diff --git a/src/plonk/keygen.rs b/src/plonk/keygen.rs
index 5df667a..c04fd42 100644
--- a/src/plonk/keygen.rs
+++ b/src/plonk/keygen.rs
@@ -76,6 +76,17 @@ where
         .max()
         .unwrap_or(1);
 
+    // The lookup argument also serves alongside the gates and must be accounted
+    // for.
+    degree = std::cmp::max(
+        degree,
+        cs.lookups
+            .iter()
+            .map(|l| l.required_degree())
+            .max()
+            .unwrap_or(1),
+    );
+
     // Account for each gate to ensure our quotient polynomial is the
     // correct degree and that our extended domain is the right size.
     for poly in cs.gates.iter() {
diff --git a/src/plonk/lookup.rs b/src/plonk/lookup.rs
index 640faef..7d588d0 100644
--- a/src/plonk/lookup.rs
+++ b/src/plonk/lookup.rs
@@ -18,6 +18,24 @@ impl Argument {
             table_columns: table_columns.to_vec(),
         }
     }
+
+    pub(crate) fn required_degree(&self) -> usize {
+        assert_eq!(self.input_columns.len(), self.table_columns.len());
+
+        // degree 2:
+        // l_0(X) * (1 - z'(X)) = 0
+        //
+        // degree 3:
+        // z'(X) (a'(X) + \beta) (s'(X) + \gamma)
+        // - z'(\omega^{-1} X) (\theta^{m-1} a_0(X) + ... + a_{m-1}(X) + \beta) (\theta^{m-1} s_0(X) + ... + s_{m-1}(X) + \gamma)
+        //
+        // degree 2:
+        // l_0(X) * (a'(X) - s'(X)) = 0
+        //
+        // degree 2:
+        // (a′(X)−s′(X))⋅(a′(X)−a′(\omega{-1} X)) = 0
+        3
+    }
 }
 
 #[derive(Clone, Debug)]
diff --git a/src/plonk/permutation.rs b/src/plonk/permutation.rs
index deda02b..0de904c 100644
--- a/src/plonk/permutation.rs
+++ b/src/plonk/permutation.rs
@@ -24,8 +24,18 @@ impl Argument {
 
     pub(crate) fn required_degree(&self) -> usize {
         // The permutation argument will serve alongside the gates, so must be
-        // accounted for.
-        self.columns.len() + 1
+        // accounted for. There are constraints of degree 2 regardless of the
+        // number of columns involved. (It doesn't make sense to make a
+        // permutation argument with zero columns but to be rigorous we account
+        // for it here.)
+
+        // degree 2:
+        // l_0(X) * (1 - z(X)) = 0
+        //
+        // degree columns + 1
+        // z(X) \prod (p(X) + \beta s_i(X) + \gamma)
+        // - z(omega^{-1} X) \prod (p(X) + \delta^i \beta X + \gamma)
+        std::cmp::max(self.columns.len() + 1, 2)
     }
 }