdef DEBUGFACTOR
+ cout << "STEP " << i << " sigma ";
+ for ( size_t p=0; p<sigma.size(); ++p ) {
+ cout << sigma[p] << " ";
+ }
+ cout << endl;
+#endif
+ if ( nterms > 1 ) {
+ z = c.op(i+1);
+ }
+ }
+ }
+#ifdef DEBUGFACTOR
+ cout << "sigma ";
+ for ( size_t i=0; i<sigma.size(); ++i ) {
+ cout << sigma[i] << " ";
+ }
+ cout << endl;
+#endif
+
+ for ( size_t i=0; i<sigma.size(); ++i ) {
+ sigma[i] = make_modular(sigma[i], R);
+ }
+
+#ifdef DEBUGFACTOR
+ cout << "sigma ";
+ for ( size_t i=0; i<sigma.size(); ++i ) {
+ cout << sigma[i] << " ";
+ }
+ cout << endl;
+#endif
+ DCOUT(END multivar_diophant);
+ return sigma;
+}
+
+ex hensel_multivar(const ex& a, const ex& x, const vector<EvalPoint>& I, unsigned int p, const cl_I& l, const UniPolyVec& u, const vector<ex>& lcU)
+{
+ DCOUT(hensel_multivar);
+ DCOUTVAR(a);
+ DCOUTVAR(x);
+ DCOUTVAR(p);
+ DCOUTVAR(l);
+ DCOUTVAR(u);
+ const size_t nu = I.size() + 1;
+ const cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),l));
+
+ DCOUTVAR(nu);
+
+ vector<ex> A(nu);
+ A[nu-1] = a;
+
+ for ( size_t j=nu; j>=2; --j ) {
+ ex x = I[j-2].x;
+ int alpha = I[j-2].evalpoint;
+ A[j-2] = A[j-1].subs(x==alpha);
+ A[j-2] = make_modular(A[j-2], R);
+ }
+
+#ifdef DEBUGFACTOR
+ cout << "A ";
+ for ( size_t i=0; i<A.size(); ++i) cout << A[i] << " ";
+ cout << endl;
+#endif
+
+ int maxdeg = a.degree(I.front().x);
+ for ( size_t i=1; i<I.size(); ++i ) {
+ int maxdeg2 = a.degree(I[i].x);
+ if ( maxdeg2 > maxdeg ) maxdeg = maxdeg2;
+ }
+ DCOUTVAR(maxdeg);
+
+ const size_t n = u.size();
+ DCOUTVAR(n);
+ vector<ex> U(n);
+ for ( size_t i=0; i<n; ++i ) {
+ U[i] = u[i].to_ex(x);
+ }
+#ifdef DEBUGFACTOR
+ cout << "U ";
+ for ( size_t i=0; i<U.size(); ++i) cout << U[i] << " ";
+ cout << endl;
+#endif
+
+ for ( size_t j=2; j<=nu; ++j ) {
+ DCOUTVAR(j);
+ vector<ex> U1 = U;
+ ex monomial = 1;
+ for ( size_t m=0; m<n; ++m) {
+ if ( lcU[m] != 1 ) {
+ ex coef = lcU[m];
+ for ( size_t i=j-1; i<nu-1; ++i ) {
+ coef = coef.subs(I[i].x == I[i].evalpoint);
+ }
+ coef = expand(coef);
+ coef = make_modular(coef, R);
+ int deg = U[m].degree(x);
+ U[m] = U[m] - U[m].lcoeff(x) * pow(x,deg) + coef * pow(x,deg);
+ }
+ }
+ ex Uprod = 1;
+ for ( size_t i=0; i<n; ++i ) {
+ Uprod *= U[i];
+ }
+ ex e = expand(A[j-1] - Uprod);
+ DCOUTVAR(e);
+
+ ex xj = I[j-2].x;
+ int alphaj = I[j-2].evalpoint;
+ size_t deg = A[j-1].degree(xj);
+ DCOUTVAR(deg);
+ for ( size_t k=1; k<=deg; ++k ) {
+ DCOUTVAR(k);
+ if ( !e.is_zero() ) {
+ DCOUTVAR(xj);
+ DCOUTVAR(alphaj);
+ monomial *= (xj - alphaj);
+ monomial = expand(monomial);
+ DCOUTVAR(monomial);
+ ex dif = e.diff(ex_to<symbol>(xj), k);
+ DCOUTVAR(dif);
+ ex c = dif.subs(xj==alphaj) / factorial(k);
+ DCOUTVAR(c);
+ if ( !c.is_zero() ) {
+ vector<EvalPoint> newI = I;
+ newI.pop_back();
+ vector<ex> deltaU = multivar_diophant(U1, x, c, newI, maxdeg, p, cl_I_to_uint(l));
+ for ( size_t i=0; i<n; ++i ) {
+ DCOUTVAR(i);
+ DCOUTVAR(deltaU[i]);
+ deltaU[i] *= monomial;
+ U[i] += deltaU[i];
+ U[i] = make_modular(U[i], R);
+ }
+ ex Uprod = 1;
+ for ( size_t i=0; i<n; ++i ) {
+ Uprod *= U[i];
+ }
+ e = expand(A[j-1] - Uprod);
+ e = make_modular(e, R);
+ DCOUTVAR(e);
+ }
+ else {
+ break;
+ }
+ }
+ }
+ }
+
+ ex acand = 1;
+ for ( size_t i=0; i<U.size(); ++i ) {
+ acand *= U[i];
+ }
+ DCOUTVAR(acand);
+ if ( expand(a-acand).is_zero() ) {
+ lst res;
+ for ( size_t i=0; i<U.size(); ++i ) {
+ res.append(U[i]);
+ }
+ return res;
+ }
+ else {
+ return lst();
+ }
+}
+
+static ex factor_multivariate(const ex& poly, const ex& x)
+{
+ // TODO
+ return 666;
+}
+
git://www.ginac.de / ginac.git / blobdiff