Implemented changes to Sec. 3.1 per Riot conversation w/@atri 120920

mult_distinct_p.tex

@@ -25,7 +25,7 @@ Given polynomial $\poly_{G}^\kElem(\prob,\ldots, \prob)$, we can write $\rpoly_{
 %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 $p^{2\kElem}$.
 Since $\rpoly_{G}^\kElem(\prob,\ldots, \prob) = \sum\limits_{i = 0}^{2\kElem} c_i \cdot \prob^i$, this implies that $\rpoly_{G}^\kElem$ is a polynomial of degree $2\kElem$ and hence $\rpoly_{G}^\kElem(\prob,\ldots, \prob)$ is a polynomial in $\prob$ of degree $2\kElem$.
 
-Given that we then have $2\kElem + 1$ distinct values of $\rpoly_{G}^\kElem(\prob,\ldots, \prob)$, it follows that we then have $2\kElem + 1$ linear equations which are distinct.  Further, by construction of the summation, the coefficient matrix of the $2\kElem + 1$ equations is the Vandermonde matrix, from which it follows that we have a matrix with full rank, and we can solve the linear system in $O(k^3)$ time to determine $\vct{c}$ exactly.
+Given that we then have $2\kElem + 1$ distinct values of $\rpoly_{G}^\kElem(\prob,\ldots, \prob)$, it follows that we then have $2\kElem + 1$ linear equations of the form $\prob_i^0\ldots\prob_i^{2\kElem}$ which are distinct.  We have then a linear system of the form $M \cdot \vct{c} = \vct{b}$ where $M$ holds the aforementioned linear equations, $\vct{c}$ is the coefficient vector ($c_0,\ldots, c_{2\kElem}$), and $\vct{b}$ is the vector containing each $\rpoly_{G}^\kElem(\prob_i,\ldots, \prob_i)$.  By construction of the summation, matrix $M$ is the Vandermonde matrix, from which it follows that we have a matrix with full rank, and we can solve the linear system in $O(k^3)$ time to determine $\vct{c}$ exactly.
 
 Note that $c_{2\kElem}$ is $\kElem! \cdot \numocc{G}{\kmatch}$.  This can be seen intuitively by looking at the original factorized representation $\poly_{G}^\kElem(\vct{X})$, where, across each of the $\kElem$ products, an arbitrary $\kElem$-matching can be selected $\prod_{i = 1}^\kElem \kElem = \kElem!$ times.  Note that each $\kElem$-matching $(X_{i_1} X_{j_1})\ldots$ $(X_{i_k} X_{j_k})$ corresponds to the unique monomial $\prod_{\ell = 1}^\kElem X_{i_\ell}X_{j_\ell}$ in $\rpoly_{G}^\kElem(\vct{X})$, where each index is distinct.  Since $\rpoly$ contains only exponents $e \leq 1$, the only degree $2\kElem$ terms that can exist in $\rpoly_{G}^\kElem$ are $\kElem$-matchings since every other monomial in $\rpoly_{G}^\kElem(\vct{X})$ has degree $< 2\kElem$.
 %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 $\poly_{G}^\kElem(\vct{X})$.