starting my pass on Lemma 5.

poly-form.tex

@@ -181,8 +181,9 @@ If we can compute $\poly(\vct{X})$ in $T(m)$ time for $O(1)$ distinct values of
 \AR{Follows from the fact that the corresponding coefficient matrix is the so called Vandermonde matrix, which has full rank.}
 \AH{This Vandermonde matrix I need to research.  Right now, the last sentences are just parrotting Atri.}
 
+\AR{Jul 31: Did not make a pass on anything above this.}
 \begin{Lemma}\label{lem:const-p}
-If we can compute $\poly(\vct{X})$ in T(m) time for $\wElem_1 =\cdots= \wElem_\numTup = \prob$, then we can count the number of triangles, 3-paths, and 3-matchings in $G$ in $T(m) + O(m)$ time.
+If we can compute $\poly(\vct{X})$ in T(m) time for $\wElem_1 =\cdots= \wElem_\numTup = \prob$,\AR{Nope, this is not correct. Either use $\tilde{Q}(p,\dots,p)$ or $\mathbb{E}_{\vct{W}\sim P^{(\vct{p})}}[Q(\vct{W})]$ for $\vct{p}=(p,\dots,p)$. Also you have to have $G$ in the notation $Q$ as the statement as the claim is you can do the computation for {\em every} $G$ with $m$ edges in $T(m)$ then you can compute the number of triangles etc. for arbitrary $G$ as well.} then we can count the number of triangles, 3-paths, and 3-matchings in $G$ in $T(m) + O(m)$ time.
 \end{Lemma}
 
 
@@ -540,4 +541,4 @@ Further algebraic manipulations result in
 
 The roots for \cref{eq:det-combine} are $p = 0, p = 1$, and $p = i$.  Thus, we have proved the lemma for fixed $p \in (0, 1)$.
 \end{proof}
-\qed
+\qed