On the set of fixed points for NRS($m$)

Abstract

Let $f(z)$ be a degree $d$ polynomial with zeros $z_i$. For arbitrary $m$ we construct explicit set of fixed points (attractors) of NRS($m$), and prove a factored formula for the Jacobian at these points. We prove that if NRS(2), when applied to $f$ with an arbitrary starting point, converges to a point $(w_0,w_1)$, then $w_0$ is of the form $z_i+z_j$ for some $i\ne j$. As a corollary, we prove a formula expressing the elementary symmetric expansion of the function \[ \prod_{1\leq i < j \leq d} (z - z_i -z_j) \] in the variables $z_i$ in terms of non-intersecting paths on certain directed graphs, using the Lindström-Gessel-Veinnot Lemma.