Refactor generic ideal method + avoid computing principal generator of ideal of PolynomialRing on construction

src/sage/arith/misc.py

@@ -1887,11 +1887,8 @@ def __GCD_sequence(v, **kwargs):
     """
     if len(v) == 0:
         return ZZ.zero()
-    if hasattr(v, 'universe'):
-        g = v.universe()(0)
-    else:
-        g = ZZ.zero()
-    for vi in v:
+    g = v[0]
+    for vi in v[1:]:
         g = vi.gcd(g, **kwargs)
     return g
 

src/sage/categories/rings.py

@@ -1000,7 +1000,7 @@ def _test_characteristic(self, **options):
             from sage.rings.integer import Integer
             tester.assertIsInstance(characteristic, Integer)
 
-        def ideal(self, *args, **kwds):
+        def ideal(self, *args, coerce=True, ideal_class=None, **kwds):
             """
             Create an ideal of this ring.
 
@@ -1096,15 +1096,9 @@ def ideal(self, *args, **kwds):
                 sage: type(ZZ.ideal((), ideal_class=CustomIdealClass))
                 <class '...CustomIdealClass'>
             """
-            if 'coerce' in kwds:
-                coerce = kwds['coerce']
-                del kwds['coerce']
-            else:
-                coerce = True
-
             from sage.rings.ideal import Ideal_generic
             if not args:
-                gens = [self(0)]
+                gens = []
             else:
                 gens = args
                 while isinstance(gens, (list, tuple, GeneratorType)) and len(gens) == 1:
@@ -1132,29 +1126,30 @@ def ideal(self, *args, **kwds):
             elif coerce:
                 gens = [self(g) for g in gens]
 
-            from sage.categories.principal_ideal_domains import PrincipalIdealDomains
-            if self in PrincipalIdealDomains():
-                # Use GCD algorithm to obtain a principal ideal
-                g = gens[0]
-                if len(gens) == 1:
-                    try:
-                        # note: we set g = gcd(g, g) to "canonicalize" the generator:
-                        # make polynomials monic, etc.
-                        g = g.gcd(g)
-                    except (AttributeError, NotImplementedError, IndexError):
-                        pass
-                else:
-                    for h in gens[1:]:
-                        g = g.gcd(h)
-                gens = [g]
-            if 'ideal_class' in kwds:
-                C = kwds['ideal_class']
-                del kwds['ideal_class']
-            else:
-                C = self._ideal_class_(len(gens))
+            # Parent classes may define eagerly_reduce_gens_by_gcd = False to opt out of this
+            # either because gcd() is expensive, checking whether self is PID is expensive,
+            # or because ideal_class constructor already have a more efficient gcd()
+            if getattr(self, 'eagerly_reduce_ideal_gens_by_gcd', True):
+                from sage.categories.principal_ideal_domains import PrincipalIdealDomains
+                if self in PrincipalIdealDomains():
+                    from sage.arith.misc import gcd
+                    # Use GCD algorithm to obtain a principal ideal
+                    if len(gens) == 1:
+                        try:
+                            # note: we set g = gcd(g, g) to "canonicalize" the generator:
+                            # make polynomials monic, etc.
+                            g = gens[0]
+                            gens = g.gcd(g),
+                        except (AttributeError, NotImplementedError, IndexError):
+                            pass
+                    else:
+                        gens = gcd(gens),
+
+            if ideal_class is None:
+                ideal_class = self._ideal_class_(len(gens))
             if len(gens) == 1 and isinstance(gens[0], (list, tuple)):
                 gens = gens[0]
-            return C(self, gens, **kwds)
+            return ideal_class(self, gens, **kwds)
 
         # Quotient rings
         def quotient(self, I, names=None, **kwds):

src/sage/rings/ideal.py

@@ -28,6 +28,7 @@
 
 from types import GeneratorType
 
+from sage.misc.cachefunc import cached_method
 from sage.categories.rings import Rings
 from sage.categories.fields import Fields
 from sage.structure.element import MonoidElement
@@ -93,7 +94,7 @@
         sage: i = ideal(1,t,t^2)
         sage: i
         Ideal (1, t, t^2) of Univariate Polynomial Ring in t over Integer Ring
         sage: ideal(1/2,t,t^2)
         Principal ideal (1) of Univariate Polynomial Ring in t over Rational Field
 
     This shows that the issues at :issue:`1104` are resolved::
@@ -159,13 +160,13 @@
         sage: R.<x> = QQ[]
         sage: I = R.ideal([x + x^2])
         sage: J = R.ideal([2*x + 2*x^2])
         sage: J
         Principal ideal (x^2 + x) of Univariate Polynomial Ring in x over Rational Field
         sage: S = R.quotient_ring(I)                                                    # needs sage.libs.pari
         sage: U = R.quotient_ring(J)                                                    # needs sage.libs.pari
         sage: I == J
         True
         sage: S == U                                                                    # needs sage.libs.pari
         True
     """
     if len(args) == 0:
@@ -519,10 +520,10 @@
             sage: J = ideal([CC['x'].0 + 1]); J
             Principal ideal (x + 1.00000000000000) of Univariate Polynomial Ring in x
              over Complex Field with 53 bits of precision
             sage: psi(J)
             Principal ideal (x - 1.00000000000000) of Univariate Polynomial Ring in x
              over Complex Field with 53 bits of precision
             sage: J.apply_morphism(psi)
             Principal ideal (x - 1.00000000000000) of Univariate Polynomial Ring in x
              over Complex Field with 53 bits of precision
 
@@ -702,6 +703,7 @@
         """
         return len(self.__gens)
 
+    @cached_method
     def gens_reduced(self):
         r"""
         Same as :meth:`gens()` for this ideal, since there is currently no
@@ -715,6 +717,11 @@
             sage: ZZ.ideal(5).gens_reduced()
             (5,)
         """
+        if not getattr(self.__ring, 'eagerly_reduce_ideal_gens_by_gcd', True):
+            from sage.categories.principal_ideal_domains import PrincipalIdealDomains
+            if self.__ring in PrincipalIdealDomains():
+                from sage.arith.misc import gcd
+                return gcd(self.gens()),
         return self.gens()
 
     def is_maximal(self):
@@ -1327,7 +1334,7 @@
             over Integer Ring
             sage: J
             Ideal (2, x) of Univariate Polynomial Ring in x over Integer Ring
             sage: K
             Principal ideal (1) of Univariate Polynomial Ring in x
             over Rational Field
             sage: I.is_principal()
@@ -1362,7 +1369,7 @@
             sage: J = R.base_extend(QQ).ideal(2,x)
             sage: a = I.gen(); a
             x
             sage: b = J.gen(); b
             1
             sage: a.base_ring()
             Integer Ring

src/sage/rings/polynomial/polynomial_ring.py

@@ -241,6 +241,7 @@ class PolynomialRing_generic(Ring):
     """
     Univariate polynomial ring over a ring.
     """
+    eagerly_reduce_ideal_gens_by_gcd = False
 
     def __init__(self, base_ring, name=None, sparse=False, implementation=None,
                  element_class=None, category=None):