gh-38942: deprecate is_generator for is_gen
    
as part of #35523

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
- [x] I have linked a relevant issue or discussion.
- [x] I have created tests covering the changes.
- [ ] I have updated the documentation and checked the documentation
preview.
    
URL: #38942
Reported by: Frédéric Chapoton
Reviewer(s): Kwankyu Lee

Author: Release Manager

src/sage/rings/polynomial/multi_polynomial.pyx

@@ -11,14 +11,14 @@ Base class for elements of multivariate polynomial rings
 # (at your option) any later version.
 #                  https://www.gnu.org/licenses/
 # ********************************************************************
+from itertools import chain
 
 from sage.rings.integer cimport Integer
 from sage.rings.integer_ring import ZZ
 from sage.structure.coerce cimport coercion_model
 from sage.misc.derivative import multi_derivative
-from itertools import chain
-
 from sage.misc.misc_c import prod
+from sage.misc.superseded import deprecated_function_alias
 
 
 def is_MPolynomial(x):
@@ -1284,28 +1284,39 @@ cdef class MPolynomial(CommutativePolynomial):
         """
         return self.base_ring().ideal(self.coefficients())
 
-    def is_generator(self):
+    def is_gen(self):
         r"""
         Return ``True`` if this polynomial is a generator of its parent.
 
         EXAMPLES::
 
             sage: R.<x,y> = ZZ[]
-            sage: x.is_generator()
+            sage: x.is_gen()
             True
-            sage: (x + y - y).is_generator()
+            sage: (x + y - y).is_gen()
             True
-            sage: (x*y).is_generator()
+            sage: (x*y).is_gen()
             False
             sage: R.<x,y> = QQ[]
-            sage: x.is_generator()
+            sage: x.is_gen()
             True
-            sage: (x + y - y).is_generator()
+            sage: (x + y - y).is_gen()
             True
-            sage: (x*y).is_generator()
+            sage: (x*y).is_gen()
             False
+
+        TESTS::
+
+            sage: R.<x,y> = ZZ[]
+            sage: x.is_generator()
+            doctest:warning...:
+            DeprecationWarning: is_generator is deprecated. Please use is_gen instead.
+            See https://github.com/sagemath/sage/issues/38942 for details.
+            True
         """
-        return (self in self.parent().gens())
+        return self in self.parent().gens()
+
+    is_generator = deprecated_function_alias(38942, is_gen)
 
     def map_coefficients(self, f, new_base_ring=None):
         r"""

src/sage/rings/polynomial/multi_polynomial_element.py

@@ -70,7 +70,7 @@
 from .multi_polynomial import MPolynomial, is_MPolynomial
 from sage.categories.morphism import Morphism
 from sage.misc.lazy_attribute import lazy_attribute
-
+from sage.misc.superseded import deprecated_function_alias
 from sage.rings.rational_field import QQ
 from sage.rings.fraction_field import FractionField
 
@@ -693,7 +693,7 @@ def degree(self, x=None, std_grading=False):
                     x = self.parent().coerce(x)
                 except TypeError:
                     raise TypeError("x must canonically coerce to parent")
-            if not x.is_generator():
+            if not x.is_gen():
                 raise TypeError("x must be one of the generators of the parent")
         else:
             raise TypeError("x must be one of the generators of the parent")
@@ -1349,27 +1349,39 @@ def _homogenize(self, var):
         R = self.parent()
         return R(X)
 
-    def is_generator(self):
+    def is_gen(self) -> bool:
         """
         Return ``True`` if ``self`` is a generator of its parent.
 
         EXAMPLES::
 
             sage: # needs sage.rings.number_field
             sage: R.<x,y> = QQbar[]
-            sage: x.is_generator()
+            sage: x.is_gen()
             True
-            sage: (x + y - y).is_generator()
+            sage: (x + y - y).is_gen()
             True
-            sage: (x*y).is_generator()
+            sage: (x*y).is_gen()
             False
+
+        TESTS::
+
+            sage: # needs sage.rings.number_field
+            sage: R.<x,y> = QQbar[]
+            sage: x.is_generator()
+            doctest:warning...:
+            DeprecationWarning: is_generator is deprecated. Please use is_gen instead.
+            See https://github.com/sagemath/sage/issues/38942 for details.
+            True
         """
         elt = self.element()
         if len(elt) == 1:
             (e, c), = elt.dict().items()
             return e.nonzero_values() == [1] and c.is_one()
         return False
 
+    is_generator = deprecated_function_alias(38942, is_gen)
+
     def is_monomial(self):
         """
         Return ``True`` if ``self`` is a monomial, which we define to be a

src/sage/rings/polynomial/multi_polynomial_libsingular.pyx

@@ -2700,7 +2700,7 @@ cdef class MPolynomial_libsingular(MPolynomial_libsingular_base):
                 x = self.parent().coerce(x)
             except TypeError:
                 raise TypeError("argument is not coercible to the parent")
-        if not x.is_generator():
+        if not x.is_gen():
             raise TypeError("argument is not a generator")
 
         return Integer(singular_polynomial_deg(p, x._poly, r))

src/sage/rings/polynomial/polynomial_element.pyx

@@ -13078,7 +13078,7 @@ cpdef bint polynomial_is_variable(x) noexcept:
     .. SEEALSO::
 
         - :meth:`sage.rings.polynomial.polynomial_element.Polynomial.is_gen`
-        - :meth:`sage.rings.polynomial.multi_polynomial.MPolynomial.is_generator`
+        - :meth:`sage.rings.polynomial.multi_polynomial.MPolynomial.is_gen`
 
     EXAMPLES::
 
@@ -13116,5 +13116,5 @@ cpdef bint polynomial_is_variable(x) noexcept:
         return (x.is_gen()
                 or (x.degree() == 1 and x[0].is_zero() and x[1].is_one()))
     if isinstance(x, MPolynomial):
-        return x.is_generator()
+        return x.is_gen()
     return False