Fallback inverse_mod to inverse_of_unit

Author: user202729

src/sage/rings/polynomial/polynomial_element.pyx

@@ -1608,6 +1608,27 @@ 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...
         """
         from sage.rings.ideal import Ideal_generic
         if isinstance(m, Ideal_generic):
@@ -1625,13 +1646,21 @@ 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:
+                    raise ValueError("Impossible inverse modulo")
+            except NotImplementedError:
+                # attempt fallback, return inverse_of_unit() if this is already an unit
+                try:
+                    return a.inverse_of_unit()
+                except (ArithmeticError, ValueError, NotImplementedError, ZeroDivisionError):
+                    pass
+                raise
         else:
             # xgcd may not converge for inexact rings.
             # Instead solve for the coefficients of