gh-36045: `sage.rings.polynomial`: Modularization fixes, `# needs`
    
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- Part of: #29705
- Cherry-picked from: #35095
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URL: #36045
Reported by: Matthias Köppe
Reviewer(s): David Coudert

Author: Release Manager

src/sage/rings/polynomial/cyclotomic.pyx

@@ -30,32 +30,34 @@ import sys
 from cysignals.memory cimport sig_malloc, check_calloc, sig_free
 from cysignals.signals cimport sig_on, sig_off
 
-from sage.structure.element cimport parent
-
 from sage.arith.misc import factor
-from sage.rings.integer_ring import ZZ
-from sage.misc.misc_c import prod
 from sage.combinat.subset import subsets
-from sage.libs.pari.all import pari
+from sage.misc.misc_c import prod
+from sage.rings.integer_ring import ZZ
+from sage.structure.element cimport parent
 
+try:
+    from sage.libs.pari.all import pari
+except ImportError:
+    pass
 
 def cyclotomic_coeffs(nn, sparse=None):
     """
-    Return the coefficients of the n-th cyclotomic polynomial
+    Return the coefficients of the `n`-th cyclotomic polynomial
     by using the formula
 
     .. MATH::
 
         \\Phi_n(x) = \\prod_{d|n} (1-x^{n/d})^{\\mu(d)}
 
-    where `\\mu(d)` is the Möbius function that is 1 if d has an even
-    number of distinct prime divisors, -1 if it has an odd number of
-    distinct prime divisors, and 0 if d is not squarefree.
+    where `\\mu(d)` is the Möbius function that is 1 if `d` has an even
+    number of distinct prime divisors, `-1` if it has an odd number of
+    distinct prime divisors, and `0` if `d` is not squarefree.
 
     Multiplications and divisions by polynomials of the
     form `1-x^n` can be done very quickly in a single pass.
 
-    If sparse is ``True``, the result is returned as a dictionary of
+    If ``sparse`` is ``True``, the result is returned as a dictionary of
     the non-zero entries, otherwise the result is returned as a list
     of python ints.
 
@@ -72,17 +74,19 @@ def cyclotomic_coeffs(nn, sparse=None):
 
     Check that it has the right degree::
 
-        sage: euler_phi(30)
+        sage: euler_phi(30)                                                             # needs sage.libs.pari
         8
-        sage: R(cyclotomic_coeffs(14)).factor()
+        sage: R(cyclotomic_coeffs(14)).factor()                                         # needs sage.libs.pari
         x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
 
     The coefficients are not always +/-1::
 
         sage: cyclotomic_coeffs(105)
-        [1, 1, 1, 0, 0, -1, -1, -2, -1, -1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, -1, -1, -2, -1, -1, 0, 0, 1, 1, 1]
+        [1, 1, 1, 0, 0, -1, -1, -2, -1, -1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, -1,
+         0, -1, 0, -1, 0, -1, 0, -1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, -1, -1, -2,
+         -1, -1, 0, 0, 1, 1, 1]
 
-    In fact the height is not bounded by any polynomial in n (Erdos),
+    In fact the height is not bounded by any polynomial in `n` (Erdos),
     although takes a while just to exceed linear::
 
         sage: v = cyclotomic_coeffs(1181895)
@@ -200,10 +204,10 @@ def cyclotomic_value(n, x):
 
     INPUT:
 
-    - `n` -- an Integer, specifying which cyclotomic polynomial is to be
+    - ``n`` -- an Integer, specifying which cyclotomic polynomial is to be
       evaluated
 
-    - `x` -- an element of a ring
+    - ``x`` -- an element of a ring
 
     OUTPUT:
 
@@ -247,9 +251,12 @@ def cyclotomic_value(n, x):
 
     TESTS::
 
-        sage: R.<x> = QQ[]
-        sage: K.<i> = NumberField(x^2 + 1)
-        sage: for y in [-1, 0, 1, 2, 1/2, Mod(3, 8), Mod(3,11), GF(9,'a').gen(), Zp(3)(54), i, x^2+2]:
+        sage: elements = [-1, 0, 1, 2, 1/2, Mod(3, 8), Mod(3,11)]
+        sage: R.<x> = QQ[]; elements += [x^2 + 2]
+        sage: K.<i> = NumberField(x^2 + 1); elements += [i]                             # needs sage.rings.number_fields
+        sage: elements += [GF(9,'a').gen()]                                             # needs sage.rings.finite_rings
+        sage: elements += [Zp(3)(54)]                                                   # needs sage.rings.padics
+        sage: for y in elements:
         ....:     for n in [1..60]:
         ....:         val1 = cyclotomic_value(n, y)
         ....:         val2 = cyclotomic_polynomial(n)(y)
@@ -258,29 +265,31 @@ def cyclotomic_value(n, x):
         ....:         if val1.parent() is not val2.parent():
         ....:             print("Wrong parent for cyclotomic_value(%s, %s) in %s"%(n,y,parent(y)))
 
-        sage: cyclotomic_value(20, I)
+        sage: cyclotomic_value(20, I)                                                   # needs sage.symbolic
         5
         sage: a = cyclotomic_value(10, mod(3, 11)); a
         6
         sage: a.parent()
         Ring of integers modulo 11
-        sage: cyclotomic_value(30, -1.0)
+        sage: cyclotomic_value(30, -1.0)                                                # needs sage.rings.real_mpfr
         1.00000000000000
+
+        sage: # needs sage.libs.pari
         sage: S.<t> = R.quotient(R.cyclotomic_polynomial(15))
         sage: cyclotomic_value(15, t)
         0
         sage: cyclotomic_value(30, t)
         2*t^7 - 2*t^5 - 2*t^3 + 2*t
         sage: S.<t> = R.quotient(x^10)
-        sage: cyclotomic_value(2^128-1, t)
+        sage: cyclotomic_value(2^128 - 1, t)
         -t^7 - t^6 - t^5 + t^2 + t + 1
-        sage: cyclotomic_value(10,mod(3,4))
+        sage: cyclotomic_value(10, mod(3,4))
         1
 
     Check that the issue with symbolic element in :trac:`14982` is fixed::
 
-        sage: a = cyclotomic_value(3, I)
-        sage: parent(a)
+        sage: a = cyclotomic_value(3, I)                                                # needs sage.rings.number_fields
+        sage: parent(a)                                                                 # needs sage.rings.number_fields
         Number Field in I with defining polynomial x^2 + 1 with I = 1*I
     """
     n = ZZ(n)
@@ -387,7 +396,7 @@ def bateman_bound(nn):
     EXAMPLES::
 
         sage: from sage.rings.polynomial.cyclotomic import bateman_bound
-        sage: bateman_bound(2**8*1234567893377)
+        sage: bateman_bound(2**8 * 1234567893377)                                       # needs sage.libs.pari
         66944986927
     """
     _, n = nn.val_unit(2)

src/sage/rings/polynomial/flatten.py

@@ -111,37 +111,41 @@ def __init__(self, domain):
 
         ::
 
+            sage: # needs sage.rings.number_field
             sage: x = polygen(ZZ, 'x')
-            sage: K.<v> = NumberField(x^3 - 2)                                          # optional - sage.rings.number_field
-            sage: R = K['x','y']['a','b']                                               # optional - sage.rings.number_field
+            sage: K.<v> = NumberField(x^3 - 2)
+            sage: R = K['x','y']['a','b']
             sage: from sage.rings.polynomial.flatten import FlatteningMorphism
-            sage: f = FlatteningMorphism(R)                                             # optional - sage.rings.number_field
-            sage: f(R('v*a*x^2 + b^2 + 1/v*y'))                                         # optional - sage.rings.number_field
+            sage: f = FlatteningMorphism(R)
+            sage: f(R('v*a*x^2 + b^2 + 1/v*y'))
             v*x^2*a + b^2 + (1/2*v^2)*y
 
         ::
 
-            sage: R = QQbar['x','y']['a','b']                                           # optional - sage.rings.number_field
+            sage: # needs sage.rings.number_field
+            sage: R = QQbar['x','y']['a','b']
             sage: from sage.rings.polynomial.flatten import FlatteningMorphism
-            sage: f = FlatteningMorphism(R)                                             # optional - sage.rings.number_field
-            sage: f(R('QQbar(sqrt(2))*a*x^2 + b^2 + QQbar(I)*y'))                       # optional - sage.rings.number_field
+            sage: f = FlatteningMorphism(R)
+            sage: f(R('QQbar(sqrt(2))*a*x^2 + b^2 + QQbar(I)*y'))                       # needs sage.symbolic
             1.414213562373095?*x^2*a + b^2 + I*y
 
         ::
 
-            sage: R.<z> = PolynomialRing(QQbar, 1)                                      # optional - sage.rings.number_field
+            sage: # needs sage.rings.number_field
+            sage: R.<z> = PolynomialRing(QQbar, 1)
             sage: from sage.rings.polynomial.flatten import FlatteningMorphism
-            sage: f = FlatteningMorphism(R)                                             # optional - sage.rings.number_field
-            sage: f.domain(), f.codomain()                                              # optional - sage.rings.number_field
+            sage: f = FlatteningMorphism(R)
+            sage: f.domain(), f.codomain()
             (Multivariate Polynomial Ring in z over Algebraic Field,
              Multivariate Polynomial Ring in z over Algebraic Field)
 
         ::
 
-            sage: R.<z> = PolynomialRing(QQbar)                                         # optional - sage.rings.number_field
+            sage: # needs sage.rings.number_field
+            sage: R.<z> = PolynomialRing(QQbar)
             sage: from sage.rings.polynomial.flatten import FlatteningMorphism
-            sage: f = FlatteningMorphism(R)                                             # optional - sage.rings.number_field
-            sage: f.domain(), f.codomain()                                              # optional - sage.rings.number_field
+            sage: f = FlatteningMorphism(R)
+            sage: f.domain(), f.codomain()
             (Univariate Polynomial Ring in z over Algebraic Field,
              Univariate Polynomial Ring in z over Algebraic Field)
 
@@ -376,8 +380,8 @@ def _call_(self, p):
 
             sage: from sage.rings.polynomial.flatten import FlatteningMorphism
             sage: rings = [ZZ['x']['y']['a,b,c']]
-            sage: rings += [GF(4)['x','y']['a','b']]                                    # optional - sage.rings.finite_rings
-            sage: rings += [AA['x']['a','b']['y'], QQbar['a1','a2']['t']['X','Y']]      # optional - sage.rings.number_field
+            sage: rings += [GF(4)['x','y']['a','b']]                                    # needs sage.rings.finite_rings
+            sage: rings += [AA['x']['a','b']['y'], QQbar['a1','a2']['t']['X','Y']]      # needs sage.rings.number_field
             sage: for R in rings:
             ....:    f = FlatteningMorphism(R)
             ....:    g = f.section()
@@ -490,8 +494,9 @@ def __init__(self, domain, D):
             sage: P.<z> = AffineSpace(R, 1)
             sage: H = End(P)
             sage: f = H([z^2 + c])
-            sage: f.specialization({c:1})
-            Scheme endomorphism of Affine Space of dimension 1 over Real Field with 53 bits of precision
+            sage: f.specialization({c:1})                                               # needs sage.modules
+            Scheme endomorphism of
+             Affine Space of dimension 1 over Real Field with 53 bits of precision
               Defn: Defined on coordinates by sending (z) to
                     (z^2 + 1.00000000000000)
         """

src/sage/rings/polynomial/groebner_fan.py

@@ -1246,51 +1246,51 @@ def render(self, file=None, larger=False, shift=0, rgbcolor=(0, 0, 0),
 
         INPUT:
 
-        -  ``file`` - a filename if you prefer the output
+        -  ``file`` -- a filename if you prefer the output
            saved to a file. This will be in xfig format.
 
-        -  ``shift`` - shift the positions of the variables in
+        -  ``shift`` -- shift the positions of the variables in
            the drawing. For example, with shift=1, the corners will be b
            (right), c (left), and d (top). The shifting is done modulo the
            number of variables in the polynomial ring. The default is 0.
 
-        -  ``larger`` - bool (default: ``False``); if ``True``, make
+        -  ``larger`` -- bool (default: ``False``); if ``True``, make
            the triangle larger so that the shape of the Groebner region
            appears. Affects the xfig file but probably not the sage graphics
            (?)
 
-        -  ``rgbcolor`` - This will not affect the saved xfig
+        -  ``rgbcolor`` -- This will not affect the saved xfig
            file, only the sage graphics produced.
 
-        -  ``polyfill`` - Whether or not to fill the cones with
+        -  ``polyfill`` -- Whether or not to fill the cones with
            a color determined by the highest degree in each reduced Groebner
            basis for that cone.
 
-        -  ``scale_colors`` - if True, this will normalize
+        -  ``scale_colors`` -- if True, this will normalize
            color values to try to maximize the range
 
 
         EXAMPLES::
 
             sage: R.<x,y,z> = PolynomialRing(QQ,3)
             sage: G = R.ideal([y^3 - x^2, y^2 - 13*x,z]).groebner_fan()
-            sage: test_render = G.render()
+            sage: test_render = G.render()                                              # needs sage.plot
 
         ::
 
             sage: R.<x,y,z> = PolynomialRing(QQ,3)
             sage: G = R.ideal([x^2*y - z, y^2*z - x, z^2*x - y]).groebner_fan()
-            sage: test_render = G.render(larger=True)
+            sage: test_render = G.render(larger=True)                                   # needs sage.plot
 
         TESTS:
 
         Testing the case where the number of generators is < 3. Currently,
-        this should raise a ``NotImplementedError`` error.
+        this should raise a :class:`NotImplementedError`.
 
         ::
 
             sage: R.<x,y> = PolynomialRing(QQ, 2)
-            sage: R.ideal([y^3 - x^2, y^2 - 13*x]).groebner_fan().render()
+            sage: R.ideal([y^3 - x^2, y^2 - 13*x]).groebner_fan().render()              # needs sage.plot
             Traceback (most recent call last):
             ...
             NotImplementedError
@@ -1460,17 +1460,17 @@ def render3d(self, verbose=False):
 
             sage: R4.<w,x,y,z> = PolynomialRing(QQ,4)
             sage: gf = R4.ideal([w^2-x,x^2-y,y^2-z,z^2-x]).groebner_fan()
-            sage: three_d = gf.render3d()
+            sage: three_d = gf.render3d()                                               # needs sage.plot
 
         TESTS:
 
         Now test the case where the number of generators is not 4. Currently,
-        this should raise a ``NotImplementedError`` error.
+        this should raise a :class:`NotImplementedError` error.
 
         ::
 
             sage: P.<a,b,c> = PolynomialRing(QQ, 3, order="lex")
-            sage: sage.rings.ideal.Katsura(P, 3).groebner_fan().render3d()
+            sage: sage.rings.ideal.Katsura(P, 3).groebner_fan().render3d()              # needs sage.plot
             Traceback (most recent call last):
             ...
             NotImplementedError

src/sage/rings/polynomial/hilbert.pyx

@@ -20,9 +20,10 @@ in any example with more than 34 variables.
 #
 #*****************************************************************************
 
+import sage.interfaces.abc
+
 from sage.rings.polynomial.polydict cimport ETuple
 from sage.rings.polynomial.polynomial_integer_dense_flint cimport Polynomial_integer_dense_flint
-from sage.interfaces.singular import Singular
 
 from cysignals.memory cimport sig_malloc
 from cpython.list cimport PyList_GET_ITEM
@@ -476,7 +477,7 @@ def first_hilbert_series(I, grading=None, return_grading=False):
     cdef Polynomial_integer_dense_flint fhs = Polynomial_integer_dense_flint.__new__(Polynomial_integer_dense_flint)
     fhs._parent = PR
     fhs._is_gen = 0
-    if isinstance(I.parent(), Singular):
+    if isinstance(I, sage.interfaces.abc.SingularElement):
         S = I._check_valid()
         # First, we need to deal with quotient rings, which also covers the case
         # of graded commutative rings that arise as cohomology rings in odd characteristic.

src/sage/rings/polynomial/ideal.py

@@ -32,9 +32,9 @@ def residue_class_degree(self):
 
         EXAMPLES::
 
-            sage: R.<t> = GF(5)[]                                                       # optional - sage.rings.finite_rings
-            sage: P = R.ideal(t^4 + t + 1)                                              # optional - sage.rings.finite_rings
-            sage: P.residue_class_degree()                                              # optional - sage.rings.finite_rings
+            sage: R.<t> = GF(5)[]
+            sage: P = R.ideal(t^4 + t + 1)
+            sage: P.residue_class_degree()
             4
         """
         return self.gen().degree()
@@ -45,8 +45,8 @@ def residue_field(self, names=None, check=True):
 
         EXAMPLES::
 
-            sage: R.<t> = GF(17)[]; P = R.ideal(t^3 + 2*t + 9)                          # optional - sage.rings.finite_rings
-            sage: k.<a> = P.residue_field(); k                                          # optional - sage.rings.finite_rings
+            sage: R.<t> = GF(17)[]; P = R.ideal(t^3 + 2*t + 9)
+            sage: k.<a> = P.residue_field(); k                                          # needs sage.rings.finite_rings
             Residue field in a of Principal ideal (t^3 + 2*t + 9) of
              Univariate Polynomial Ring in t over Finite Field of size 17
         """
@@ -75,11 +75,11 @@ def groebner_basis(self, algorithm=None):
 
             sage: R.<x> = QQ[]
             sage: I = R.ideal([x^2 - 1, x^3 - 1])
-            sage: G = I.groebner_basis(); G                                             # optional - sage.libs.singular
+            sage: G = I.groebner_basis(); G
             [x - 1]
-            sage: type(G)                                                               # optional - sage.libs.singular
+            sage: type(G)
             <class 'sage.rings.polynomial.multi_polynomial_sequence.PolynomialSequence_generic'>
-            sage: list(G)                                                               # optional - sage.libs.singular
+            sage: list(G)
             [x - 1]
         """
         gb = self.gens_reduced()