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arXiv · 2509.14176

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

Mario DeFranco
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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 $i \neq j$. 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 Lindstr\"om-Gessel-Veinnot Lemma.

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Mario DeFranco. 2025-09-17. On the set of fixed points for NRS($m$). https://arxiv.org/abs/2509.14176

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