gh-39402: Simplify __invert__ for polynomial quotient ring element
    
Previously `__invert__` uses `xgcd`, while the method `inverse_mod`
already exists, it's duplication of code. Besides, while simplifying
this some missing features in `inverse_mod` is uncovered.

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

### ⌛ Dependencies

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URL: #39402
Reported by: user202729
Reviewer(s):

Author: Release Manager

src/sage/rings/polynomial/polynomial_element.pyx

@@ -1609,6 +1609,43 @@ cdef class Polynomial(CommutativePolynomial):
         AUTHORS:
 
         - Robert Bradshaw (2007-05-31)
+
+        TESTS:
+
+        At the time of writing :meth:`inverse_mod` is not implemented in general for the following ring,
+        but it should fallback to :meth:`inverse_of_unit` when possible::
+
+            sage: R.<u,v> = ZZ[]
+            sage: S = R.localization(u)
+            sage: T.<x> = S[]
+            sage: T
+            Univariate Polynomial Ring in x over Multivariate Polynomial Ring in u, v over Integer Ring localized at (u,)
+            sage: T(u).is_unit()
+            True
+            sage: T(u).inverse_of_unit()
+            1/u
+            sage: T(u).inverse_mod(T(v))
+            1/u
+            sage: T(v).inverse_mod(T(v^2-1))
+            Traceback (most recent call last):
+            ...
+            NotImplementedError: Multivariate Polynomial Ring in u, v over Integer Ring localized at (u,) does not provide...
+
+        The behavior of ``xgcd`` over rings like ``ZZ`` are nonstandard, we check the behavior::
+
+            sage: R.<x> = ZZ[]
+            sage: a = 2*x^2+1
+            sage: b = -2*x^2+1
+            sage: a.inverse_mod(b)
+            Traceback (most recent call last):
+            ...
+            NotImplementedError: The base ring (=Integer Ring) is not a field
+            sage: a.xgcd(b)
+            (16, 8, 8)
+            sage: a*x^2+b*(x^2+1)
+            1
+            sage: a*x^2%b  # shows the result exists, thus we cannot raise ValueError, only NotImplementedError
+            1
         """
         from sage.rings.ideal import Ideal_generic
         if isinstance(m, Ideal_generic):
@@ -1626,13 +1663,25 @@ cdef class Polynomial(CommutativePolynomial):
                 return a.parent()(~u)
         if a.parent().is_exact():
             # use xgcd
-            g, s, _ = a.xgcd(m)
-            if g == 1:
-                return s
-            elif g.is_unit():
-                return g.inverse_of_unit() * s
-            else:
-                raise ValueError("Impossible inverse modulo")
+            try:
+                g, s, _ = a.xgcd(m)
+                if g == 1:
+                    return s
+                elif g.is_unit():
+                    return g.inverse_of_unit() * s
+                else:
+                    R = m.base_ring()
+                    if R.is_field():
+                        raise ValueError("Impossible inverse modulo")
+                    else:
+                        raise NotImplementedError(f"The base ring (={R}) is not a field")
+            except NotImplementedError:
+                # attempt fallback, return inverse_of_unit() if this is already an unit
+                try:
+                    return a.inverse_of_unit()
+                except (ArithmeticError, ValueError, NotImplementedError):
+                    pass
+                raise
         else:
             # xgcd may not converge for inexact rings.
             # Instead solve for the coefficients of

src/sage/rings/polynomial/polynomial_integer_dense_ntl.pyx

@@ -619,7 +619,7 @@ cdef class Polynomial_integer_dense_ntl(Polynomial):
         since they need not exist.  Instead, over the integers, we
         first multiply `g` by a divisor of the resultant of `a/g` and
         `b/g`, up to sign, and return ``g, u, v`` such that
-        ``g = s*self + s*right``.  But note that this `g` may be a
+        ``g = u*self + v*right``.  But note that this `g` may be a
         multiple of the gcd.
 
         If ``self`` and ``right`` are coprime as polynomials over the

src/sage/rings/polynomial/polynomial_quotient_ring_element.py

@@ -386,10 +386,6 @@ def __invert__(self):
         """
         Return the inverse of this element.
 
-        .. WARNING::
-
-            Only implemented when the base ring is a field.
-
         EXAMPLES::
 
             sage: R.<x> = QQ[]
@@ -414,35 +410,27 @@ def __invert__(self):
             sage: (2*y)^(-1)
             Traceback (most recent call last):
             ...
-            NotImplementedError: The base ring (=Ring of integers modulo 16) is not a field
+            NotImplementedError
+            sage: (2*y+1)^(-1)  # this cannot raise ValueError because...
+            Traceback (most recent call last):
+            ...
+            NotImplementedError
+            sage: (2*y+1) * (10*y+5)  # the element is in fact invertible
+            1
 
         Check that :issue:`29469` is fixed::
 
             sage: ~S(3)
             11
         """
-        if self._polynomial.is_zero():
-            raise ZeroDivisionError("element %s of quotient polynomial ring not invertible" % self)
-        if self._polynomial.is_one():
-            return self
-
-        parent = self.parent()
-
+        P = self.parent()
         try:
-            if self._polynomial.is_unit():
-                inv_pol = self._polynomial.inverse_of_unit()
-                return parent(inv_pol)
-        except (TypeError, NotImplementedError):
-            pass
-
-        base = parent.base_ring()
-        if not base.is_field():
-            raise NotImplementedError("The base ring (=%s) is not a field" % base)
-        g, _, a = parent.modulus().xgcd(self._polynomial)
-        if g.degree() != 0:
-            raise ZeroDivisionError("element %s of quotient polynomial ring not invertible" % self)
-        c = g[0]
-        return self.__class__(self.parent(), (~c)*a, check=False)
+            return type(self)(P, self._polynomial.inverse_mod(P.modulus()), check=False)
+        except ValueError as e:
+            if e.args[0] == "Impossible inverse modulo":
+                raise ZeroDivisionError(f"element {self} of quotient polynomial ring not invertible")
+            else:
+                raise NotImplementedError
 
     def field_extension(self, names):
         r"""