Deprecate is_FunctionFieldElement, is_FunctionFieldElement

src/sage/dynamics/arithmetic_dynamics/projective_ds.py

@@ -89,7 +89,6 @@ class initialization directly.
 from sage.rings.finite_rings.integer_mod_ring import Zmod
 from sage.rings.fraction_field import FractionField, FractionField_generic, FractionField_1poly_field
 from sage.rings.fraction_field_element import FractionFieldElement
-from sage.rings.function_field.function_field import is_FunctionField
 from sage.rings.integer import Integer
 from sage.rings.integer_ring import ZZ
 from sage.rings.polynomial.flatten import FlatteningMorphism, UnflatteningMorphism
@@ -4539,7 +4538,7 @@ def preperiodic_points(self, m, n, **kwds):
         else:
             f_sub = self.change_ring(R)
             R = f_sub.base_ring() #in the case when R is an embedding
-        if isinstance(R, FractionField_1poly_field) or is_FunctionField(R):
+        if isinstance(R, FractionField_1poly_field) or R in FunctionFields():
             raise NotImplementedError('Periodic points not implemented for function fields; '
                 'clear denominators and use the polynomial ring instead')
         CR = f_sub.coordinate_ring()
@@ -4877,7 +4876,7 @@ def periodic_points(self, n, minimal=True, formal=False, R=None, algorithm='vari
         else:
             f_sub = self.change_ring(R)
             R = f_sub.base_ring() #in the case when R is an embedding
-        if isinstance(R, FractionField_1poly_field) or is_FunctionField(R):
+        if isinstance(R, FractionField_1poly_field) or R in FunctionFields():
             raise NotImplementedError('periodic points not implemented for fraction function fields; '
                 'clear denominators and use the polynomial ring instead')
         if isinstance(R, FractionField_generic):

src/sage/rings/function_field/element.pyx

@@ -64,8 +64,9 @@ AUTHORS:
 #                  https://www.gnu.org/licenses/
 # *****************************************************************************
 
-from sage.structure.element cimport FieldElement
+from sage.categories.function_fields import FunctionFields
 from sage.misc.cachefunc import cached_method
+from sage.structure.element cimport FieldElement
 
 
 def is_FunctionFieldElement(x):
@@ -76,14 +77,24 @@ def is_FunctionFieldElement(x):
 
         sage: t = FunctionField(QQ,'t').gen()
         sage: sage.rings.function_field.element.is_FunctionFieldElement(t)
+        doctest:warning...
+        DeprecationWarning: The function is_FunctionFieldElement is deprecated;
+        use '....parent() in FunctionFields()' instead.
+        See https://github.com/sagemath/sage/issues/38289 for details.
         True
         sage: sage.rings.function_field.element.is_FunctionFieldElement(0)
         False
     """
+    from sage.misc.superseded import deprecation_cython
+    deprecation_cython(38289,
+                       "The function is_FunctionFieldElement is deprecated; "
+                       "use '....parent() in FunctionFields()' instead.")
     if isinstance(x, FunctionFieldElement):
         return True
-    from sage.rings.function_field.function_field import is_FunctionField
-    return is_FunctionField(x.parent())
+    from sage.rings.function_field.function_field import FunctionField
+    if isinstance(x.parent(), FunctionField):
+        return True
+    return x.parent() in FunctionFields()
 
 
 def make_FunctionFieldElement(parent, element_class, representing_element):

src/sage/rings/function_field/function_field.py

@@ -263,10 +263,17 @@ def is_FunctionField(x):
 
         sage: from sage.rings.function_field.function_field import is_FunctionField
         sage: is_FunctionField(QQ)
+        doctest:warning...
+        DeprecationWarning: The function is_FunctionField is deprecated; use '... in FunctionFields()' instead.
+        See https://github.com/sagemath/sage/issues/38289 for details.
         False
         sage: is_FunctionField(FunctionField(QQ, 't'))
         True
     """
+    from sage.misc.superseded import deprecation
+    deprecation(38289,
+                "The function is_FunctionField is deprecated; "
+                "use '... in FunctionFields()' instead.")
     if isinstance(x, FunctionField):
         return True
     return x in FunctionFields()
@@ -846,7 +853,7 @@ def _intermediate_fields(self, base):
             ...
             TypeError: base must be a function field
         """
-        if not is_FunctionField(base):
+        if base not in FunctionFields():
             raise TypeError("base must be a function field")
 
         ret = [self]

src/sage/schemes/berkovich/berkovich_cp_element.py

@@ -35,11 +35,12 @@
 
 import sage.rings.abc
 
+from sage.categories.function_fields import FunctionFields
 from sage.misc.lazy_import import lazy_import
 from sage.rings.infinity import Infinity
 from sage.rings.integer_ring import ZZ
 from sage.rings.rational_field import QQ
-from sage.rings.real_mpfr import RR, RealNumber
+from sage.rings.real_mpfr import RealNumber, RR
 from sage.schemes.projective.projective_point import SchemeMorphism_point_projective_field
 from sage.schemes.projective.projective_space import ProjectiveSpace
 from sage.structure.element import Element, Expression
@@ -84,7 +85,6 @@ def __init__(self, parent, center, radius=None, power=None, prec=20, space_type=
             sage: B(4)
             Type I point centered at 4 + O(5^20)
         """
-        from sage.rings.function_field.element import is_FunctionFieldElement
         from sage.rings.polynomial.polynomial_element import Polynomial
         from sage.rings.fraction_field_element import FractionFieldElement_1poly_field
         self._type = None
@@ -109,8 +109,7 @@ def __init__(self, parent, center, radius=None, power=None, prec=20, space_type=
             if not error_check:
                 return
 
-        # is_FunctionFieldElement calls .parent
-        elif hasattr(center, "parent") and hasattr(radius, 'parent'):
+        elif isinstance(center, Element) and isinstance(radius, Element):
             from sage.rings.polynomial.multi_polynomial import MPolynomial
             if isinstance(center, MPolynomial):
                 try:
@@ -119,10 +118,10 @@ def __init__(self, parent, center, radius=None, power=None, prec=20, space_type=
                     raise TypeError('center was %s, a multivariable polynomial' % center)
 
             # check if the radius and the center are functions
-            center_func_check = is_FunctionFieldElement(center) or isinstance(center, Polynomial) or\
-                isinstance(center, FractionFieldElement_1poly_field) or isinstance(center, Expression)
-            radius_func_check = is_FunctionFieldElement(radius) or isinstance(radius, Polynomial) or\
-                isinstance(radius, FractionFieldElement_1poly_field) or isinstance(radius, Expression)
+            center_func_check = center.parent() in FunctionFields() or \
+                isinstance(center, (Polynomial, FractionFieldElement_1poly_field, Expression))
+            radius_func_check = radius.parent() in FunctionFields() or \
+                isinstance(radius, (Polynomial, FractionFieldElement_1poly_field, Expression))
 
             if center_func_check:
                 # check that both center and radii are supported univariate function