Let $n$ be an even positive integer. Let $e_{\frac{n}{2}}(x_1, x_2, \ldots ,x_n)$ denote the ${\frac{n}{2}}^{\textrm{th}}$ elementary symmetric polynomial in $n$ variables $x_1,x_2,\cdots,x_n$, defined as \begin{align*} e_{\frac{n}{2}}(x_1,x_2, \ldots ,x_n) = \sum_{ 1\leq j_1 < j_2 < \ldots <j_{\frac{n}{2}} \leq n } x_{j_1}x_{j_2}\ldots x_{j_{\frac{n}{2}}}. \end{align*} Now let $r_0, r_1, \cdots, r_n$ be $n+1$ positive real numbers. What is the complexity of the best possible algorithm to compute the following $n+1$ numbers? \begin{align*} e_{\frac{n}{2}}(r_1, r_2, & \ldots ,r_n) \\ e_{\frac{n}{2}}(r_0, r_2, & \ldots ,r_n) \\ & \vdots \\ e_{\frac{n}{2}}(r_0, r_1, & \ldots ,r_{n-1}). \end{align*}
I can compute, for example, $e_{\frac{n}{2}}(r_1, r_2, \ldots ,r_n)$ with complexity $O(n^2)$ and therefore the $n+1$ numbers above with complexity $O(n^3)$. Is it possible to do better?