We first argue that $\rpoly_{G}^\kElem(\prob,\ldots, \prob)=\sum\limits_{i =0}^{2\kElem} c_i \cdot\prob^i$. First, since $\poly_G(\vct{X})$ has degree $2$, it follows that $\poly_G^\kElem(\vct{X})$ has degree $2\kElem$. By definition, $\rpoly_{G}^{\kElem}(\vct{X})$ sets every exponent $e > 1$ to $e =1$, which means that $\degree(\rpoly_{G}^\kElem)\le\degree(\poly_G^\kElem)=2k$. Thus, if we think of $\prob$ as a variable, then $\rpoly_{G}^{\kElem}(\prob,\dots,\prob)$ is a univariate polynomial of degree at most $\degree(\rpoly_{G}^\kElem)\le2k$. Thus, we can write
We note that $c_i$ is {\em exactly} the number of monomials in the SMB expansion of $\poly_{G}^{\kElem}(\vct{X})$ composed of $i$ distinct variables.\footnote{Since $\rpoly_G^\kElem(\vct{X})$ does not have any monomial with degree $< 2$, it is the case that $c_0= c_1=0$ but for the sake of simplcity we will ignore this observation.}
Given that we then have $2\kElem+1$ distinct values of $\rpoly_{G}^\kElem(\prob,\ldots, \prob)$ for $0\leq i\leq2\kElem$, it follows that we have a linear system of the form $\vct{M}\cdot\vct{c}=\vct{b}$ where the $i$th row of $\vct{M}$ is $\inparen{\prob_i^0\ldots\prob_i^{2\kElem}}$, $\vct{c}$ is the coefficient vector $\inparen{c_0,\ldots, c_{2\kElem}}$, and $\vct{b}$ is the vector such that $\vct{b}[i]=\rpoly_{G}^\kElem(\prob_i,\ldots, \prob_i)$. In other words, matrix $\vct{M}$ is the Vandermonde matrix, from which it follows that we have a matrix with full rank (the $p_i$'s are distinct), and we can solve the linear system in $O(k^3)$ time (e.g., using Gaussian Elimination) to determine $\vct{c}$ exactly.
Thus, after $O(k^3)$ work, we know $\vct{c}$ and in particular, $c_{2k}$ exactly.
We claim that $c_{2\kElem}$ is $\kElem!\cdot\numocc{G}{\kmatch}$. This can be seen intuitively by looking at the expansion of the original factorized representation
where a unique $\kElem$-matching in the multi-set of product terms has $\kElem$ distinct $\inparen{i_\ell, j_\ell}$ index pairs. Further, any monomial composed of such a distinct set of $2\kElem$ variables will be produced $\kElem!$ times in a $\kElem$-wise product of the sum of a set of elements with itself. This is true because each (identical) product term contains each of the $\kElem$ distinct elements, giving us $\kElem\cdot\kElem-1\cdots1=\kElem!$ permutations of a distinct $\kElem$-matching. %Such must be the case since each product term has since this is the number of permutations for a given monomial.can be selected $\prod_{i = 1}^\kElem i = \kElem!$ times.
%Indeed, note that each $\kElem$-matching $(i_1, j_1)\ldots$ $(i_k, j_k)$ in $G$ corresponds to the monomial $\prod_{\ell = 1}^\kElem X_{i_\ell}X_{j_\ell}$ in $\poly_{G}^\kElem(\vct{X})$, with distinct indexes, and this implies that each distinct $\kElem$-matching appears the exact number of permutations that exist for its particular set of $\kElem$ edges, or $k!$.
Since, as noted earlier, $c_{2\kElem}$ represents the number of monomials with $2\kElem$ distinct variables, then it must be that $c_{2\kElem}$ is the overall number of $\kElem$-matchings. And since we have $\kElem!$ copies of each distinct $\kElem$-matching, it follows that