diff --git a/src/sage/dynamics/arithmetic_dynamics/projective_ds.py b/src/sage/dynamics/arithmetic_dynamics/projective_ds.py
index a04cf57a28d..3767bafed6a 100644
--- a/src/sage/dynamics/arithmetic_dynamics/projective_ds.py
+++ b/src/sage/dynamics/arithmetic_dynamics/projective_ds.py
@@ -8437,6 +8437,14 @@ def normal_form(self, return_conjugation=False):
               To:   Finite Field in z2 of size 3^2
               Defn: 1 |--> 1
 
+        Fixes :issue:`38012` by not forcing univariate polynomial to be univariate::
+
+            sage: R.<z> = PolynomialRing(QQ)
+            sage: f = DynamicalSystem_affine(z^2 + z + 1).homogenize(1)
+            sage: f.normal_form()
+            Dynamical System of Projective Space of dimension 1 over Rational Field
+             Defn: Defined on coordinates by sending (x0 : x1) to
+                   (x0^2 + 5/4*x1^2 : x1^2)
         """
         # defines the field of fixed points
         if self.codomain().dimension_relative() != 1:
@@ -8465,7 +8473,8 @@ def normal_form(self, return_conjugation=False):
         #we find one and not go all the way to the splitting field
         i = 0
         if G.degree() != 0:
-            G = G.polynomial(G.variable(0))
+            if is_MPolynomialRing(G.parent()):
+                G = G.polynomial(G.variable(0))
         else:
             #no other fixed points
             raise NotImplementedError("map is not a polynomial")