diff --git a/book/src/background/polynomials.md b/book/src/background/polynomials.md
index 5e4c164..257bf1f 100644
--- a/book/src/background/polynomials.md
+++ b/book/src/background/polynomials.md
@@ -7,7 +7,16 @@ A(X) = a_0 + a_1 X + a_2 X^2 + a_3 X^3
 $$
 
 defines a degree-$3$ polynomial. $a_0$ is referred to as the constant term. Polynomials of
-degree $n-1$ have $n$ coefficients.
+degree $n-1$ have $n$ coefficients. We will often want to compute the result of replacing
+the formal indeterminate $X$ with some concrete value $x$, which we denote by $A(x)$.
+
+> In mathematics this is commonly referred to as "evaluating $A(X)$ at a point $x$".
+> The word "point" here stems from the geometrical usage of polynomials in the form
+> $y = A(x)$, where $(x, y)$ is the coordinate of a point in two-dimensional space.
+> However, the polynomials we deal with are almost always constrained to equal zero, and
+> $x$ will be an [element of some group](fields.md#groups). This should not be confused
+> with points on an [elliptic curve](curves.md), which we also make use of, but never in
+> the context of polynomial evaluation.
 
 Important notes:
 
diff --git a/book/src/background/upa.md b/book/src/background/upa.md
index 3a4000b..260b1a5 100644
--- a/book/src/background/upa.md
+++ b/book/src/background/upa.md
@@ -5,11 +5,11 @@ We call the field over which the circuit is defined $\mathbb{F} = \mathbb{F}_p$.
 Let $n = 2^k$, and assume that $\omega$ is a primitive root of unity of order $n$ in
 $\mathbb{F}^\times$, so that $\mathbb{F}^\times$ has a multiplicative subgroup
 $\mathcal{H} = \{1, \omega, \omega^2, \cdots, \omega^{n-1}\}$. This forms a Lagrange
-basis corresponding to the points in the subgroup.
+basis corresponding to the elements in the subgroup.
 
 ## Polynomial rules
 A polynomial rule defines a constraint that must hold between its specified columns at
-every row (i.e. at every point in the multiplicative subgroup).
+every row (i.e. at every element in the multiplicative subgroup).
 
 e.g.
 
@@ -59,7 +59,7 @@ Reference: [Generic Lookups with PLONK (DRAFT)](/LTPc5f-3S0qNF6MtwD-Tdg?view)
 
 ### Vanishing argument
 We want to check that the expressions defined by the gate constraints, permutation
-constraints and loookup constraints evaluate to zero at all points in the multiplicative
+constraints and loookup constraints evaluate to zero at all elements in the multiplicative
 subgroup. To do this, the prover collapses all the expressions into one polynomial 
 $$H(X) = \sum_{i=0}^e y^i E_i(X),$$
 where $e$ is the number of expressions and $y$ is a random challenge used to keep the