From c23ec600afe80da40f5d57016f42ac478acb26d3 Mon Sep 17 00:00:00 2001
From: DavidAyotte <davidayotte94@outlook.com>
Date: Tue, 7 Mar 2023 19:39:44 -0500
Subject: [PATCH 01/26] inital commit

---
 .../drinfeld_modules/drinfeld_module.py       | 24 +++++++++++++++++++
 1 file changed, 24 insertions(+)

diff --git a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
index 1a24f09d66d..37a2de8fb7c 100644
--- a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
+++ b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
@@ -28,8 +28,10 @@
 from sage.categories.homset import Hom
 from sage.misc.latex import latex
 from sage.misc.latex import latex_variable_name
+from sage.misc.lazy_import import lazy_import
 from sage.misc.lazy_string import _LazyString
 from sage.rings.integer import Integer
+from sage.rings.integer_ring import ZZ
 from sage.rings.polynomial.ore_polynomial_element import OrePolynomial
 from sage.rings.polynomial.polynomial_ring import PolynomialRing_general
 from sage.rings.ring_extension import RingExtension_generic
@@ -38,6 +40,8 @@
 from sage.structure.sequence import Sequence
 from sage.structure.unique_representation import UniqueRepresentation
 
+lazy_import('sage.rings.lazy_series_ring', 'LazyPowerSeriesRing')
+
 
 class DrinfeldModule(Parent, UniqueRepresentation):
     r"""
@@ -1103,6 +1107,26 @@ def j_invariant(self):
         q = self._Fq.order()
         return (g**(q+1)) / delta
 
+    def _compute_coefficient(self, k):
+        if k not in ZZ:
+            raise TypeError("input must be an integer")
+        if k == 0:
+            return self._base.one()
+        r = self._gen.degree()
+        T = self._gen[0]
+        q = self._Fq.cardinality()
+        c = self._base.zero()
+        for i in range(k):
+            j = k - i
+            if j < r + 1:
+                c += self._compute_coefficient(i)*self._gen[j]**(q**i)
+        return c/(T - T**(q**k))
+
+    def logarithm(self, name='u'):
+        L = LazyPowerSeriesRing(self._base, name)
+        return L(self._compute_coefficient)
+
+
     def morphism(self):
         r"""
         Return the morphism object that defines the Drinfeld module.

From 1f608cf7e32a8d35191954209d018c71e9965654 Mon Sep 17 00:00:00 2001
From: DavidAyotte <davidayotte94@outlook.com>
Date: Fri, 10 Mar 2023 01:51:01 -0500
Subject: [PATCH 02/26] enhance_compute_coeffcient_log

---
 .../drinfeld_modules/drinfeld_module.py       | 21 ++++++++++++-------
 1 file changed, 13 insertions(+), 8 deletions(-)

diff --git a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
index 37a2de8fb7c..3fa7eccab97 100644
--- a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
+++ b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
@@ -1107,24 +1107,29 @@ def j_invariant(self):
         q = self._Fq.order()
         return (g**(q+1)) / delta
 
-    def _compute_coefficient(self, k):
+    def _compute_coefficient_log(self, k):
         if k not in ZZ:
             raise TypeError("input must be an integer")
+        k = ZZ(k)
         if k == 0:
+            return self._base.zero()
+        if k == 1:
             return self._base.one()
         r = self._gen.degree()
         T = self._gen[0]
         q = self._Fq.cardinality()
+        if not k.is_power_of(q):
+            return self._base.zero()
         c = self._base.zero()
-        for i in range(k):
-            j = k - i
+        for i in range(k.log(q)):
+            j = k.log(q) - i
             if j < r + 1:
-                c += self._compute_coefficient(i)*self._gen[j]**(q**i)
-        return c/(T - T**(q**k))
+                c += self._compute_coefficient_log(q**i)*self._gen[j]**(q**i)
+        return c/(T - T**k)
 
-    def logarithm(self, name='u'):
-        L = LazyPowerSeriesRing(self._base, name)
-        return L(self._compute_coefficient)
+    def logarithm(self, name='z'):
+        L = LazyPowerSeriesRing(self._base, name, sparse=False)
+        return L(self._compute_coefficient_log)
 
 
     def morphism(self):

From 3daba4a90ccab30920bfb452347d8ab01b207ea0 Mon Sep 17 00:00:00 2001
From: DavidAyotte <davidayotte94@outlook.com>
Date: Fri, 10 Mar 2023 02:18:52 -0500
Subject: [PATCH 03/26] add documentation

---
 .../drinfeld_modules/drinfeld_module.py       | 50 +++++++++++++++++++
 1 file changed, 50 insertions(+)

diff --git a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
index 3fa7eccab97..45d257c0068 100644
--- a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
+++ b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
@@ -1108,6 +1108,30 @@ def j_invariant(self):
         return (g**(q+1)) / delta
 
     def _compute_coefficient_log(self, k):
+        r"""
+        Return the `k`-th coefficient of the logarithm of ``self``.
+
+        TESTS::
+
+            sage: A = GF(2)['T']
+            sage: K.<T> = Frac(A)
+            sage: phi = DrinfeldModule(A, [T, 1])
+            sage: q = A.base_ring().cardinality()
+            sage: phi._compute_coefficient_log(0)
+            0
+            sage: phi._compute_coefficient_log(1)
+            1
+            sage: phi._compute_coefficient_log(2)
+            1/(T^2 + T)
+            sage: phi._compute_coefficient_log(3)
+            0
+            sage: phi._compute_coefficient_log(2^4)
+            1/(T^30 + T^29 + T^27 + T^26 + T^23 + T^22 + T^20 + T^19 + T^15 + T^14 + T^12 + T^11 + T^8 + T^7 + T^5 + T^4)
+            sage: phi._compute_coefficient_log(T)
+            Traceback (most recent call last):
+            ...
+            TypeError: input must be an integer
+        """
         if k not in ZZ:
             raise TypeError("input must be an integer")
         k = ZZ(k)
@@ -1128,6 +1152,32 @@ def _compute_coefficient_log(self, k):
         return c/(T - T**k)
 
     def logarithm(self, name='z'):
+        r"""
+        Return the logarithm of the given Drinfeld module.
+
+        EXAMPLES::
+
+            sage: A = GF(2)['T']
+            sage: K.<T> = Frac(A)
+            sage: phi = DrinfeldModule(A, [T, 1])
+            sage: q = A.base_ring().cardinality()
+            sage: log = phi.logarithm(); log
+            z + ((1/(T^2+T))*z^2) + ((1/(T^6+T^5+T^3+T^2))*z^4) + O(z^7)
+            sage: log[q] == -1/((T**q - T))
+            True
+            sage: log[q**2] == 1/((T**q - T)*(T**(q**2) - T))
+            True
+            sage: log[q**3] == -1/((T**q - T)*(T**(q**2) - T)*(T**(q**3) - T))
+            True
+
+        ::
+
+            sage: A = GF(5)['T']
+            sage: K.<T> = Frac(A)
+            sage: phi = DrinfeldModule(A, [T, T^2, T + T^2 + T^4, 1])
+            sage: phi.logarithm()
+            z + ((4*T/(T^4+4))*z^5) + O(z^7)
+        """
         L = LazyPowerSeriesRing(self._base, name, sparse=False)
         return L(self._compute_coefficient_log)
 

From 451cf1f670b5837041caf5a9f6218075588fad3f Mon Sep 17 00:00:00 2001
From: DavidAyotte <davidayotte94@outlook.com>
Date: Sat, 11 Mar 2023 10:28:50 -0500
Subject: [PATCH 04/26] implement exponential

---
 .../drinfeld_modules/drinfeld_module.py       | 29 ++++++++++++++++---
 1 file changed, 25 insertions(+), 4 deletions(-)

diff --git a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
index 45d257c0068..d302b129bb6 100644
--- a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
+++ b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
@@ -962,6 +962,27 @@ def coefficients(self, sparse=True):
         """
         return self._gen.coefficients(sparse=sparse)
 
+    def _compute_coefficient_exp(self, k):
+        if k not in ZZ:
+            raise ValueError
+        k = ZZ(k)
+        if k.is_zero():
+            return self._base.zero()
+        if k.is_one():
+            return self._base.one()
+        q = self._Fq.cardinality()
+        if not k.is_power_of(q):
+            return self._base.zero()
+        c = self._base.zero()
+        for i in range(k.log(q)):
+            j = k.log(q) - i
+            c += self._compute_coefficient_exp(q**i)*self._compute_coefficient_log(q**j)**(q**i)
+        return -c
+
+    def exponential(self, name='z'):
+        L = LazyPowerSeriesRing(self._base, name)
+        return L(self._compute_coefficient_exp, valuation=1)
+
     def gen(self):
         r"""
         Return the generator of the Drinfeld module.
@@ -1135,9 +1156,9 @@ def _compute_coefficient_log(self, k):
         if k not in ZZ:
             raise TypeError("input must be an integer")
         k = ZZ(k)
-        if k == 0:
+        if k.is_zero():
             return self._base.zero()
-        if k == 1:
+        if k.is_one():
             return self._base.one()
         r = self._gen.degree()
         T = self._gen[0]
@@ -1178,8 +1199,8 @@ def logarithm(self, name='z'):
             sage: phi.logarithm()
             z + ((4*T/(T^4+4))*z^5) + O(z^7)
         """
-        L = LazyPowerSeriesRing(self._base, name, sparse=False)
-        return L(self._compute_coefficient_log)
+        L = LazyPowerSeriesRing(self._base, name)
+        return L(self._compute_coefficient_log, valuation=1)
 
 
     def morphism(self):

From bfe9a0c7961fe4c8e6ccc2fdab23e27c26c32bb8 Mon Sep 17 00:00:00 2001
From: DavidAyotte <davidayotte94@outlook.com>
Date: Sat, 11 Mar 2023 11:15:35 -0500
Subject: [PATCH 05/26] document exponential method

---
 .../drinfeld_modules/drinfeld_module.py       | 77 ++++++++++++++++++-
 1 file changed, 76 insertions(+), 1 deletion(-)

diff --git a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
index d302b129bb6..9e0faf0fc68 100644
--- a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
+++ b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
@@ -963,8 +963,36 @@ def coefficients(self, sparse=True):
         return self._gen.coefficients(sparse=sparse)
 
     def _compute_coefficient_exp(self, k):
+        r"""
+        Return the `k`-th coefficient of the logarithm of ``self``.
+
+        INPUT:
+
+        - ``k`` (integer) -- the index of the coefficient
+
+        TESTS::
+
+            sage: A = GF(2)['T']
+            sage: K.<T> = Frac(A)
+            sage: phi = DrinfeldModule(A, [T, 1])
+            sage: q = A.base_ring().cardinality()
+            sage: phi._compute_coefficient_exp(0)
+            0
+            sage: phi._compute_coefficient_exp(1)
+            1
+            sage: phi._compute_coefficient_exp(2)
+            1/(T^2 + T)
+            sage: phi._compute_coefficient_exp(3)
+            0
+            sage: phi._compute_coefficient_exp(2^4)
+            1/(T^64 + T^56 + T^52 + T^50 + T^49 + T^44 + T^42 + T^41 + T^38 + T^37 + T^35 + T^30 + T^29 + T^27 + T^23 + T^15)
+            sage: phi._compute_coefficient_exp(T)
+            Traceback (most recent call last):
+            ...
+            TypeError: input must be an integer
+        """
         if k not in ZZ:
-            raise ValueError
+            raise TypeError("input must be an integer")
         k = ZZ(k)
         if k.is_zero():
             return self._base.zero()
@@ -980,6 +1008,53 @@ def _compute_coefficient_exp(self, k):
         return -c
 
     def exponential(self, name='z'):
+        r"""
+        Return the exponential of the given Drinfeld module.
+
+        INPUT:
+
+        - ``name`` (string, default: ``'z'``) -- the name of the generator of
+          the lazy power series ring.
+
+        EXAMPLES::
+
+            sage: A = GF(2)['T']
+            sage: K.<T> = Frac(A)
+            sage: phi = DrinfeldModule(A, [T, 1])
+            sage: q = A.base_ring().cardinality()
+            sage: exp = phi.exponential(); exp
+            z + ((1/(T^2+T))*z^2) + ((1/(T^8+T^6+T^5+T^3))*z^4) + O(z^8)
+
+        The exponential is returned as a lazy power series, meaning that any of
+        its coefficients can be computed on demands::
+
+            sage: exp[2^4]
+            1/(T^64 + T^56 + T^52 + T^50 + T^49 + T^44 + T^42 + T^41 + T^38 + T^37 + T^35 + T^30 + T^29 + T^27 + T^23 + T^15)
+            sage: exp[2^5]
+            1/(T^160 + T^144 + T^136 + T^132 + T^130 + T^129 + T^120 + T^116 + T^114 + T^113 + T^108 + T^106 + T^105 + T^102 + T^101 + T^99 + T^92 + T^90 + T^89 + T^86 + T^85 + T^83 + T^78 + T^77 + T^75 + T^71 + T^62 + T^61 + T^59 + T^55 + T^47 + T^31)
+
+        Example in higher rank::
+
+            sage: A = GF(5)['T']
+            sage: K.<T> = Frac(A)
+            sage: phi = DrinfeldModule(A, [T, T^2, T + T^2 + T^4, 1])
+            sage: exp = phi.exponential(); exp
+            z + ((T/(T^4+4))*z^5) + O(z^8)
+
+        The exponential is the compositional inverse of the logarithm
+        (see :meth:`logarithm`)::
+
+            sage: log = phi.logarithm(); log
+            z + ((4*T/(T^4+4))*z^5) + O(z^8)
+            sage: exp.compose(log)
+            z + O(z^8)
+            sage: log.compose(exp)
+            z + O(z^8)
+
+        REFERENCE:
+
+        See section 4.6 of [Gos1998]_ for the definition of the exponential.
+        """
         L = LazyPowerSeriesRing(self._base, name)
         return L(self._compute_coefficient_exp, valuation=1)
 

From 01d620dd1b8491bf72b93e9cbd40b39e4007cb9f Mon Sep 17 00:00:00 2001
From: DavidAyotte <davidayotte94@outlook.com>
Date: Sat, 11 Mar 2023 13:42:27 -0500
Subject: [PATCH 06/26] enhance documentation of logarithm

---
 .../drinfeld_modules/drinfeld_module.py       | 50 +++++++++++++++----
 1 file changed, 40 insertions(+), 10 deletions(-)

diff --git a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
index 9e0faf0fc68..d74dad8463e 100644
--- a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
+++ b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
@@ -1051,6 +1051,19 @@ def exponential(self, name='z'):
             sage: log.compose(exp)
             z + O(z^8)
 
+        TESTS::
+
+            sage: A = GF(2)['T']
+            sage: K.<T> = Frac(A)
+            sage: phi = DrinfeldModule(A, [T, 1])
+            sage: exp = phi.exponential()
+            sage: exp[2] == 1/(T**q - T)  # expected value
+            True
+            sage: exp[2^2] == 1/((T**(q**2) - T)*(T**q - T)**q)  # expected value
+            True
+            sage: exp[2^3] == 1/((T**(q**3) - T)*(T**(q**2) - T)**q*(T**q - T)**(q**2))  # expected value
+            True
+
         REFERENCE:
 
         See section 4.6 of [Gos1998]_ for the definition of the exponential.
@@ -1251,28 +1264,45 @@ def logarithm(self, name='z'):
         r"""
         Return the logarithm of the given Drinfeld module.
 
+        By definition, the logarithm is the compositional inverse of the
+        exponential (see :meth:`exponential`).
+
         EXAMPLES::
 
             sage: A = GF(2)['T']
             sage: K.<T> = Frac(A)
             sage: phi = DrinfeldModule(A, [T, 1])
-            sage: q = A.base_ring().cardinality()
             sage: log = phi.logarithm(); log
-            z + ((1/(T^2+T))*z^2) + ((1/(T^6+T^5+T^3+T^2))*z^4) + O(z^7)
-            sage: log[q] == -1/((T**q - T))
-            True
-            sage: log[q**2] == 1/((T**q - T)*(T**(q**2) - T))
-            True
-            sage: log[q**3] == -1/((T**q - T)*(T**(q**2) - T)*(T**(q**3) - T))
-            True
+            z + ((1/(T^2+T))*z^2) + ((1/(T^6+T^5+T^3+T^2))*z^4) + O(z^8)
 
-        ::
+        The logarithm is returned as a lazy power series, meaning that any of
+        its coefficients can be computed on demands::
+
+            sage: log[2^4]
+            1/(T^30 + T^29 + T^27 + T^26 + T^23 + T^22 + T^20 + T^19 + T^15 + T^14 + T^12 + T^11 + T^8 + T^7 + T^5 + T^4)
+            sage: log[2^5]
+            1/(T^62 + T^61 + T^59 + T^58 + T^55 + T^54 + T^52 + T^51 + T^47 + T^46 + T^44 + T^43 + T^40 + T^39 + T^37 + T^36 + T^31 + T^30 + T^28 + T^27 + T^24 + T^23 + T^21 + T^20 + T^16 + T^15 + T^13 + T^12 + T^9 + T^8 + T^6 + T^5)
+
+        Example in higher rank::
 
             sage: A = GF(5)['T']
             sage: K.<T> = Frac(A)
             sage: phi = DrinfeldModule(A, [T, T^2, T + T^2 + T^4, 1])
             sage: phi.logarithm()
-            z + ((4*T/(T^4+4))*z^5) + O(z^7)
+            z + ((4*T/(T^4+4))*z^5) + O(z^8)
+
+        TESTS::
+
+            sage: A = GF(2)['T']
+            sage: K.<T> = Frac(A)
+            sage: phi = DrinfeldModule(A, [T, 1])
+            sage: q = 2
+            sage: log[2] == -1/((T**q - T))  # expected value
+            True
+            sage: log[2**2] == 1/((T**q - T)*(T**(q**2) - T))  # expected value
+            True
+            sage: log[2**3] == -1/((T**q - T)*(T**(q**2) - T)*(T**(q**3) - T))  # expected value
+            True
         """
         L = LazyPowerSeriesRing(self._base, name)
         return L(self._compute_coefficient_log, valuation=1)

From 643cfde38388eac3d0b131c34003908a7cf7958b Mon Sep 17 00:00:00 2001
From: DavidAyotte <davidayotte94@outlook.com>
Date: Thu, 16 Mar 2023 11:56:44 -0400
Subject: [PATCH 07/26] fix some line length and add some details to the doc

---
 .../drinfeld_modules/drinfeld_module.py       | 28 ++++++++++++++-----
 1 file changed, 21 insertions(+), 7 deletions(-)

diff --git a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
index d74dad8463e..9fcdfdc52dd 100644
--- a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
+++ b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
@@ -1013,8 +1013,12 @@ def exponential(self, name='z'):
 
         INPUT:
 
-        - ``name`` (string, default: ``'z'``) -- the name of the generator of
-          the lazy power series ring.
+        - ``name`` (string, default: ``'z'``) -- the name of the
+          generator of the lazy power series ring.
+
+        OUTPUT:
+
+        A lazy power series over the base field.
 
         EXAMPLES::
 
@@ -1025,8 +1029,8 @@ def exponential(self, name='z'):
             sage: exp = phi.exponential(); exp
             z + ((1/(T^2+T))*z^2) + ((1/(T^8+T^6+T^5+T^3))*z^4) + O(z^8)
 
-        The exponential is returned as a lazy power series, meaning that any of
-        its coefficients can be computed on demands::
+        The exponential is returned as a lazy power series, meaning that
+        any of its coefficients can be computed on demands::
 
             sage: exp[2^4]
             1/(T^64 + T^56 + T^52 + T^50 + T^49 + T^44 + T^42 + T^41 + T^38 + T^37 + T^35 + T^30 + T^29 + T^27 + T^23 + T^15)
@@ -1066,7 +1070,8 @@ def exponential(self, name='z'):
 
         REFERENCE:
 
-        See section 4.6 of [Gos1998]_ for the definition of the exponential.
+        See section 4.6 of [Gos1998]_ for the definition of the
+        exponential.
         """
         L = LazyPowerSeriesRing(self._base, name)
         return L(self._compute_coefficient_exp, valuation=1)
@@ -1267,6 +1272,15 @@ def logarithm(self, name='z'):
         By definition, the logarithm is the compositional inverse of the
         exponential (see :meth:`exponential`).
 
+        INPUT:
+
+        - ``name`` (string, default: ``'z'``) -- the name of the
+          generator of the lazy power series ring.
+
+        OUTPUT:
+
+        A lazy power series over the base field.
+
         EXAMPLES::
 
             sage: A = GF(2)['T']
@@ -1275,8 +1289,8 @@ def logarithm(self, name='z'):
             sage: log = phi.logarithm(); log
             z + ((1/(T^2+T))*z^2) + ((1/(T^6+T^5+T^3+T^2))*z^4) + O(z^8)
 
-        The logarithm is returned as a lazy power series, meaning that any of
-        its coefficients can be computed on demands::
+        The logarithm is returned as a lazy power series, meaning that
+        any of its coefficients can be computed on demands::
 
             sage: log[2^4]
             1/(T^30 + T^29 + T^27 + T^26 + T^23 + T^22 + T^20 + T^19 + T^15 + T^14 + T^12 + T^11 + T^8 + T^7 + T^5 + T^4)

From 5356e394fc0e35ac909fd38bbbb6aa362b1126d4 Mon Sep 17 00:00:00 2001
From: DavidAyotte <davidayotte94@outlook.com>
Date: Fri, 24 Mar 2023 13:44:00 -0400
Subject: [PATCH 08/26] fix mistake

---
 .../rings/function_field/drinfeld_modules/drinfeld_module.py    | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
index 9fcdfdc52dd..467923eda44 100644
--- a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
+++ b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
@@ -964,7 +964,7 @@ def coefficients(self, sparse=True):
 
     def _compute_coefficient_exp(self, k):
         r"""
-        Return the `k`-th coefficient of the logarithm of ``self``.
+        Return the `k`-th coefficient of the exponential of ``self``.
 
         INPUT:
 

From 9c8d4a75ff23ed049ccba544d28e505df121a41c Mon Sep 17 00:00:00 2001
From: DavidAyotte <davidayotte94@outlook.com>
Date: Wed, 12 Apr 2023 17:33:08 -0400
Subject: [PATCH 09/26] small updates

---
 .../drinfeld_modules/drinfeld_module.py          | 16 +++++++++-------
 1 file changed, 9 insertions(+), 7 deletions(-)

diff --git a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
index 903481f65c5..982b13a2378 100644
--- a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
+++ b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
@@ -1027,7 +1027,7 @@ def exponential(self, name='z'):
             sage: phi = DrinfeldModule(A, [T, 1])
             sage: q = A.base_ring().cardinality()
             sage: exp = phi.exponential(); exp
-            z + ((1/(T^2+T))*z^2) + ((1/(T^8+T^6+T^5+T^3))*z^4) + O(z^8)
+            z + ((1/(T^2+T))*z^2) + ((1/(T^8+T^6+T^5+T^3))*z^4) + O(z^7)
 
         The exponential is returned as a lazy power series, meaning that
         any of its coefficients can be computed on demands::
@@ -1043,13 +1043,13 @@ def exponential(self, name='z'):
             sage: K.<T> = Frac(A)
             sage: phi = DrinfeldModule(A, [T, T^2, T + T^2 + T^4, 1])
             sage: exp = phi.exponential(); exp
-            z + ((T/(T^4+4))*z^5) + O(z^8)
+            z + ((T/(T^4+4))*z^5) + O(z^7)
 
         The exponential is the compositional inverse of the logarithm
         (see :meth:`logarithm`)::
 
             sage: log = phi.logarithm(); log
-            z + ((4*T/(T^4+4))*z^5) + O(z^8)
+            z + ((4*T/(T^4+4))*z^5) + O(z^7)
             sage: exp.compose(log)
             z + O(z^8)
             sage: log.compose(exp)
@@ -1074,7 +1074,8 @@ def exponential(self, name='z'):
         exponential.
         """
         L = LazyPowerSeriesRing(self._base, name)
-        return L(self._compute_coefficient_exp, valuation=1)
+        exp = lambda k: self._compute_coefficient_exp(k)
+        return L(exp)
 
     def gen(self):
         r"""
@@ -1287,7 +1288,7 @@ def logarithm(self, name='z'):
             sage: K.<T> = Frac(A)
             sage: phi = DrinfeldModule(A, [T, 1])
             sage: log = phi.logarithm(); log
-            z + ((1/(T^2+T))*z^2) + ((1/(T^6+T^5+T^3+T^2))*z^4) + O(z^8)
+            z + ((1/(T^2+T))*z^2) + ((1/(T^6+T^5+T^3+T^2))*z^4) + O(z^7)
 
         The logarithm is returned as a lazy power series, meaning that
         any of its coefficients can be computed on demands::
@@ -1303,7 +1304,7 @@ def logarithm(self, name='z'):
             sage: K.<T> = Frac(A)
             sage: phi = DrinfeldModule(A, [T, T^2, T + T^2 + T^4, 1])
             sage: phi.logarithm()
-            z + ((4*T/(T^4+4))*z^5) + O(z^8)
+            z + ((4*T/(T^4+4))*z^5) + O(z^7)
 
         TESTS::
 
@@ -1319,7 +1320,8 @@ def logarithm(self, name='z'):
             True
         """
         L = LazyPowerSeriesRing(self._base, name)
-        return L(self._compute_coefficient_log, valuation=1)
+        log = lambda k: self._compute_coefficient_log(k)
+        return L(log)
 
 
     def morphism(self):

From 644cafc88d9916afd622fffab1678ca5043315a7 Mon Sep 17 00:00:00 2001
From: DavidAyotte <davidayotte94@outlook.com>
Date: Wed, 12 Apr 2023 23:23:38 -0400
Subject: [PATCH 10/26] (re)add the valuation=1 flag

---
 .../rings/function_field/drinfeld_modules/drinfeld_module.py  | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
index 982b13a2378..9f728cf715e 100644
--- a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
+++ b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
@@ -1075,7 +1075,7 @@ def exponential(self, name='z'):
         """
         L = LazyPowerSeriesRing(self._base, name)
         exp = lambda k: self._compute_coefficient_exp(k)
-        return L(exp)
+        return L(exp, valuation=1)
 
     def gen(self):
         r"""
@@ -1321,7 +1321,7 @@ def logarithm(self, name='z'):
         """
         L = LazyPowerSeriesRing(self._base, name)
         log = lambda k: self._compute_coefficient_log(k)
-        return L(log)
+        return L(log, valuation=1)
 
 
     def morphism(self):

From 8323f6f0bbd3bd1095efc6a2d9f83d7331cb6dab Mon Sep 17 00:00:00 2001
From: DavidAyotte <davidayotte94@outlook.com>
Date: Sat, 15 Apr 2023 08:47:48 -0400
Subject: [PATCH 11/26] fix failing doctests

---
 .../function_field/drinfeld_modules/drinfeld_module.py | 10 +++++-----
 1 file changed, 5 insertions(+), 5 deletions(-)

diff --git a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
index 9f728cf715e..5f41e874c6f 100644
--- a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
+++ b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
@@ -1027,7 +1027,7 @@ def exponential(self, name='z'):
             sage: phi = DrinfeldModule(A, [T, 1])
             sage: q = A.base_ring().cardinality()
             sage: exp = phi.exponential(); exp
-            z + ((1/(T^2+T))*z^2) + ((1/(T^8+T^6+T^5+T^3))*z^4) + O(z^7)
+            z + ((1/(T^2+T))*z^2) + ((1/(T^8+T^6+T^5+T^3))*z^4) + O(z^8)
 
         The exponential is returned as a lazy power series, meaning that
         any of its coefficients can be computed on demands::
@@ -1043,13 +1043,13 @@ def exponential(self, name='z'):
             sage: K.<T> = Frac(A)
             sage: phi = DrinfeldModule(A, [T, T^2, T + T^2 + T^4, 1])
             sage: exp = phi.exponential(); exp
-            z + ((T/(T^4+4))*z^5) + O(z^7)
+            z + ((T/(T^4+4))*z^5) + O(z^8)
 
         The exponential is the compositional inverse of the logarithm
         (see :meth:`logarithm`)::
 
             sage: log = phi.logarithm(); log
-            z + ((4*T/(T^4+4))*z^5) + O(z^7)
+            z + ((4*T/(T^4+4))*z^5) + O(z^8)
             sage: exp.compose(log)
             z + O(z^8)
             sage: log.compose(exp)
@@ -1288,7 +1288,7 @@ def logarithm(self, name='z'):
             sage: K.<T> = Frac(A)
             sage: phi = DrinfeldModule(A, [T, 1])
             sage: log = phi.logarithm(); log
-            z + ((1/(T^2+T))*z^2) + ((1/(T^6+T^5+T^3+T^2))*z^4) + O(z^7)
+            z + ((1/(T^2+T))*z^2) + ((1/(T^6+T^5+T^3+T^2))*z^4) + O(z^8)
 
         The logarithm is returned as a lazy power series, meaning that
         any of its coefficients can be computed on demands::
@@ -1304,7 +1304,7 @@ def logarithm(self, name='z'):
             sage: K.<T> = Frac(A)
             sage: phi = DrinfeldModule(A, [T, T^2, T + T^2 + T^4, 1])
             sage: phi.logarithm()
-            z + ((4*T/(T^4+4))*z^5) + O(z^7)
+            z + ((4*T/(T^4+4))*z^5) + O(z^8)
 
         TESTS::
 

From eb781112d67a93cba6371a39fd1b110ed6f8a6f1 Mon Sep 17 00:00:00 2001
From: David Ayotte <34245930+DavidAyotte@users.noreply.github.com>
Date: Thu, 27 Apr 2023 09:40:22 -0400
Subject: [PATCH 12/26] Update
 src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py

Co-authored-by: Xavier Caruso <xcaruso@users.noreply.github.com>
---
 .../rings/function_field/drinfeld_modules/drinfeld_module.py    | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
index 5f41e874c6f..c7f12431921 100644
--- a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
+++ b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
@@ -1009,7 +1009,7 @@ def _compute_coefficient_exp(self, k):
 
     def exponential(self, name='z'):
         r"""
-        Return the exponential of the given Drinfeld module.
+        Return the exponential of this Drinfeld module.
 
         INPUT:
 

From b73d1ae52e15ed0b639a41dbf99da5c5feb27226 Mon Sep 17 00:00:00 2001
From: DavidAyotte <davidayotte94@outlook.com>
Date: Wed, 26 Apr 2023 19:58:18 -0400
Subject: [PATCH 13/26] remove unnecessary integer type conversion

---
 .../drinfeld_modules/drinfeld_module.py            | 14 --------------
 1 file changed, 14 deletions(-)

diff --git a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
index c7f12431921..95d52f06e1c 100644
--- a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
+++ b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
@@ -986,14 +986,7 @@ def _compute_coefficient_exp(self, k):
             0
             sage: phi._compute_coefficient_exp(2^4)
             1/(T^64 + T^56 + T^52 + T^50 + T^49 + T^44 + T^42 + T^41 + T^38 + T^37 + T^35 + T^30 + T^29 + T^27 + T^23 + T^15)
-            sage: phi._compute_coefficient_exp(T)
-            Traceback (most recent call last):
-            ...
-            TypeError: input must be an integer
         """
-        if k not in ZZ:
-            raise TypeError("input must be an integer")
-        k = ZZ(k)
         if k.is_zero():
             return self._base.zero()
         if k.is_one():
@@ -1242,14 +1235,7 @@ def _compute_coefficient_log(self, k):
             0
             sage: phi._compute_coefficient_log(2^4)
             1/(T^30 + T^29 + T^27 + T^26 + T^23 + T^22 + T^20 + T^19 + T^15 + T^14 + T^12 + T^11 + T^8 + T^7 + T^5 + T^4)
-            sage: phi._compute_coefficient_log(T)
-            Traceback (most recent call last):
-            ...
-            TypeError: input must be an integer
         """
-        if k not in ZZ:
-            raise TypeError("input must be an integer")
-        k = ZZ(k)
         if k.is_zero():
             return self._base.zero()
         if k.is_one():

From ba57245bd0cc68ddbc6bf5d8e34d01b5c1cb7305 Mon Sep 17 00:00:00 2001
From: DavidAyotte <davidayotte94@outlook.com>
Date: Wed, 26 Apr 2023 20:00:46 -0400
Subject: [PATCH 14/26] add cached_method decorator

---
 .../rings/function_field/drinfeld_modules/drinfeld_module.py   | 3 +++
 1 file changed, 3 insertions(+)

diff --git a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
index 95d52f06e1c..b69314aa908 100644
--- a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
+++ b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
@@ -26,6 +26,7 @@
 
 from sage.categories.drinfeld_modules import DrinfeldModules
 from sage.categories.homset import Hom
+from sage.misc.cachefunc import cached_method
 from sage.misc.latex import latex
 from sage.misc.latex import latex_variable_name
 from sage.misc.lazy_import import lazy_import
@@ -962,6 +963,7 @@ def coefficients(self, sparse=True):
         """
         return self._gen.coefficients(sparse=sparse)
 
+    @cached_method
     def _compute_coefficient_exp(self, k):
         r"""
         Return the `k`-th coefficient of the exponential of ``self``.
@@ -1215,6 +1217,7 @@ def j_invariant(self):
         q = self._Fq.order()
         return (g**(q+1)) / delta
 
+    @cached_method
     def _compute_coefficient_log(self, k):
         r"""
         Return the `k`-th coefficient of the logarithm of ``self``.

From 364fd558f459b581015efebbc62bbf6420b2196a Mon Sep 17 00:00:00 2001
From: DavidAyotte <davidayotte94@outlook.com>
Date: Wed, 26 Apr 2023 20:05:09 -0400
Subject: [PATCH 15/26] use ellipsis

---
 .../function_field/drinfeld_modules/drinfeld_module.py    | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
index b69314aa908..a1ac0f0a1ef 100644
--- a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
+++ b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
@@ -1028,9 +1028,9 @@ def exponential(self, name='z'):
         any of its coefficients can be computed on demands::
 
             sage: exp[2^4]
-            1/(T^64 + T^56 + T^52 + T^50 + T^49 + T^44 + T^42 + T^41 + T^38 + T^37 + T^35 + T^30 + T^29 + T^27 + T^23 + T^15)
+            1/(T^64 + T^56 + T^52 + ... + T^27 + T^23 + T^15)
             sage: exp[2^5]
-            1/(T^160 + T^144 + T^136 + T^132 + T^130 + T^129 + T^120 + T^116 + T^114 + T^113 + T^108 + T^106 + T^105 + T^102 + T^101 + T^99 + T^92 + T^90 + T^89 + T^86 + T^85 + T^83 + T^78 + T^77 + T^75 + T^71 + T^62 + T^61 + T^59 + T^55 + T^47 + T^31)
+            1/(T^160 + T^144 + T^136 + ... + T^55 + T^47 + T^31)
 
         Example in higher rank::
 
@@ -1283,9 +1283,9 @@ def logarithm(self, name='z'):
         any of its coefficients can be computed on demands::
 
             sage: log[2^4]
-            1/(T^30 + T^29 + T^27 + T^26 + T^23 + T^22 + T^20 + T^19 + T^15 + T^14 + T^12 + T^11 + T^8 + T^7 + T^5 + T^4)
+            1/(T^30 + T^29 + T^27 + ... + T^7 + T^5 + T^4)
             sage: log[2^5]
-            1/(T^62 + T^61 + T^59 + T^58 + T^55 + T^54 + T^52 + T^51 + T^47 + T^46 + T^44 + T^43 + T^40 + T^39 + T^37 + T^36 + T^31 + T^30 + T^28 + T^27 + T^24 + T^23 + T^21 + T^20 + T^16 + T^15 + T^13 + T^12 + T^9 + T^8 + T^6 + T^5)
+            1/(T^62 + T^61 + T^59 + ... + T^8 + T^6 + T^5)
 
         Example in higher rank::
 

From 8f292e9c7e2e217855ef53adf6a4b8fdf9dc3fbe Mon Sep 17 00:00:00 2001
From: DavidAyotte <davidayotte94@outlook.com>
Date: Wed, 26 Apr 2023 20:10:25 -0400
Subject: [PATCH 16/26] fix failing doctets

---
 .../rings/function_field/drinfeld_modules/drinfeld_module.py    | 2 ++
 1 file changed, 2 insertions(+)

diff --git a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
index a1ac0f0a1ef..302f541ed85 100644
--- a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
+++ b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
@@ -989,6 +989,7 @@ def _compute_coefficient_exp(self, k):
             sage: phi._compute_coefficient_exp(2^4)
             1/(T^64 + T^56 + T^52 + T^50 + T^49 + T^44 + T^42 + T^41 + T^38 + T^37 + T^35 + T^30 + T^29 + T^27 + T^23 + T^15)
         """
+        k = ZZ(k)
         if k.is_zero():
             return self._base.zero()
         if k.is_one():
@@ -1239,6 +1240,7 @@ def _compute_coefficient_log(self, k):
             sage: phi._compute_coefficient_log(2^4)
             1/(T^30 + T^29 + T^27 + T^26 + T^23 + T^22 + T^20 + T^19 + T^15 + T^14 + T^12 + T^11 + T^8 + T^7 + T^5 + T^4)
         """
+        k = ZZ(k)
         if k.is_zero():
             return self._base.zero()
         if k.is_one():

From 1504a19b48623084ad60ce8a4550c931758993a2 Mon Sep 17 00:00:00 2001
From: DavidAyotte <davidayotte94@outlook.com>
Date: Thu, 27 Apr 2023 12:41:41 -0400
Subject: [PATCH 17/26] add checks for finite Drinfeld modules

---
 .../drinfeld_modules/drinfeld_module.py       | 30 ++++++++++++++++++-
 1 file changed, 29 insertions(+), 1 deletion(-)

diff --git a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
index 302f541ed85..2d2ddf31098 100644
--- a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
+++ b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
@@ -1007,6 +1007,9 @@ def exponential(self, name='z'):
         r"""
         Return the exponential of this Drinfeld module.
 
+        Note that the exponential is only defined for Drinfeld modules
+        over nonfinite field.
+
         INPUT:
 
         - ``name`` (string, default: ``'z'``) -- the name of the
@@ -1051,6 +1054,16 @@ def exponential(self, name='z'):
             sage: log.compose(exp)
             z + O(z^8)
 
+        ::
+
+            sage: Fq.<w> = GF(3)
+            sage: A = Fq['T']
+            sage: phi = DrinfeldModule(A, [w, 1])
+            sage: phi.exponential()
+            Traceback (most recent call last):
+            ...
+            ValueError: base field must not be finite
+
         TESTS::
 
             sage: A = GF(2)['T']
@@ -1069,6 +1082,8 @@ def exponential(self, name='z'):
         See section 4.6 of [Gos1998]_ for the definition of the
         exponential.
         """
+        if self.is_finite():
+            raise ValueError("base field must not be finite")
         L = LazyPowerSeriesRing(self._base, name)
         exp = lambda k: self._compute_coefficient_exp(k)
         return L(exp, valuation=1)
@@ -1262,7 +1277,8 @@ def logarithm(self, name='z'):
         Return the logarithm of the given Drinfeld module.
 
         By definition, the logarithm is the compositional inverse of the
-        exponential (see :meth:`exponential`).
+        exponential (see :meth:`exponential`). Note that the logarithm
+        is only defined for Drinfeld modules over nonfinite field.
 
         INPUT:
 
@@ -1309,7 +1325,19 @@ def logarithm(self, name='z'):
             True
             sage: log[2**3] == -1/((T**q - T)*(T**(q**2) - T)*(T**(q**3) - T))  # expected value
             True
+
+        ::
+
+            sage: Fq.<w> = GF(3)
+            sage: A = Fq['T']
+            sage: phi = DrinfeldModule(A, [w, 1])
+            sage: phi.logarithm()
+            Traceback (most recent call last):
+            ...
+            ValueError: base field must not be finite
         """
+        if self.is_finite():
+            raise ValueError("base field must not be finite")
         L = LazyPowerSeriesRing(self._base, name)
         log = lambda k: self._compute_coefficient_log(k)
         return L(log, valuation=1)

From 706ded593f7cb06b51e9ec7e6a4bb10b07cb7290 Mon Sep 17 00:00:00 2001
From: DavidAyotte <davidayotte94@outlook.com>
Date: Thu, 27 Apr 2023 13:42:29 -0400
Subject: [PATCH 18/26] update error message

---
 .../function_field/drinfeld_modules/drinfeld_module.py    | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
index 2d2ddf31098..e58d0189230 100644
--- a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
+++ b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
@@ -1062,7 +1062,7 @@ def exponential(self, name='z'):
             sage: phi.exponential()
             Traceback (most recent call last):
             ...
-            ValueError: base field must not be finite
+            ValueError: characteristic must be zero (=T + 2)
 
         TESTS::
 
@@ -1083,7 +1083,7 @@ def exponential(self, name='z'):
         exponential.
         """
         if self.is_finite():
-            raise ValueError("base field must not be finite")
+            raise ValueError(f"characteristic must be zero (={self.characteristic()})")
         L = LazyPowerSeriesRing(self._base, name)
         exp = lambda k: self._compute_coefficient_exp(k)
         return L(exp, valuation=1)
@@ -1334,10 +1334,10 @@ def logarithm(self, name='z'):
             sage: phi.logarithm()
             Traceback (most recent call last):
             ...
-            ValueError: base field must not be finite
+            ValueError: characteristic must be zero (=T + 2)
         """
         if self.is_finite():
-            raise ValueError("base field must not be finite")
+            raise ValueError(f"characteristic must be zero (={self.characteristic()})")
         L = LazyPowerSeriesRing(self._base, name)
         log = lambda k: self._compute_coefficient_log(k)
         return L(log, valuation=1)

From 26498b8f2e11369837ddad6d0a3b90f60506c9f7 Mon Sep 17 00:00:00 2001
From: DavidAyotte <davidayotte94@outlook.com>
Date: Thu, 4 May 2023 10:33:38 -0400
Subject: [PATCH 19/26] refactor _compute_coefficient methods

---
 .../drinfeld_modules/drinfeld_module.py       | 50 ++++++++-----------
 1 file changed, 21 insertions(+), 29 deletions(-)

diff --git a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
index e58d0189230..27d5c4ee5f3 100644
--- a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
+++ b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
@@ -979,28 +979,22 @@ def _compute_coefficient_exp(self, k):
             sage: phi = DrinfeldModule(A, [T, 1])
             sage: q = A.base_ring().cardinality()
             sage: phi._compute_coefficient_exp(0)
-            0
-            sage: phi._compute_coefficient_exp(1)
             1
-            sage: phi._compute_coefficient_exp(2)
+            sage: phi._compute_coefficient_exp(1)
             1/(T^2 + T)
+            sage: phi._compute_coefficient_exp(2)
+            1/(T^8 + T^6 + T^5 + T^3)
             sage: phi._compute_coefficient_exp(3)
-            0
-            sage: phi._compute_coefficient_exp(2^4)
-            1/(T^64 + T^56 + T^52 + T^50 + T^49 + T^44 + T^42 + T^41 + T^38 + T^37 + T^35 + T^30 + T^29 + T^27 + T^23 + T^15)
+            1/(T^24 + T^20 + T^18 + T^17 + T^14 + T^13 + T^11 + T^7)
         """
         k = ZZ(k)
         if k.is_zero():
-            return self._base.zero()
-        if k.is_one():
             return self._base.one()
         q = self._Fq.cardinality()
-        if not k.is_power_of(q):
-            return self._base.zero()
         c = self._base.zero()
-        for i in range(k.log(q)):
-            j = k.log(q) - i
-            c += self._compute_coefficient_exp(q**i)*self._compute_coefficient_log(q**j)**(q**i)
+        for i in range(k):
+            j = k - i
+            c += self._compute_coefficient_exp(i)*self._compute_coefficient_log(j)**(q**i)
         return -c
 
     def exponential(self, name='z'):
@@ -1085,7 +1079,9 @@ def exponential(self, name='z'):
         if self.is_finite():
             raise ValueError(f"characteristic must be zero (={self.characteristic()})")
         L = LazyPowerSeriesRing(self._base, name)
-        exp = lambda k: self._compute_coefficient_exp(k)
+        zero = self._base.zero()
+        q = self._Fq.cardinality()
+        exp = lambda k: self._compute_coefficient_exp(ZZ(k).log(q)) if ZZ(k).is_power_of(q) or k == 0 else zero
         return L(exp, valuation=1)
 
     def gen(self):
@@ -1245,32 +1241,26 @@ def _compute_coefficient_log(self, k):
             sage: phi = DrinfeldModule(A, [T, 1])
             sage: q = A.base_ring().cardinality()
             sage: phi._compute_coefficient_log(0)
-            0
-            sage: phi._compute_coefficient_log(1)
             1
-            sage: phi._compute_coefficient_log(2)
+            sage: phi._compute_coefficient_log(1)
             1/(T^2 + T)
+            sage: phi._compute_coefficient_log(2)
+            1/(T^6 + T^5 + T^3 + T^2)
             sage: phi._compute_coefficient_log(3)
-            0
-            sage: phi._compute_coefficient_log(2^4)
-            1/(T^30 + T^29 + T^27 + T^26 + T^23 + T^22 + T^20 + T^19 + T^15 + T^14 + T^12 + T^11 + T^8 + T^7 + T^5 + T^4)
+            1/(T^14 + T^13 + T^11 + T^10 + T^7 + T^6 + T^4 + T^3)
         """
         k = ZZ(k)
         if k.is_zero():
-            return self._base.zero()
-        if k.is_one():
             return self._base.one()
         r = self._gen.degree()
         T = self._gen[0]
         q = self._Fq.cardinality()
-        if not k.is_power_of(q):
-            return self._base.zero()
         c = self._base.zero()
-        for i in range(k.log(q)):
-            j = k.log(q) - i
+        for i in range(k):
+            j = k - i
             if j < r + 1:
-                c += self._compute_coefficient_log(q**i)*self._gen[j]**(q**i)
-        return c/(T - T**k)
+                c += self._compute_coefficient_log(i)*self._gen[j]**(q**i)
+        return c/(T - T**(q**k))
 
     def logarithm(self, name='z'):
         r"""
@@ -1339,7 +1329,9 @@ def logarithm(self, name='z'):
         if self.is_finite():
             raise ValueError(f"characteristic must be zero (={self.characteristic()})")
         L = LazyPowerSeriesRing(self._base, name)
-        log = lambda k: self._compute_coefficient_log(k)
+        zero = self._base.zero()
+        q = self._Fq.cardinality()
+        log = lambda k: self._compute_coefficient_log(ZZ(k).log(q)) if ZZ(k).is_power_of(q) or k == 0 else zero
         return L(log, valuation=1)
 
 

From 005acd55c0cf3954f448067f8e61bfad6ac26feb Mon Sep 17 00:00:00 2001
From: David Ayotte <34245930+DavidAyotte@users.noreply.github.com>
Date: Thu, 4 May 2023 14:01:54 -0400
Subject: [PATCH 20/26] Update
 src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py

Co-authored-by: Xavier Caruso <xcaruso@users.noreply.github.com>
---
 .../rings/function_field/drinfeld_modules/drinfeld_module.py    | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
index 27d5c4ee5f3..efad8883ea3 100644
--- a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
+++ b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
@@ -966,7 +966,7 @@ def coefficients(self, sparse=True):
     @cached_method
     def _compute_coefficient_exp(self, k):
         r"""
-        Return the `k`-th coefficient of the exponential of ``self``.
+        Return the `q^k`-th coefficient of the exponential of this Drinfeld module.
 
         INPUT:
 

From 7f72a218408df30b7bfc4387490bc90f7a4fe04b Mon Sep 17 00:00:00 2001
From: David Ayotte <34245930+DavidAyotte@users.noreply.github.com>
Date: Thu, 4 May 2023 14:02:16 -0400
Subject: [PATCH 21/26] Update
 src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py

Co-authored-by: Xavier Caruso <xcaruso@users.noreply.github.com>
---
 .../rings/function_field/drinfeld_modules/drinfeld_module.py    | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
index efad8883ea3..99aa500a6dd 100644
--- a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
+++ b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
@@ -1232,7 +1232,7 @@ def j_invariant(self):
     @cached_method
     def _compute_coefficient_log(self, k):
         r"""
-        Return the `k`-th coefficient of the logarithm of ``self``.
+        Return the `q^k`-th coefficient of the logarithm of this Drinfeld module.
 
         TESTS::
 

From 962e70db82de734aa023123aa1d8798c9f1a5946 Mon Sep 17 00:00:00 2001
From: David Ayotte <34245930+DavidAyotte@users.noreply.github.com>
Date: Fri, 5 May 2023 13:33:48 -0400
Subject: [PATCH 22/26] Update
 src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py

Co-authored-by: Xavier Caruso <xcaruso@users.noreply.github.com>
---
 .../rings/function_field/drinfeld_modules/drinfeld_module.py    | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
index 99aa500a6dd..694e0529697 100644
--- a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
+++ b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
@@ -1076,7 +1076,7 @@ def exponential(self, name='z'):
         See section 4.6 of [Gos1998]_ for the definition of the
         exponential.
         """
-        if self.is_finite():
+        if not self.category()._characteristic:
             raise ValueError(f"characteristic must be zero (={self.characteristic()})")
         L = LazyPowerSeriesRing(self._base, name)
         zero = self._base.zero()

From 3e873b95ba45cf11ccde74c6e65612faac6e435b Mon Sep 17 00:00:00 2001
From: David Ayotte <34245930+DavidAyotte@users.noreply.github.com>
Date: Fri, 5 May 2023 13:34:00 -0400
Subject: [PATCH 23/26] Update
 src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py

Co-authored-by: Xavier Caruso <xcaruso@users.noreply.github.com>
---
 .../rings/function_field/drinfeld_modules/drinfeld_module.py    | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
index 694e0529697..b4f6ee79b60 100644
--- a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
+++ b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
@@ -1326,7 +1326,7 @@ def logarithm(self, name='z'):
             ...
             ValueError: characteristic must be zero (=T + 2)
         """
-        if self.is_finite():
+        if not self.category()._characteristic:
             raise ValueError(f"characteristic must be zero (={self.characteristic()})")
         L = LazyPowerSeriesRing(self._base, name)
         zero = self._base.zero()

From f8c867d032c6d80ba2fb89373b86dab3ecf7beea Mon Sep 17 00:00:00 2001
From: DavidAyotte <davidayotte94@outlook.com>
Date: Fri, 5 May 2023 12:52:32 -0400
Subject: [PATCH 24/26] fix failing doctest

---
 .../rings/function_field/drinfeld_modules/drinfeld_module.py  | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
index b4f6ee79b60..21b2dfcb56e 100644
--- a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
+++ b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
@@ -1076,7 +1076,7 @@ def exponential(self, name='z'):
         See section 4.6 of [Gos1998]_ for the definition of the
         exponential.
         """
-        if not self.category()._characteristic:
+        if self.category()._characteristic:
             raise ValueError(f"characteristic must be zero (={self.characteristic()})")
         L = LazyPowerSeriesRing(self._base, name)
         zero = self._base.zero()
@@ -1326,7 +1326,7 @@ def logarithm(self, name='z'):
             ...
             ValueError: characteristic must be zero (=T + 2)
         """
-        if not self.category()._characteristic:
+        if self.category()._characteristic:
             raise ValueError(f"characteristic must be zero (={self.characteristic()})")
         L = LazyPowerSeriesRing(self._base, name)
         zero = self._base.zero()

From e85227e63b6c4671c6bb8584b2b9a7acc2d653cc Mon Sep 17 00:00:00 2001
From: DavidAyotte <davidayotte94@outlook.com>
Date: Fri, 5 May 2023 13:48:37 -0400
Subject: [PATCH 25/26] minor update to documentation of exponential and
 logarithm

---
 .../function_field/drinfeld_modules/drinfeld_module.py     | 7 ++++---
 1 file changed, 4 insertions(+), 3 deletions(-)

diff --git a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
index 21b2dfcb56e..8bcadf4dbac 100644
--- a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
+++ b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
@@ -1001,8 +1001,8 @@ def exponential(self, name='z'):
         r"""
         Return the exponential of this Drinfeld module.
 
-        Note that the exponential is only defined for Drinfeld modules
-        over nonfinite field.
+        Note that the exponential is only defined when the
+        `\mathbb{F}_q[T]`-characteristic is zero.
 
         INPUT:
 
@@ -1268,7 +1268,8 @@ def logarithm(self, name='z'):
 
         By definition, the logarithm is the compositional inverse of the
         exponential (see :meth:`exponential`). Note that the logarithm
-        is only defined for Drinfeld modules over nonfinite field.
+        is only defined when the `\mathbb{F}_q[T]`-characteristic is
+        zero.
 
         INPUT:
 

From 2b242fbf012de88bb4851c62c312403e2d30f864 Mon Sep 17 00:00:00 2001
From: DavidAyotte <davidayotte94@outlook.com>
Date: Sun, 14 May 2023 18:19:22 -0400
Subject: [PATCH 26/26] refactor exponential and logarithm

---
 .../drinfeld_modules/drinfeld_module.py       | 22 +++++++++++++++----
 1 file changed, 18 insertions(+), 4 deletions(-)

diff --git a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
index 8bcadf4dbac..9c8b2ecde94 100644
--- a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
+++ b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
@@ -1081,8 +1081,15 @@ def exponential(self, name='z'):
         L = LazyPowerSeriesRing(self._base, name)
         zero = self._base.zero()
         q = self._Fq.cardinality()
-        exp = lambda k: self._compute_coefficient_exp(ZZ(k).log(q)) if ZZ(k).is_power_of(q) or k == 0 else zero
-        return L(exp, valuation=1)
+
+        def coeff_exp(k):
+            # Return the k-th coefficient of the exponential.
+            k = ZZ(k)
+            if k.is_power_of(q):
+                return self._compute_coefficient_exp(k.log(q))
+            else:
+                return zero
+        return L(coeff_exp, valuation=1)
 
     def gen(self):
         r"""
@@ -1332,8 +1339,15 @@ def logarithm(self, name='z'):
         L = LazyPowerSeriesRing(self._base, name)
         zero = self._base.zero()
         q = self._Fq.cardinality()
-        log = lambda k: self._compute_coefficient_log(ZZ(k).log(q)) if ZZ(k).is_power_of(q) or k == 0 else zero
-        return L(log, valuation=1)
+
+        def coeff_log(k):
+            # Return the k-th coefficient of the logarithm
+            k = ZZ(k)
+            if k.is_power_of(q):
+                return self._compute_coefficient_log(k.log(q))
+            else:
+                return self._base.zero()
+        return L(coeff_log, valuation=1)
 
 
     def morphism(self):