sagemathgh-39419: trying to get rid of coerce_c_impl
    
not seriously tested so far

### 📝 Checklist

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- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
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- [ ] I have created tests covering the changes.
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preview.
    
URL: sagemath#39419
Reported by: Frédéric Chapoton
Reviewer(s): Frédéric Chapoton, Michael Orlitzky, user202729

Author: Release Manager

src/sage/rings/polynomial/multi_polynomial_libsingular.pyx

@@ -499,8 +499,8 @@ cdef class MPolynomialRing_libsingular(MPolynomialRing_base):
             if self is other:
                 return True
             n = other.ngens()
-            if(other.base_ring is base_ring and self.ngens() >= n and
-               self.variable_names()[:n] == other.variable_names()):
+            if (other.base_ring is base_ring and self.ngens() >= n and
+                    self.variable_names()[:n] == other.variable_names()):
                 return True
             elif base_ring.has_coerce_map_from(other._mpoly_base_ring(self.variable_names())):
                 return True

src/sage/rings/polynomial/multi_polynomial_ring_base.pxd

@@ -9,8 +9,6 @@ cdef class MPolynomialRing_base(CommutativeRing):
     cdef public object _magma_gens
     cdef public dict _magma_cache
 
-    cdef _coerce_c_impl(self, x)
-
 
 cdef class BooleanPolynomialRing_base(MPolynomialRing_base):
     pass

src/sage/rings/polynomial/multi_polynomial_ring_base.pyx

@@ -571,25 +571,28 @@ cdef class MPolynomialRing_base(CommutativeRing):
         else:
             return self._generic_coerce_map(self.base_ring())
 
-    cdef _coerce_c_impl(self, x):
+    cpdef _coerce_map_from_(self, other):
         """
-        Return the canonical coercion of x to this multivariate
-        polynomial ring, if one is defined, or raise a :exc:`TypeError`.
+        Return whether there is canonical coercion map
+        from the ring ``other`` to this multivariate polynomial ring `R`.
 
-        The rings that canonically coerce to this polynomial ring are:
+        The rings that canonically coerce to the polynomial ring `R` are:
 
-        - this ring itself
-        - polynomial rings in the same variables over any base ring that
-          canonically coerces to the base ring of this ring
-        - polynomial rings in a subset of the variables over any base ring that
-          canonically coerces to the base ring of this ring
-        - any ring that canonically coerces to the base ring of this polynomial
-          ring.
+        - the ring `R` itself,
+
+        - the base ring of `R`,
+
+        - any ring that canonically coerces to the base ring of `R`.
+
+        - polynomial rings in an initial subset of the variables of `R`
+          over any base ring that canonically coerces to the base ring of `R`,
+
+        - polynomial rings in one of the variables of `R`
+          over any base ring that canonically coerces to the base ring of `R`,
 
         TESTS:
 
-        This fairly complicated code (from Michel Vandenbergh) ends up
-        implicitly calling ``_coerce_c_impl``::
+        Fairly complicated code (from Michel Vandenbergh)::
 
             sage: # needs sage.rings.number_field
             sage: z = polygen(QQ, 'z')
@@ -602,27 +605,40 @@ cdef class MPolynomialRing_base(CommutativeRing):
             sage: x + 1/u
             x + 1/u
         """
-        try:
-            P = x.parent()
-            # polynomial rings in the same variable over the any base
-            # that coerces in:
-            if isinstance(P, MPolynomialRing_base):
-                if P.variable_names() == self.variable_names():
-                    if self.has_coerce_map_from(P.base_ring()):
-                        return self(x)
-                elif self.base_ring().has_coerce_map_from(P._mpoly_base_ring(self.variable_names())):
-                    return self(x)
-
-            elif isinstance(P, polynomial_ring.PolynomialRing_generic):
-                if P.variable_name() in self.variable_names():
-                    if self.has_coerce_map_from(P.base_ring()):
-                        return self(x)
-
-        except AttributeError:
-            pass
-
-        # any ring that coerces to the base ring of this polynomial ring.
-        return self(self.base_ring().coerce(x))
+        base_ring = self.base_ring()
+        if other is base_ring:
+            # Because this parent class is a Cython class, the method
+            # UnitalAlgebras.ParentMethods.__init_extra__(), which normally
+            # registers the coercion map from the base ring, is called only
+            # when inheriting from this class in Python (cf. Issue #26958).
+            return self._coerce_map_from_base_ring()
+
+        f = self._coerce_map_via([base_ring], other)
+        if f is not None:
+            return f
+
+        # polynomial rings in an initial subset of variables
+        # over the any base that coerces in
+        if isinstance(other, MPolynomialRing_base):
+            if self is other:
+                return True
+            n = other.ngens()
+            check = (self.ngens() >= n and
+                     self.variable_names()[:n] == other.variable_names())
+            if other.base_ring is base_ring and check:
+                return True
+            elif base_ring.has_coerce_map_from(other._mpoly_base_ring(self.variable_names())):
+                return True
+
+        # polynomial rings in one of the variables
+        # over the any base that coerces in
+        elif isinstance(other, polynomial_ring.PolynomialRing_generic):
+            if other.variable_name() in self.variable_names():
+                if self.has_coerce_map_from(other.base_ring()):
+                    return True
+
+        # any ring that coerces to the base ring of this polynomial ring
+        return self.base_ring().has_coerce_map_from(other)
 
     def _extract_polydict(self, x):
         """