sagemathgh-37159: Fix change_ring method of multi-variate polynomials

    
<!-- ^^^^^
Please provide a concise, informative and self-explanatory title.
Don't put issue numbers in there, do this in the PR body below.
For example, instead of "Fixes sagemath#1234" use "Introduce new method to
calculate 1+1"
-->
<!-- Describe your changes here in detail -->

Fixes sagemath#36832.

<!-- Why is this change required? What problem does it solve? -->
<!-- If this PR resolves an open issue, please link to it here. For
example "Fixes sagemath#12345". -->
<!-- If your change requires a documentation PR, please link it
appropriately. -->

### 📝 Checklist

<!-- Put an `x` in all the boxes that apply. -->
<!-- If your change requires a documentation PR, please link it
appropriately -->
<!-- If you're unsure about any of these, don't hesitate to ask. We're
here to help! -->
<!-- Feel free to remove irrelevant items. -->

- [x] The title is concise, informative, and self-explanatory.
- [ ] The description explains in detail what this PR is about.
- [x] I have linked a relevant issue or discussion.
- [ ] I have created tests covering the changes.
- [ ] I have updated the documentation accordingly.

### ⌛ Dependencies

<!-- List all open PRs that this PR logically depends on
- sagemath#12345: short description why this is a dependency
- sagemath#34567: ...
-->

<!-- If you're unsure about any of these, don't hesitate to ask. We're
here to help! -->
    
URL: sagemath#37159
Reported by: Kwankyu Lee
Reviewer(s): Gonzalo Tornaría, Kwankyu Lee

src/sage/rings/polynomial/multi_polynomial.pyx

@@ -2,13 +2,15 @@ r"""
 Base class for elements of multivariate polynomial rings
 """
 
-#*****************************************************************************
+# ********************************************************************
+#       Copyright (C) 2005 William Stein <wstein@gmail.com>
+#
 # This program is free software: you can redistribute it and/or modify
 # it under the terms of the GNU General Public License as published by
 # the Free Software Foundation, either version 2 of the License, or
 # (at your option) any later version.
 #                  https://www.gnu.org/licenses/
-#*****************************************************************************
+# ********************************************************************
 
 from sage.rings.integer cimport Integer
 from sage.rings.integer_ring import ZZ
@@ -33,9 +35,9 @@ from sage.rings.polynomial.polynomial_element cimport Polynomial
 
 cdef class MPolynomial(CommutativePolynomial):
 
-    ####################
+    # -------------------------
     # Some standard conversions
-    ####################
+    # -------------------------
     def _scalar_conversion(self, R):
         r"""
         TESTS::
@@ -360,7 +362,6 @@ cdef class MPolynomial(CommutativePolynomial):
         """
         return multi_derivative(self, args)
 
-
     def polynomial(self, var):
         r"""
         Let ``var`` be one of the variables of the parent of ``self``.  This
@@ -840,26 +841,43 @@ cdef class MPolynomial(CommutativePolynomial):
 
     def change_ring(self, R):
         r"""
-        Return a copy of this polynomial but with coefficients in ``R``,
-        if at all possible.
+        Return this polynomial with coefficients converted to ``R``.
 
         INPUT:
 
-        - ``R`` -- a ring or morphism.
+        - ``R`` -- a ring or morphism; if a morphism, the coefficients
+          are mapped to the codomain of ``R``
+
+        OUTPUT: a new polynomial with the base ring changed to ``R``.
 
         EXAMPLES::
 
             sage: R.<x,y> = QQ[]
             sage: f = x^3 + 3/5*y + 1
             sage: f.change_ring(GF(7))
             x^3 + 2*y + 1
+            sage: g = x^2 + 5*y
+            sage: g.change_ring(GF(5))
+            x^2
 
         ::
 
-            sage: R.<x,y> = GF(9,'a')[]                                                 # needs sage.rings.finite_rings
+            sage: # needs sage.rings.finite_rings
+            sage: R.<x,y> = GF(9,'a')[]
             sage: (x+2*y).change_ring(GF(3))
             x - y
 
+        ::
+
+            sage: # needs sage.rings.finite_rings
+            sage: F.<a> = GF(7^2)
+            sage: R.<x,y> = F[]
+            sage: f = x^2 + a^2*y^2 + a*x + a^3*y
+            sage: g = f.change_ring(F.frobenius_endomorphism()); g
+            x^2 + (-a - 2)*y^2 + (-a + 1)*x + (2*a + 2)*y
+            sage: g.change_ring(F.frobenius_endomorphism()) == f
+            True
+
         ::
 
             sage: # needs sage.rings.number_field
@@ -869,21 +887,35 @@ cdef class MPolynomial(CommutativePolynomial):
             sage: f.change_ring(K.embeddings(CC)[1])
             x^2 + (-0.500000000000000 - 0.866025403784438*I)*y
 
+        ::
+
+            sage: # needs sage.rings.number_field
+            sage: K.<w> = CyclotomicField(5)
+            sage: R.<x,y> = K[]
+            sage: f = x^2 + w*y
+            sage: f.change_ring(K.embeddings(QQbar)[1])
+            x^2 + (-0.8090169943749474? + 0.5877852522924731?*I)*y
+
         TESTS:
 
-        Check that :trac:`25022` is fixed::
+        Check that :issue:`25022` is fixed::
 
+            sage: # needs sage.rings.number_field sage.symbolic
             sage: K.<x,y> = ZZ[]
-            sage: (x*y).change_ring(SR).monomials()                                     # needs sage.rings.number_field sage.symbolic
+            sage: (x*y).change_ring(SR).monomials()
             [x*y]
+
+        Check that :issue:`36832` is fixed::
+
+            sage: F = GF(11)
+            sage: phi = Hom(F,F).an_element()
+            sage: R.<x,y> = F[]
+            sage: x.change_ring(phi)
+            x
         """
         if isinstance(R, Map):
-        #if we're given a hom of the base ring extend to a poly hom
-            if R.domain() == self.base_ring():
-                R = self.parent().hom(R, self.parent().change_ring(R.codomain()))
-            return R(self)
-        else:
-            return self.parent().change_ring(R)(self.dict())
+            return self.map_coefficients(R)
+        return self.parent().change_ring(R)(self.dict())
 
     def is_symmetric(self, group=None):
         r"""

src/sage/rings/polynomial/multi_polynomial_element.py

@@ -412,40 +412,6 @@ def __rpow__(self, n):
     def element(self):
         return self.__element
 
-    def change_ring(self, R):
-        r"""
-        Change the base ring of this polynomial to ``R``.
-
-        INPUT:
-
-        - ``R`` -- ring or morphism.
-
-        OUTPUT: a new polynomial converted to ``R``.
-
-        EXAMPLES::
-
-            sage: R.<x,y> = QQ[]
-            sage: f = x^2 + 5*y
-            sage: f.change_ring(GF(5))
-            x^2
-
-        ::
-
-            sage: # needs sage.rings.number_field
-            sage: K.<w> = CyclotomicField(5)
-            sage: R.<x,y> = K[]
-            sage: f = x^2 + w*y
-            sage: f.change_ring(K.embeddings(QQbar)[1])
-            x^2 + (-0.8090169943749474? + 0.5877852522924731?*I)*y
-        """
-        if isinstance(R, Morphism):
-            #if we're given a hom of the base ring extend to a poly hom
-            if R.domain() == self.base_ring():
-                R = self.parent().hom(R, self.parent().change_ring(R.codomain()))
-            return R(self)
-        else:
-            return self.parent().change_ring(R)(self)
-
 
 class MPolynomial_polydict(Polynomial_singular_repr, MPolynomial_element):
     r"""

src/sage/rings/polynomial/multi_polynomial_libsingular.pyx

@@ -149,13 +149,15 @@ AUTHORS:
 
 """
 
-#*****************************************************************************
+# ********************************************************************
+#       Copyright (C) 2005 William Stein <wstein@gmail.com>
+#
 # This program is free software: you can redistribute it and/or modify
 # it under the terms of the GNU General Public License as published by
 # the Free Software Foundation, either version 2 of the License, or
 # (at your option) any later version.
-#                  http://www.gnu.org/licenses/
-#*****************************************************************************
+#                  https://www.gnu.org/licenses/
+# ********************************************************************
 
 # The Singular API is as follows:
 #
@@ -1530,9 +1532,9 @@ cdef class MPolynomialRing_libsingular(MPolynomialRing_base):
 
         return old
 
-    ### The following methods are handy for implementing Groebner
-    ### basis algorithms. They do only superficial type/sanity checks
-    ### and should be called carefully.
+    # The following methods are handy for implementing Groebner
+    # basis algorithms. They do only superficial type/sanity checks
+    # and should be called carefully.
 
     def monomial_quotient(self, MPolynomial_libsingular f, MPolynomial_libsingular g, coeff=False):
         r"""
@@ -1677,7 +1679,6 @@ cdef class MPolynomialRing_libsingular(MPolynomialRing_base):
         else:
             return True
 
-
     def monomial_lcm(self, MPolynomial_libsingular f, MPolynomial_libsingular g):
         """
         LCM for monomials. Coefficients are ignored.
@@ -5242,7 +5243,6 @@ cdef class MPolynomial_libsingular(MPolynomial_libsingular_base):
             y += base_map(c)*mul([ im_gens[i]**m[i] for i in range(n) if m[i]])
         return y
 
-
     def _derivative(self, MPolynomial_libsingular var):
         """
         Differentiates this polynomial with respect to the provided
@@ -5694,7 +5694,6 @@ cdef class MPolynomial_libsingular(MPolynomial_libsingular_base):
             i.append( new_MP(self._parent, pDiff(self._poly, k)))
         return i
 
-
     def numerator(self):
         """
         Return a numerator of self computed as self * self.denominator()