In the following definitions and examples, we use the following polynomial as an example:
\begin{equation}
\label{eq:poly-eg}
\poly(X, Y) = 2X^2 + 3XY - 2Y^2.
\end{equation}
\begin{Definition}[Pure Expansion]
The pure expansion of a polynomial $\poly$ is formed by computing all product of sums occurring in $\poly$, without combining like monomials. The pure expansion of $\poly$ generalizes \Cref{def:smb} by allowing monomials $m_i = m_j$ for $i \neq j$.
Note that similar in spirit to \Cref{def:reduced-bi-poly}, $\expansion{\circuit}$\Cref{def:expand-circuit} reduces all variable exponents $e > 1$ to $e =1$.
The pure expansion of the product is $2X^2- XY +4XY -2Y^2$ and the $\expansion{\circuit}$ is $[(X, 2), (XY, -1), (XY, 4), (Y, -2)]$.
\end{Example}
$\expansion{\circuit}$ effectively\footnote{The minor difference here is that $\expansion{\circuit}$ encodes the \emph{reduced} form over the SOP expansion of the compressed representation, as opposed to the \abbrSMB representation} encodes the \emph{reduced} form of $\polyf\inparen{\circuit}$, decoupling each monomial into a set of variables $\monom$ and a real coefficient $\coef$.
However, unlike the constraint on the input to compute $\rpoly$, the input circuit $\circuit$ does not need to be in \abbrSMB/SOP form.
\begin{Example}[Example for \Cref{def:positive-circuit}]\label{ex:def-pos-circ}
Using the same factorization from \Cref{example:expr-tree-T}, $\polyf(\abs{\circuit})=(X +2Y)(2X + Y)=2X^2+XY +4XY +2Y^2=2X^2+5XY +2Y^2$. Note that this \textit{is not} the same as the polynomial from \Cref{eq:poly-eg}.
Given a circuit $\circuit$ and a valuation $\vct{a}\in\mathbb{R}^\numvar$, we define the evaluation of $\circuit$ on $\vct{a}$ as $\circuit(\vct{a})=\polyf(\circuit)(\vct{a})$.
\end{Definition}
\begin{Definition}[Subcircuit]
A subcircuit of a circuit $\circuit$ is a circuit \subcircuit such that \subcircuit is a DAG \textit{subgraph} of the DAG representing \circuit. The sink of \subcircuit has exactly one gate \gate.