Ticket #10910: factor_nfinit_free.patch

sage/rings/polynomial/polynomial_element.pyx

@@ -2709 +2709 @@
             sage: pol.factor()
             x^5 * (x^5 + (4/7*a - 6/7)*x^4 + (9/49*a^2 - 3/7*a + 15/49)*x^3 + (8/343*a^3 - 32/343*a^2 + 40/343*a - 20/343)*x^2 + (5/2401*a^4 - 20/2401*a^3 + 40/2401*a^2 - 5/343*a + 15/2401)*x - 6/16807*a^4 + 12/16807*a^3 - 18/16807*a^2 + 12/16807*a - 6/16807)
 
+        Check that we do not compute nfinit::
+
+            sage: x = var('x')
+            sage: N = NumberField(x^2-2,'x')
+            sage: K = N['x']
+            sage: f1 = K.random_element(3)
+            sage: f2 = K.random_element(3)
+            sage: f = f1*f2
+            sage: hasattr(N, '_NumberField_generic__pari_nf')
+            False
+            sage: F1 = factor(f)
+            sage: hasattr(N, '_NumberField_generic__pari_nf')
+            False
+            sage: _ = N.pari_nf()
+            sage: F2 = factor(f)
+            sage: hasattr(N, '_NumberField_generic__pari_nf')
+            True
+            sage: F1 -f
+            0
+            sage: F2 -f
+            0
+
+        Next example would be very slow if we had to compute a nfinit             structure. See #10910::
+
+            sage: x = var('x')
+            sage: N = NumberField(x^8+1212/413*x^7+1222*x+1/17,'a')
+            sage: K.<t> = N[]
+            sage: f = K.random_element(4)
+            sage: g = K.random_element(4)
+            sage: F = factor(f*g)
+            sage: F - f*g
+            0
+            sage: hasattr(N, '_NumberField_generic__pari_nf')
+            False
+
+        Check that the factorization is correct. See #10910::
+
+            sage: K=QQ[sqrt(2**40+1), 3^(1/3)]
+            sage: x = polygen(K)
+            sage: f = x^3 - 3
+            sage: len(factor(f))
+            2
         """
         # PERFORMANCE NOTE:
         #     In many tests with SMALL degree PARI is substantially
@@ -2777 +2819 @@
             g = M['x']([to_M(x) for x in self.list()])
             F = g.factor()
             S = self.parent()
-            v = [(S([from_M(x) for x in f.list()]), e) for f, e in g.factor()]
+            v = [(S([from_M(x) for x in f.list()]), e) for f, e in F]
             return Factorization(v, from_M(F.unit()))
 
         elif is_FiniteField(R):
@@ -2811 +2853 @@
             if R.defining_polynomial().denominator() == 1:
                 v = [ x._pari_("a") for x in self.list() ]
                 f = pari(v).Polrev()
-                Rpari = R.pari_nf()
-                if (Rpari.variable() != "a"):
-                    Rpari = copy.copy(Rpari)
-                    Rpari[0] = Rpari[0]("a")
-                    Rpari[6] = [ x("a") for x in Rpari[6] ]
-                G = list(Rpari.nffactor(f))
-                # PARI's nffactor() ignores the unit, _factor_pari_helper()
-                # adds back the unit of the factorization.
+                if hasattr(R, '_NumberField_generic__pari_nf'):
+                    # If we have a nf structure, this should be faster
+                    Rpari = R.pari_nf()
+                    if (Rpari.variable() != "a"):
+                        Rpari = copy.copy(Rpari)
+                        Rpari[0] = Rpari[0]("a")
+                        Rpari[6] = [ x("a") for x in Rpari[6] ]
+                    G = list(Rpari.nffactor(f))
+                    # PARI's nffactor() ignores the unit, _factor_pari_helper()
+                    # adds back the unit of the factorization.
+
+                else:
+                    # We do not compute a nf structure, since it may be
+                    # expensive
+                    nf = R.pari_polynomial('a')
+                    G = list(nf.nffactor(f))
+
                 return self._factor_pari_helper(G)
 
             else:
 
                 Rdenom = R.defining_polynomial().denominator()
-
-                new_Rpoly = (R.defining_polynomial() * Rdenom).change_variable_name("a")
-
-                Rpari, Rdiff = new_Rpoly._pari_().nfinit(3)
-
-                AZ = QQ['z']
-                Raux = NumberField(AZ(Rpari[0]),'alpha')
-
-                S, gSRaux, fRauxS = Raux.change_generator(Raux(Rdiff))
-
-                phi_RS = R.Hom(S)([S.gen(0)])
-                phi_SR = S.Hom(R)([R.gen(0)])
-
+                d = R.degree()
+                Rdenom_d = Rdenom**d
+                x = R.polynomial_ring().gen()
+                new_Rpoly = R.defining_polynomial()(x/Rdenom)*Rdenom_d
+
+                S = NumberField(new_Rpoly, 'alpha')
+                phi_RS = R.Hom(S)([S.gen()/Rdenom])
+                phi_SR = S.Hom(R)([R.gen()*Rdenom])
                 unit = self.leading_coefficient()
-                temp_f = self * 1/unit
-
-                v = [ gSRaux(phi_RS(x))._pari_("a") for x in temp_f.list() ]
+                temp_f = self * ~unit
+                v = [ (phi_RS(x))._pari_("a") for x in temp_f.list() ]
                 f = pari(v).Polrev()
-
-                pari_factors = Rpari.nffactor(f)
-
-                factors = [ ( self.parent([ phi_SR(fRauxS(Raux(pari_factors[0][i][j])))
-                                            for j in range(len(pari_factors[0][i])) ]) ,
-                             int(pari_factors[1][i]) )
+                nf = S.pari_polynomial('a')
+                pari_factors = nf.nffactor(f)
+                factors = [ ( self.parent([ phi_SR(S(pari_factors[0][i][j]))
+                            for j in range(len(pari_factors[0][i])) ]) ,
+                            int(pari_factors[1][i]) )
                             for i in range(pari_factors.nrows()) ]
 
                 return Factorization(factors, unit)
-        
 
         elif is_RealField(R):
             n = pari.set_real_precision(int(3.5*R.prec()) + 1)