On the set of fixed points for NRS($m$)
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915499638521856 |
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| author | DeFranco, Mario |
| author_facet | DeFranco, Mario |
| contents | 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 Lindström-Gessel-Veinnot Lemma. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_14176 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the set of fixed points for NRS($m$) DeFranco, Mario Combinatorics 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 Lindström-Gessel-Veinnot Lemma. |
| title | On the set of fixed points for NRS($m$) |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.14176 |