On the set of fixed points for NRS(m)

2025/09/17 by Mario DeFranco, DeFranco, Mario
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Combinatorics (math.CO) #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2509.14176

openalex publication_date 2025/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 (w₀, w₁), then w₀ is of the form z_i+z_j for some i ≠ j. As a corollary, we prove a formula expressing the elementary symmetric expansion of the function ∏_{1≤ i lt; j ≤ 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.

Citations

• On Taylor series of zeros of complex-exponent polynomials
• On Generalizations of the Newton-Raphson-Simpson Method

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