34212: log embedding in number_field.py

Author: guojing0

src/sage/rings/number_field/number_field.py

@@ -9292,6 +9292,66 @@ def minkowski_embedding(self, B=None, prec=None):
 
         return sage.matrix.all.matrix(d)
 
+def logarithmic_embedding(self, prec=53):
+    """
+    Return the image of ``self`` under the logarithmic embedding.
+
+    The logarithmic embedding is defined as a map from the number field ``self`` to `\RR^n`.
+
+    It is defined under Definition 4.9.6 in [Cohen1993]_.
+
+    INPUT:
+
+        - ``prec`` -- desired floating point precision.
+
+        OUTPUT:
+
+        - a tuple of real numbers.
+
+        EXAMPLES::
+
+            sage: CF.<a> = CyclotomicField(97)
+            sage: log_map = logarithmic_embedding(CF)
+            sage: log_map(0)
+            (-1)
+            sage: log_map(7)
+            (1.94591014905531)
+
+        ::
+
+            sage: F.<a> = NumberField(x^3 + 5)
+            sage: K.<b> = F.extension(x^2 + 2)
+            sage: log_map = logarithmic_embedding(K)
+            sage: log_map(0)
+            (-1, -1)
+            sage: log_map(7)
+            (1.94591014905531, 3.89182029811063)
+    """
+    def closure_map(x):
+        """
+        The function closure of the logarithmic embedding.
+        """
+        K = self.base_ring()
+        K_embeddings = K.places(prec)
+        r1, r2 = K.signature()
+        r = r1 + r2 - 1
+
+        Reals = RealField(prec)
+
+        if x == 0:
+            return vector([-1 for _ in range(r + 1)])
+
+        x_logs = []
+        for i in range(r1):
+            sigma = K_embeddings[i]
+            x_logs.append(Reals(abs(sigma(x))).log())
+        for i in range(r1, r + 1):
+            tau = K_embeddings[i]
+            x_logs.append(2 * Reals(abs(tau(x))).log())
+
+        return vector(x_logs)
+    return closure_map
+
     def places(self, all_complex=False, prec=None):
         r"""
         Return the collection of all infinite places of self.