The key technical fact exploited in this section relates the sum of any low-degree polynomial over a potentially large subset of inputs $H$ to the polynomial's evaluation at a single input, namely 0 . Below, a non-empty subset $H \subseteq \mathbb{F}$ is said to be a multiplicative subgroup of field $\mathbb{F}$ if $H$ is closed under multiplication and inverses, i.e., for any $a, b \in H, a \cdot b \in H$, and $a^{-1}, b^{-1} \in H$. Fact 10.1. Let $\mathbb{F}$ be a finite field and suppose that $H$ is a multiplicative subgroup of $\mathbb{F}$ of size $n$. Then for any polynomial $q$ of degree less than $|H|=n, \sum_{a \in H} q(a)=q(0) \cdot|H|$. It follows that $\sum_{a \in H} q(a)$ is 0 if and only if $q(0)=0$.