We would like to argue for a compressed version of $\poly(\vct{w})$, in general $\expct_{\vct{w}}\pbox{\poly(\vct{w})}$ cannot be computed in linear time.
\AR{Added the hardness result below.}
The hardness result is based on the following hardness result:
\begin{theorem}[\cite{k-match}]
\label{thm:k-match-hard}
Given a positive integer $k$ and an undirected graph $G$ with no self-loops of parallel edges, couting the number of $k$-matchings in $G$ is $\#W[1]$-hard.
\end{theorem}
The above result means that we cannot hope to count the number of $k$-matchings in $G=(V,E)$ in time $f(k)\cdot |V|^{O(1)}$ for any function $f$. In fact, all known algorithms to solve this problem takes time $|V|^{\Omega(k)}$.
Given polynomial $\poly_{G}^\kElem(\prob,\ldots, \prob)$, we can write $\rpoly_{G}^\kElem$ as $\rpoly_{G}^\kElem(\prob,\ldots, \prob)=\sum\limits_{i =0}^{2\kElem} c_i \cdot\prob^i$ for some fixed terms $\vct{c}$. Given $2\kElem+1$ distinct $\prob$ values, one can compute each $c_i$ in $\vct{c}$ exactly. Additionally, the number of $\kElem$-matchings can be computed exactly.
It is trivial to see that one can readily expand the exponential expression by performing the $n^\kElem$ product operations, yielding the polynomial in the sum of products form of the lemma statement. By definition $\rpoly_{G}^\kElem$ reduces all variable exponents greater than $1$ to $1$. Thus, a monomial such as $X_i^\kElem X_j^\kElem$ is $X_iX_j$ in $\rpoly_{G}^\kElem$, and the value after substitution is $p_i\cdot p_j = p^2$. Further, that the number of terms in the sum is no greater than $2\kElem+1$, can be easily justified by the fact that each edge has two endpoints, and the most endpoints occur when we have $\kElem$ distinct edges (such a subgraph is also known as a $\kElem$-matching), with non-intersecting points, a case equivalent to $p^{2\kElem}$.
Given that we have $2\kElem+1$ distinct values of $\prob$ by the lemma statement, it follows that we then have $2\kElem+1$ linear equations which are distinct. Further, by construction of the summation, these $2\kElem+1$ equations collectively form the Vandermonde matrix, from which it follows that we have a matrix with full rank, and we can solve the linear system to determine $\vct{c}$ exactly.
It has already been established above that a $\kElem$-matching ($\kmatch$) has coefficient $c_{2\kElem}$. As noted, a $\kElem$-matching occurs when there are $\kElem$ edges, $e_1, e_2,\ldots, e_\kElem$, such that all of them are disjoint, i.e., $e_1\neq e_2\neq\cdots\neq e_\kElem$. In all $\kElem$ factors of $\poly_{G}^\kElem(\vct{X})$ there are $k$ choices from the first factor to select an edge for a given $\kElem$ matching, $\kElem-1$ choices in the second factor, and so on throughout all the factors, yielding $\kElem!$ duplicate terms for each $\kElem$ matching in the expansion of $\poly_{G}^\kElem(\vct{X})$.
Thus, the product $\kElem!\cdot\numocc{G}{\kmatch}$ is the exact number of $\kElem$-matchings in $\poly_{G}^\kElem(\vct{X})$.
By ~\cref{lem:qEk-multi-p}, the term $c_{2\kElem}$ can be exactly computed. Additionally we know that $c_{2\kElem}$ can be broken into two factors, and by dividing $c_{2\kElem}$ by the factor $\kElem!$, it follows that the resulting value is indeed $\numocc{G}{\kmatch}$.
