%root: main.tex
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 $i \neq j$.
\end{Definition}
Note that similar in spirit to \Cref{def:reduced-bi-poly}, $\expansion{\circuit}$ \Cref{def:expand-circuit} reduces all variable exponents $e > 1$ to $e = 1$.  Further, it is true that $\expansion{\circuit}$ is the pure expansion of $\circuit$.

In the following, recall by \cref{def:expand-circuit} that we use $\encMon$ to denote the monomial composed of the variables in $\monom$.

\begin{Example}[Example for \Cref{def:expand-circuit}]\label{example:expr-tree-T}
Consider the factorized representation $(X+ 2Y)(2X - Y)$ of the polynomial in \Cref{eq:poly-eg}.
Its circuit $\circuit$ is illustrated in \Cref{fig:circuit}.
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 pure 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 $\poly$ 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}.
\end{Example}

%\begin{Definition}[Evaluation]\label{def:exp-poly-eval}
%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}
%
%\AH{Do we use this anywhere \cref{def:exp-poly-eval}?}

\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.
\end{Definition}
