From 430b69fd76a40cb1f1426e030320476be945f273 Mon Sep 17 00:00:00 2001
From: Aaron Huber <ahuber@buffalo.edu>
Date: Sat, 19 Dec 2020 12:31:35 -0500
Subject: [PATCH] Small changes to S2.4

---
 ra-to-poly.tex | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/ra-to-poly.tex b/ra-to-poly.tex
index ff72a1e..6e3f7f1 100644
--- a/ra-to-poly.tex
+++ b/ra-to-poly.tex
@@ -91,7 +91,7 @@ A \emph{\ti} is a \bi where each block contains exactly one tuple.
 \subsection{Expression Trees}\label{sec:expression-trees}
 
 In this section, we formally define expression trees, an encoding of polynomials that we use throughout much of the paper before generalizing to circuits in~\Cref{sec:gen}.
-For illustrative purposes consider the polynomial $\poly(\vct{X}) = 2X_1^2 + 3X_1X_2 - 2X_2^2$ over $\vct{X} = [X_1, X_2]$.
+For illustrative purposes consider the polynomial $\poly(\vct{X}) = 2X^2 + 3XY - 2Y^2$ over $\vct{X} = [X, Y]$.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{Definition}[Expression Tree]\label{def:express-tree}
@@ -101,11 +101,11 @@ tree, whose internal nodes are from the set $\{+, \times\}$, with leaf nodes bei
 \end{Definition}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-We ignore the remaining fields (\vari{partial} and \vari{weight}) until \Cref{sec:algo}. Note that $\etree$ need not encode an expression in standard monomial basis.  For instance, $\etree$ could represent a compressed form of the running example, such as $(X_1 + 2X_2)(2X_1 - X_2)$.
+We ignore the remaining fields (\vari{partial} and \vari{weight}) until \Cref{sec:algo}. Note that $\etree$ need not encode an expression in standard monomial basis.  For instance, $\etree$ could represent a compressed form of the running example, such as $(X + 2Y)(2X - Y)$.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{Definition}[poly$(\cdot)$]\label{def:poly-func}
-Denote $poly(\etree)$ to be the function that takes as input expression tree $\etree$ and outputs its corresponding polynomial in \abbrSMB.  $poly(\cdot)$ is recursively defined on $\etree$ as follows, where $\etree_\lchild$ and $\etree_\rchild$ denote the left and right child of $\etree$ respectively.
+Denote $poly(\etree)$ to be the function that takes as input expression tree $\etree$ and outputs its corresponding polynomial in \abbrSMB.  We define $poly(\cdot)$ recursively on $\etree$ as follows, where $\etree_\lchild$ and $\etree_\rchild$ denote the left/right child of $\etree$ respectively.
 %
 %	\begin{align*}
 %		&\etree.\type = +\mapsto&& \polyf(\etree_\lchild) + \polyf(\etree_\rchild)\\
@@ -131,7 +131,7 @@ Note that addition and multiplication above follow the standard interpretation o
 \end{Definition}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-For our running example, $\etreeset{\smb} = \{2X_1^2 + 3X_1X_2 - 2X_2^2, (X_1 + 2X_2)(2X_1 - X_2), X_1(2X_1 - X_2) + 2X_2(2X_1 - X_2), 2X_1(X_1 + 2X_2) - X_2(X_1 + 2X_2)\}$.  Note that \cref{def:express-tree-set} implies that $\etree \in \etreeset{poly(\etree)}$.
+For our running example, $\etreeset{\smb} = \{2X^2 + 3XY - 2Y^2, (X + 2Y)(2X - Y), X(2X - Y) + 2Y(2X - Y), 2X(X + 2Y) - Y(X + 2Y)\}$.  Note that \cref{def:express-tree-set} implies that $\etree \in \etreeset{poly(\etree)}$.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Problem Definition}\label{sec:problem-definition}