gh-39958: some care for unused variables in rings/polynomial/ pyx files
    
This fixes all suggestions of cython-lint about unused variables in pyx
files in `rings/polynomial/`

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
    
URL: #39958
Reported by: Frédéric Chapoton
Reviewer(s): Travis Scrimshaw

Author: Release Manager

src/sage/rings/polynomial/hilbert.pyx

@@ -131,7 +131,6 @@ cdef list quotient_by_var(list L, size_t index):
     Return the quotient of the ideal represented by ``L`` and the
     variable number ``index``.
     """
-    cdef ETuple m_j
     cdef list result = L[:len(L)] # creates a copy
     cdef size_t i
     for i in range(len(L)):
@@ -166,7 +165,6 @@ cdef bint HilbertBaseCase(Polynomial_integer_dense_flint fhs, Node D, tuple w) n
     Otherwiese, ``False`` is returned.
     """
     cdef size_t i, j, exp
-    cdef int e
     # First, the easiest cases:
     if not D.Id: # The zero ideal
         fmpz_poly_set_coeff_si(fhs._poly, 0, 1) # = PR(1)
@@ -227,7 +225,6 @@ cdef bint HilbertBaseCase(Polynomial_integer_dense_flint fhs, Node D, tuple w) n
 
     easy = True
     cdef ETuple m2
-    cdef list v
     for j in range(i+1, len(D.Id)):
         if (<ETuple>PyList_GET_ITEM(D.Id,j))._nonzero > 1:
             # i.e., another generator contains more than a single var
@@ -296,7 +293,7 @@ cdef make_children(Node D, tuple w):
     if ``D.Right`` is not ``None``.
     """
     cdef size_t j, m
-    cdef int i, ii
+    cdef int i
     # Determine the variable that appears most often in the monomials.
     # If "most often" means "only once", then instead we choose a variable that is
     # guaranteed to appear in a composed monomial.

src/sage/rings/polynomial/multi_polynomial.pyx

@@ -884,7 +884,7 @@ cdef class MPolynomial(CommutativePolynomial):
         except AttributeError:
             raise NotImplementedError
         else:
-            q, r = quo_rem(other)
+            _, r = quo_rem(other)
             return r
 
     def change_ring(self, R):
@@ -2566,7 +2566,6 @@ cdef class MPolynomial(CommutativePolynomial):
 
         prec = kwds.get('prec', 300)
         return_conjugation = kwds.get('return_conjugation', True)
-        error_limit = kwds.get('error_limit', 0.000001)
         emb = kwds.get('emb', None)
 
         # getting a numerical approximation of the roots of our polynomial
@@ -2606,7 +2605,8 @@ cdef class MPolynomial(CommutativePolynomial):
             # since we are searching anyway, don't need the 'true' reduced covariant
             from sage.rings.polynomial.binary_form_reduce import smallest_poly
             norm_type = kwds.get('norm_type', 'norm')
-            sm_F, sm_m = smallest_poly(self(tuple(M * vector([x,y]))), prec=prec, norm_type=norm_type, emb=emb)
+            _, sm_m = smallest_poly(self(tuple(M * vector([x, y]))), prec=prec,
+                                    norm_type=norm_type, emb=emb)
             M = M*sm_m
         else:
             # solve the minimization problem for 'true' covariant
@@ -2631,7 +2631,7 @@ cdef class MPolynomial(CommutativePolynomial):
             return (self(tuple(M * vector([x,y]))), M)
         return self(tuple(M * vector([x,y])))
 
-    def is_unit(self):
+    def is_unit(self) -> bool:
         r"""
         Return ``True`` if ``self`` is a unit, that is, has a
         multiplicative inverse.
@@ -2892,7 +2892,7 @@ cdef class MPolynomial(CommutativePolynomial):
         return result(True)
 
 
-def _is_M_convex_(points):
+def _is_M_convex_(points) -> bool:
     r"""
     Return whether ``points`` represents a set of integer lattice points
     which are M-convex.
@@ -2961,7 +2961,6 @@ def _is_M_convex_(points):
         for p2 in points_set:
             if p2 == p1:
                 continue
-            delta = list(x2 - x1 for x1, x2 in zip(p1, p2))
             for i in range(dim):
                 if p2[i] > p1[i]:
                     # modify list_p1 to represent point p1 + e_i - e_j for various i, j

src/sage/rings/polynomial/multi_polynomial_ideal_libsingular.pyx

@@ -121,7 +121,6 @@ cdef ideal *sage_ideal_to_singular_ideal(I) except NULL:
         gens = I.gens()
     except AttributeError:
         gens = I
-    cdef ideal *result
     cdef ring *r
     cdef ideal *i
     cdef int j = 0
@@ -245,7 +244,6 @@ def slimgb_libsingular(I):
     """
     global singular_options
 
-    cdef tHomog hom=testHomog
     cdef ideal *i
     cdef ring *r
     cdef ideal *result

src/sage/rings/polynomial/multi_polynomial_libsingular.pyx

@@ -4555,14 +4555,14 @@ cdef class MPolynomial_libsingular(MPolynomial_libsingular_base):
             (x - y) * (x + y) * (x^2 + y^2) * (x^4 + y^4) * (x^8 + y^8) * (x^16 + y^16) * (x^32 + y^32) * (x^64 + y^64) * (x^128 + y^128) * (x^256 + y^256) * (x^512 + y^512) * (x^1024 + y^1024)
         """
         cdef ring *_ring = self._parent_ring
-        cdef poly *ptemp
         cdef intvec *iv
         cdef int *ivv
         cdef ideal *I = NULL
         cdef MPolynomialRing_libsingular parent = self._parent
         cdef int i
 
-        if _ring!=currRing: rChangeCurrRing(_ring)
+        if _ring != currRing:
+            rChangeCurrRing(_ring)
 
         if p_IsConstant(self._poly, _ring):
             return self.constant_coefficient().factor()
@@ -4687,14 +4687,13 @@ cdef class MPolynomial_libsingular(MPolynomial_libsingular_base):
         cdef ideal *_I
         cdef MPolynomialRing_libsingular parent = self._parent
         cdef int i = 0
-        cdef int j
         cdef ring *r = self._parent_ring
         cdef ideal *res = NULL
 
         if isinstance(I, MPolynomialIdeal):
             I = I.gens()
 
-        _I = idInit(len(I),1)
+        _I = idInit(len(I), 1)
 
         for f in I:
             if not (isinstance(f,MPolynomial_libsingular)

src/sage/rings/polynomial/ore_polynomial_element.pyx

@@ -906,7 +906,7 @@ cdef class OrePolynomial(AlgebraElement):
             raise TypeError("the base ring must be a field")
         cdef OrePolynomial G = self
         cdef OrePolynomial U = self._parent.one()
-        cdef OrePolynomial V, V1, V2, Q, R, T
+        cdef OrePolynomial V, V1, Q, R, T
         cdef Morphism m
         if not other:
             V = self._parent.zero()
@@ -2568,7 +2568,7 @@ cdef class OrePolynomial_generic_dense(OrePolynomial):
             sage: a + a == 2*a
             True
         """
-        cdef Py_ssize_t i, min
+        cdef Py_ssize_t i
         cdef list x = (<OrePolynomial_generic_dense>self)._coeffs
         cdef list y = (<OrePolynomial_generic_dense>right)._coeffs
         cdef Py_ssize_t dx = len(x), dy = len(y)
@@ -2599,7 +2599,7 @@ cdef class OrePolynomial_generic_dense(OrePolynomial):
             sage: c - c == 0
             True
         """
-        cdef Py_ssize_t i, min
+        cdef Py_ssize_t i
         cdef list x = (<OrePolynomial_generic_dense>self)._coeffs
         cdef list y = (<OrePolynomial_generic_dense>right)._coeffs
         cdef Py_ssize_t dx = len(x), dy = len(y)
@@ -2867,16 +2867,16 @@ cdef class OrePolynomial_generic_dense(OrePolynomial):
             sage: a.coefficients(sparse=False)
             [t^2 + 1, 0, t + 1, 0, 1]
         """
-        zero = self.parent().base_ring().zero()
         if sparse:
             return [c for c in self._coeffs if not c.is_zero()]
         else:
             return self._coeffs
 
     def hilbert_shift(self, s, var=None):
         r"""
-        Return this Ore polynomial with variable shifted by `s`,
-        i.e. if this Ore polynomial is `P(x)`, return `P(x+s)`.
+        Return this Ore polynomial with variable shifted by `s`.
+
+        If this Ore polynomial is `P(x)`, this returns `P(x+s)`.
 
         INPUT:
 

src/sage/rings/polynomial/pbori/pbori.pyx

@@ -2130,7 +2130,6 @@ class BooleanMonomialMonoid(UniqueRepresentation, Monoid_class):
             x*z
         """
         cdef BooleanMonomial m
-        cdef PBMonom t
 
         # this is needed for the PolyBoRi python code
         if other is None:

src/sage/rings/polynomial/polynomial_element.pyx

@@ -2134,7 +2134,7 @@ cdef class Polynomial(CommutativePolynomial):
             (x^3 + z8^7 + z8^4 + z8^3 + z8^2 + z8, 3)]
         """
         q = self.base_ring().order() # p^k
-        R, x = self.parent().objgen()
+        _, x = self.parent().objgen()
 
         # Initialise values
         v = self
@@ -2983,7 +2983,6 @@ cdef class Polynomial(CommutativePolynomial):
                 # see github issue 24308
                 p = -1
             if 0 < p <= right and (self.base_ring() in sage.categories.integral_domains.IntegralDomains() or p.is_prime()):
-                x = self.parent().gen()
                 ret = self.parent().one()
                 e = 1
                 q = right
@@ -3782,7 +3781,6 @@ cdef class Polynomial(CommutativePolynomial):
         with a single term.
         """
         cdef Py_ssize_t d = term.degree()
-        cdef Py_ssize_t i
         cdef list coeffs, output
         c = term.get_unsafe(d)
         if term_on_right:
@@ -6471,7 +6469,7 @@ cdef class Polynomial(CommutativePolynomial):
             from sage.rings.qqbar import number_field_elements_from_algebraics
             from sage.schemes.projective.projective_space import ProjectiveSpace
 
-            K_pre, P, phi = number_field_elements_from_algebraics(self.coefficients())
+            K_pre, P, _ = number_field_elements_from_algebraics(self.coefficients())
             Pr = ProjectiveSpace(K_pre, len(P) - 1)
             return Pr.point(P).global_height(prec=prec)
         raise TypeError("Must be over a Numberfield or a Numberfield Order.")
@@ -12306,7 +12304,7 @@ cdef class Polynomial_generic_dense(Polynomial):
             2*y*x^3 + (y + 3)*x^2 + (-2*y + 1)*x + 1
         """
         cdef Polynomial_generic_dense res
-        cdef Py_ssize_t check=0, i, min
+        cdef Py_ssize_t i, min
         x = (<Polynomial_generic_dense>self)._coeffs
         y = (<Polynomial_generic_dense>right)._coeffs
         if len(x) > len(y):
@@ -12327,7 +12325,7 @@ cdef class Polynomial_generic_dense(Polynomial):
 
     cpdef _sub_(self, right):
         cdef Polynomial_generic_dense res
-        cdef Py_ssize_t check=0, i, min
+        cdef Py_ssize_t i, min
         x = (<Polynomial_generic_dense>self)._coeffs
         y = (<Polynomial_generic_dense>right)._coeffs
         if len(x) > len(y):

src/sage/rings/polynomial/polynomial_integer_dense_flint.pyx

@@ -402,7 +402,6 @@ cdef class Polynomial_integer_dense_flint(Polynomial):
         cdef RealBall arb_a, arb_z
         cdef ComplexBall acb_a, acb_z
 
-        cdef unsigned long limbs
         cdef fmpz_t a_fmpz
         cdef fmpz_t z_fmpz
 
@@ -1642,16 +1641,15 @@ cdef class Polynomial_integer_dense_flint(Polynomial):
             sage: f = -30*x; f.factor()
             (-1) * 2 * 3 * 5 * x
         """
-        cdef int i
         cdef long deg = fmpz_poly_degree(self._poly)
         # it appears that pari has a window from about degrees 30 and 300
         # in which it beats NTL.
         c = self.content()
-        g = self//c
+        g = self // c
         if deg < 30 or deg > 300:
-            return c.factor()*g._factor_ntl()
+            return c.factor() * g._factor_ntl()
         else:
-            return c.factor()*g._factor_pari()
+            return c.factor() * g._factor_pari()
 
     def factor_mod(self, p):
         """

src/sage/rings/polynomial/polynomial_modn_dense_ntl.pyx

@@ -629,7 +629,7 @@ def small_roots(self, X=None, beta=1.0, epsilon=None, **kwds):
 
     f = self.change_ring(ZZ)
 
-    P,(x,) = f.parent().objgens()
+    _, (x,) = f.parent().objgens()
 
     delta = f.degree()
 
@@ -649,21 +649,21 @@ def small_roots(self, X=None, beta=1.0, epsilon=None, **kwds):
 
     # we could do this much faster, but this is a cheap step
     # compared to LLL
-    g  = [x**j * N**(m-i) * f**i for i in range(m) for j in range(delta) ]
-    g.extend([x**i * f**m for i in range(t)]) # h
+    g  = [x**j * N**(m-i) * f**i for i in range(m) for j in range(delta)]
+    g.extend([x**i * f**m for i in range(t)])  # h
 
     B = Matrix(ZZ, len(g), delta*m + max(delta,t) )
     for i in range(B.nrows()):
         for j in range( g[i].degree()+1 ):
-            B[i,j] = g[i][j]*X**j
+            B[i, j] = g[i][j]*X**j
 
     B = B.LLL(**kwds)
 
-    f = sum([ZZ(B[0,i]//X**i)*x**i for i in range(B.ncols())])
+    f = sum([ZZ(B[0, i]//X**i)*x**i for i in range(B.ncols())])
     R = f.roots()
 
     ZmodN = self.base_ring()
-    roots = set([ZmodN(r) for r,m in R if abs(r) <= X])
+    roots = set(ZmodN(r) for r, m in R if abs(r) <= X)
     Nbeta = N**beta
     return [root for root in roots if N.gcd(ZZ(self(root))) >= Nbeta]
 

src/sage/rings/polynomial/polynomial_zmod_flint.pyx

@@ -190,7 +190,6 @@ cdef class Polynomial_zmod_flint(Polynomial_template):
         """
         cdef list l_in = x
         cdef unsigned long length = len(l_in)
-        cdef unsigned long modulus = nmod_poly_modulus(&self.x)
         cdef int i
         if length == 0:
             nmod_poly_zero(&self.x)