gh-35594: Remove some circular imports in `sage.rings`, `sage.symbolic`
    
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Fixes #35580.
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URL: #35594
Reported by: Matthias Köppe
Reviewer(s): Dima Pasechnik

Author: Release Manager

src/.relint.yml

@@ -49,7 +49,7 @@
     namespace package. Type import_statements("SOME_IDENTIFIER") to find a more specific import,
     or use 'sage --fiximports' to fix automatically in the source file.
   # Keep in sync with SAGE_ROOT/src/sage/misc/replace_dot_all.py
-  pattern: 'from\s+sage(|[.](arith|categories|combinat|crypto|databases|data_structures|dynamics|ext|game_theory|games|graphs|groups|interfaces|manifolds|matrix|matroids|misc|modules|monoids|numerical|probability|quadratic_forms|quivers|rings|sat|schemes|sets|stats|tensor)[a-z0-9_.]*|[.]libs)[.]all\s+import'
+  pattern: 'from\s+sage(|[.](arith|categories|combinat|crypto|databases|data_structures|dynamics|ext|game_theory|games|graphs|groups|interfaces|manifolds|matrix|matroids|misc|modules|monoids|numerical|probability|quadratic_forms|quivers|rings|sat|schemes|sets|stats|symbolic|tensor)[a-z0-9_.]*|[.]libs)[.]all\s+import'
   # imports from .all are allowed in all.py; also allow in some modules that need sage.all
   filePattern: '(.*/|)(?!(all|benchmark|dev_tools|parsing|sage_eval))[^/.]*[.](py|pyx|pxi)$'
 

src/sage/categories/classical_crystals.py

@@ -180,7 +180,7 @@ def demazure_character(self, w, f=None):
             u = self.algebra(ZZ).sum_of_monomials(self.module_generators)
             u = self.demazure_operator(u, word)
             if f is None:
-                from sage.symbolic.all import SR as P
+                from sage.symbolic.ring import SR as P
                 x = [P.var('x%s' % (i+1)) for i in range(n)]
                 # TODO: use P.linear_combination when PolynomialRing will be a ModulesWithBasis
                 return sum((coeff*prod((x[i]**(c.weight()[i]) for i in range(n)), P.one()) for c, coeff in u), P.zero())

src/sage/functions/error.py

@@ -46,10 +46,10 @@
 from sage.symbolic.expression import Expression
 from sage.functions.all import exp
 from sage.misc.functional import sqrt
-from sage.symbolic.constants import pi
+from sage.symbolic.constants import I, pi
 from sage.rings.rational import Rational
 from sage.rings.infinity import unsigned_infinity
-from sage.symbolic.expression import I
+
 
 class Function_erf(BuiltinFunction):
     r"""

src/sage/misc/functional.py

@@ -1881,8 +1881,7 @@ def _do_sqrt(x, prec=None, extend=True, all=False):
             from sage.rings.complex_mpfr import ComplexField
             return ComplexField(prec)(x).sqrt(all=all)
     if x == -1:
-        from sage.symbolic.expression import I
-        z = I
+        from sage.symbolic.constants import I as z
     else:
         from sage.symbolic.ring import SR
         z = SR(x).sqrt()

src/sage/modular/modform_hecketriangle/element.py

@@ -285,7 +285,7 @@ def lseries(self, num_prec=None, max_imaginary_part=0, max_asymp_coeffs=40):
             -23.9781792831...
         """
         from sage.rings.integer_ring import ZZ
-        from sage.symbolic.all import pi
+        from sage.symbolic.constants import pi
         from sage.misc.functional import sqrt
         from sage.lfunctions.dokchitser import Dokchitser
 

src/sage/modules/vector_symbolic_dense.py

@@ -55,7 +55,7 @@
 #*****************************************************************************
 
 from . import free_module_element
-from sage.symbolic.all import Expression
+from sage.symbolic.expression import Expression
 
 
 def apply_map(phi):

src/sage/rings/infinity.py

@@ -216,15 +216,17 @@
 #*****************************************************************************
 
 from sys import maxsize
+
+import sage.rings.abc
+
+from sage.misc.fast_methods import Singleton
+from sage.misc.lazy_import import lazy_import
 from sage.rings.ring import Ring
 from sage.structure.element import RingElement, InfinityElement
 from sage.structure.richcmp import rich_to_bool, richcmp
-from sage.misc.fast_methods import Singleton
-import sage.rings.abc
-import sage.rings.integer
-import sage.rings.rational
 
-import sage.rings.integer_ring
+lazy_import('sage.rings.integer', 'Integer')
+
 
 _obj = {}
 class _uniq():
@@ -327,12 +329,11 @@ def __pari__(self):
             sage: pari(oo)
             +oo
         """
-        # For some reason, it seems problematic to import sage.libs.all.pari,
-        # so we call it directly.
+        from sage.libs.pari.all import pari
         if self._sign >= 0:
-            return sage.libs.all.pari('oo')
+            return pari('oo')
         else:
-            return sage.libs.all.pari('-oo')
+            return pari('-oo')
 
     def _latex_(self):
         r"""
@@ -850,7 +851,7 @@ def _div_(self, other):
             0
         """
         if isinstance(other, UnsignedInfinity):
-            return sage.rings.integer_ring.ZZ(0)
+            return Integer(0)
         raise ValueError("quotient of number < oo by number < oo not defined")
 
     def _richcmp_(self, other, op):
@@ -1390,7 +1391,7 @@ def _mul_(self, other):
         if other.is_zero():
             if isinstance(self, InfinityElement):
                 raise SignError("cannot multiply infinity by zero")
-            return sage.rings.integer_ring.ZZ(0)
+            return Integer(0)
         if self.value < 0:
             if isinstance(other, InfinityElement):
                 return -other
@@ -1402,7 +1403,7 @@ def _mul_(self, other):
         if self.value == 0:
             if isinstance(other, InfinityElement):
                 raise SignError("cannot multiply infinity by zero")
-            return sage.rings.integer_ring.ZZ(0)
+            return Integer(0)
 
     def _div_(self, other):
         """

src/sage/rings/number_field/number_field.py

@@ -8285,7 +8285,8 @@ def _coerce_map_from_(self, R):
             sage: K.coerce(sqrt(2))
             Traceback (most recent call last):
             ...
-            TypeError: no canonical coercion from Symbolic Ring to Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095?
+            TypeError: no canonical coercion from Symbolic Ring to Number Field in a
+            with defining polynomial x^2 - 2 with a = 1.414213562373095?
 
         TESTS::
 

src/sage/rings/polynomial/polynomial_element.pyx

@@ -8234,7 +8234,7 @@ cdef class Polynomial(CommutativePolynomial):
         if isinstance(L, sage.rings.abc.SymbolicRing):
             if self.degree() == 2:
                 from sage.misc.functional import sqrt
-                from sage.symbolic.expression import I
+                from sage.symbolic.constants import I
                 coeffs = self.list()
                 D = coeffs[1]*coeffs[1] - 4*coeffs[0]*coeffs[2]
                 l = None

src/sage/rings/real_mpfr.pyx

@@ -153,10 +153,6 @@ from .rational cimport Rational
 
 from sage.categories.map cimport Map
 
-cdef ZZ, QQ, RDF
-from .integer_ring import ZZ
-from .rational_field import QQ
-from .real_double import RDF
 from .real_double cimport RealDoubleElement
 
 import sage.rings.rational_field
@@ -729,6 +725,10 @@ cdef class RealField_class(sage.rings.abc.RealField):
             sage: R.get_action(ZZ)
             Right scalar multiplication by Integer Ring on Univariate Polynomial Ring in x over Real Field with 53 bits of precision
         """
+        from .integer_ring import ZZ
+        from .rational_field import QQ
+        from .real_double import RDF
+
         if S is ZZ:
             return ZZtoRR(ZZ, self)
         elif S is QQ:
@@ -1872,6 +1872,8 @@ cdef class RealNumber(sage.structure.element.RingElement):
             sage: RealField(100)(2).imag()
             0
         """
+        from .integer_ring import ZZ
+
         return ZZ(0)
 
     def str(self, int base=10, size_t digits=0, *, no_sci=None,