sagemathgh-38061: `sage.rings`: Modularization fixes (imports)
    
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Replacing unconditional imports from `sage.libs.pari`,
`sage.misc.sage_eval`, `sage.rings.number_field`,
`sage.rings.polynomial.pbori`, `sage.symbolic` using standard
modularization techniques.

- Cherry-picked from sagemath#35095

### 📝 Checklist

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- [ ] The description explains in detail what this PR is about.
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- [ ] I have created tests covering the changes.
- [ ] I have updated the documentation and checked the documentation
preview.

### ⌛ Dependencies

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URL: sagemath#38061
Reported by: Matthias Köppe
Reviewer(s): Kwankyu Lee

Author: Release Manager

src/sage/rings/asymptotic/asymptotics_multivariate_generating_functions.py

@@ -199,18 +199,20 @@
 from functools import total_ordering
 from itertools import combinations_with_replacement
 
-from sage.structure.element import RingElement
-from sage.structure.unique_representation import UniqueRepresentation
-from sage.structure.parent import Parent
-from sage.calculus.var import var
-from sage.calculus.functional import diff
-from sage.symbolic.ring import SR
+from sage.categories.rings import Rings
+from sage.misc.lazy_import import lazy_import
 from sage.misc.misc_c import prod
 from sage.rings.integer import Integer
 from sage.rings.integer_ring import ZZ
 from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
-from sage.categories.rings import Rings
+from sage.structure.element import RingElement
+from sage.structure.parent import Parent
 from sage.structure.richcmp import richcmp_by_eq_and_lt
+from sage.structure.unique_representation import UniqueRepresentation
+
+lazy_import("sage.calculus.var", "var")
+lazy_import("sage.calculus.functional", "diff")
+lazy_import("sage.symbolic.ring", "SR")
 
 
 @total_ordering

src/sage/rings/complex_arb.pyx

@@ -179,6 +179,8 @@ from sage.libs.gsl.complex cimport gsl_complex_rect
 from sage.rings.real_double cimport RealDoubleElement
 from sage.rings.complex_double cimport ComplexDoubleElement
 from sage.rings.integer cimport Integer
+from sage.rings.rational_field import QQ
+from sage.rings.number_field.number_field_base import NumberField
 from sage.rings.polynomial.polynomial_complex_arb cimport Polynomial_complex_arb
 from sage.rings.real_arb cimport mpfi_to_arb, arb_to_mpfi
 from sage.rings.real_arb import RealBallField
@@ -568,8 +570,7 @@ class ComplexBallField(UniqueRepresentation, sage.rings.abc.ComplexBallField):
         elif isinstance(other, ComplexBallField):
             return other._prec >= self._prec
 
-        import sage.rings.number_field.number_field as number_field
-        if isinstance(other, number_field.NumberField_generic):
+        if other is not QQ and isinstance(other, NumberField):
             emb = other.coerce_embedding()
             return emb is not None and self.has_coerce_map_from(emb.codomain())
 

src/sage/rings/complex_mpc.pyx

@@ -72,7 +72,11 @@ from sage.structure.parent cimport Parent
 from sage.structure.element cimport Element
 from sage.structure.richcmp cimport rich_to_bool
 from sage.categories.map cimport Map
-from sage.libs.pari.all import pari
+
+try:
+    from sage.libs.pari.all import pari, pari_gen, PariError
+except ImportError:
+    pari_gen = PariError = ()
 
 from sage.rings.integer cimport Integer
 from sage.rings.complex_mpfr cimport ComplexNumber
@@ -842,7 +846,7 @@ cdef class MPComplexNumber(sage.structure.element.FieldElement):
             elif isinstance(z, ComplexNumber):
                 mpc_set_fr_fr(self.value, (<ComplexNumber>z).__re, (<ComplexNumber>z).__im, rnd)
                 return
-            elif isinstance(z, sage.libs.pari.all.pari_gen):
+            elif isinstance(z, pari_gen):
                 real, imag = z.real(), z.imag()
             elif isinstance(z, (list, tuple)):
                 real, imag = z
@@ -2238,7 +2242,7 @@ cdef class MPComplexNumber(sage.structure.element.FieldElement):
         """
         try:
             return self._parent(self.__pari__().eta(not omit_frac))
-        except sage.libs.pari.all.PariError:
+        except PariError:
             raise ValueError("value must be in the upper half plane")
 
     def gamma(self):
@@ -2264,7 +2268,7 @@ cdef class MPComplexNumber(sage.structure.element.FieldElement):
         """
         try:
             return self._parent(self.__pari__().gamma())
-        except sage.libs.pari.all.PariError:
+        except PariError:
             from sage.rings.infinity import UnsignedInfinityRing
             return UnsignedInfinityRing.gen()
 

src/sage/rings/complex_mpfr.pyx

@@ -65,13 +65,9 @@ except ImportError:
 
 # Some objects that are not imported at startup in order to break
 # circular imports
-NumberFieldElement_quadratic = None
-AlgebraicNumber_base = None
-AlgebraicNumber = None
-AlgebraicReal = None
+NumberFieldElement_quadratic = ()
 AA = None
 QQbar = None
-SR = None
 CDF = CLF = RLF = None
 def late_import():
     """
@@ -82,25 +78,14 @@ def late_import():
         sage: sage.rings.complex_mpfr.late_import()
     """
     global NumberFieldElement_quadratic
-    global AlgebraicNumber_base
-    global AlgebraicNumber
-    global AlgebraicReal
-    global UniversalCyclotomicField
-    global AA, QQbar, SR
+    global AA, QQbar
     global CLF, RLF, CDF
-    if NumberFieldElement_quadratic is None:
-        import sage.rings.number_field.number_field
-        import sage.rings.number_field.number_field_element_quadratic as nfeq
-        NumberFieldElement_quadratic = nfeq.NumberFieldElement_quadratic
-        import sage.rings.qqbar
-        AlgebraicNumber_base = sage.rings.qqbar.AlgebraicNumber_base
-        AlgebraicNumber = sage.rings.qqbar.AlgebraicNumber
-        AlgebraicReal = sage.rings.qqbar.AlgebraicReal
-        from sage.rings.universal_cyclotomic_field import UniversalCyclotomicField
-        AA = sage.rings.qqbar.AA
-        QQbar = sage.rings.qqbar.QQbar
-        import sage.symbolic.ring
-        SR = sage.symbolic.ring.SR
+    if CLF is None:
+        try:
+            from sage.rings.number_field.number_field_element_quadratic import NumberFieldElement_quadratic
+            from sage.rings.qqbar import AA, QQbar
+        except ImportError:
+            pass
         from sage.rings.real_lazy import CLF, RLF
         from sage.rings.complex_double import CDF
 
@@ -586,7 +571,7 @@ class ComplexField_class(sage.rings.abc.ComplexField):
         # parts of real elements) that get picked for conversion from UCF both
         # to CC and to other types of complex fields depend in which order the
         # coercions are discovered.
-        if isinstance(S, UniversalCyclotomicField):
+        if isinstance(S, sage.rings.abc.UniversalCyclotomicField):
             return self._generic_coerce_map(S)
         return self._coerce_map_via([CLF], S)
 

src/sage/rings/fraction_field.py

@@ -680,7 +680,11 @@ def _element_constructor_(self, x, y=None, coerce=True):
         x = py_scalar_to_element(x)
         y = py_scalar_to_element(y)
 
-        from sage.libs.pari.all import pari_gen
+        try:
+            from sage.libs.pari.all import pari_gen
+        except ImportError:
+            pari_gen = ()
+
         if isinstance(x, pari_gen) and x.type() == 't_POL':
             # This recursive approach is needed because PARI
             # represents multivariate polynomials as iterated

src/sage/rings/infinity.py

@@ -1285,7 +1285,10 @@ def _pushout_(self, other):
             sage: QQbar(-2*i)*infinity                                                  # needs sage.rings.number_field sage.symbolic
             (-I)*Infinity
         """
-        from sage.symbolic.ring import SR
+        try:
+            from sage.symbolic.ring import SR
+        except ImportError:
+            return None
         if SR.has_coerce_map_from(other):
             return SR
 
@@ -1801,10 +1804,15 @@ def test_comparison(ring):
         ...
         AssertionError: testing -1000.0 in Symbolic Ring: id = ...
     """
-    from sage.symbolic.ring import SR
+
     from sage.rings.rational_field import QQ
-    elements = [-1e3, 99.9999, -SR(2).sqrt(), 0, 1,
-                3 ** (-QQ.one() / 3), SR.pi(), 100000]
+    elements = [-1e3, 99.9999, 0, 1, 100000]
+    try:
+        from sage.symbolic.ring import SR
+    except ImportError:
+        pass
+    else:
+        elements += [-SR(2).sqrt(), SR.pi(), 3 ** (-QQ.one() / 3)]
     elements.append(ring.an_element())
     elements.extend(ring.some_elements())
     for z in elements:

src/sage/rings/integer_ring.pyx

@@ -54,8 +54,6 @@ import sage.rings.infinity
 import sage.rings.rational
 import sage.rings.rational_field
 import sage.rings.ideal
-import sage.libs.pari.all
-import sage.rings.ideal
 from sage.categories.basic import EuclideanDomains, DedekindDomains
 from sage.categories.infinite_enumerated_sets import InfiniteEnumeratedSets
 from sage.categories.noetherian_rings import NoetherianRings

src/sage/rings/polynomial/multi_polynomial_libsingular.pyx

@@ -215,16 +215,18 @@ from sage.libs.singular.ring cimport singular_ring_new, singular_ring_reference,
 
 # polynomial imports
 from sage.rings.polynomial.multi_polynomial_ring import MPolynomialRing_polydict, MPolynomialRing_polydict_domain
+from sage.rings.polynomial.multi_polynomial_ring_base import BooleanPolynomialRing_base
 from sage.rings.polynomial.multi_polynomial_element import MPolynomial_polydict
 from sage.rings.polynomial.multi_polynomial_ideal import MPolynomialIdeal
 from sage.rings.polynomial.polydict cimport ETuple
 from sage.rings.polynomial.polynomial_ring import is_PolynomialRing
 
 # base ring imports
 import sage.rings.abc
+import sage.structure.element
+
 from sage.rings.rational cimport Rational
 from sage.rings.rational_field import QQ
-import sage.rings.abc
 from sage.rings.integer_ring import is_IntegerRing, ZZ
 from sage.rings.integer cimport Integer
 from sage.rings.number_field.number_field_base cimport NumberField
@@ -950,8 +952,7 @@ cdef class MPolynomialRing_libsingular(MPolynomialRing_base):
                     raise
             return new_MP(self, _p)
 
-        from sage.rings.polynomial.pbori.pbori import BooleanPolynomial
-        if isinstance(element, BooleanPolynomial):
+        if isinstance(element, sage.structure.element.Element) and isinstance(element.parent(), BooleanPolynomialRing_base):
             if element.constant():
                 if element:
                     return self._one_element

src/sage/rings/power_series_ring.py

@@ -137,7 +137,7 @@
 import sage.categories.commutative_rings as commutative_rings
 import sage.misc.latex as latex
 from sage.interfaces.abc import MagmaElement
-from sage.misc.sage_eval import sage_eval
+from sage.misc.lazy_import import lazy_import
 from sage.rings import (
     integer,
     laurent_series_ring,
@@ -167,6 +167,8 @@
 
 from sage.categories.complete_discrete_valuation import CompleteDiscreteValuationRings
 
+lazy_import('sage.misc.sage_eval', 'sage_eval')
+
 try:
     from .laurent_series_ring import LaurentSeriesRing
     from .laurent_series_ring_element import LaurentSeries

src/sage/rings/quotient_ring.py

@@ -307,7 +307,7 @@ def QuotientRing(R, I, names=None, **kwds):
         kwds.pop('implementation')
         return BooleanPolynomialRing(R.ngens(), names=names, **kwds)
     # workaround to silence warning from #34806
-    from sage.rings.number_field.order import Order
+    from sage.rings.abc import Order
     if isinstance(R, Order):
         if not R.is_maximal():
             raise NotImplementedError('only implemented for maximal orders')
@@ -480,7 +480,7 @@ def __init__(self, R, I, names, category=None):
         if R not in _Rings:
             raise TypeError("The first argument must be a ring, but %s is not" % R)
         # workaround to silence warning from #34806
-        from sage.rings.number_field.order import Order
+        from sage.rings.abc import Order
         if isinstance(R, Order):
             M = R.number_field().ideal_monoid()
         else: