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On recurrence relations arising from NRS(2) applied to a cubic polynomial

Mario DeFranco

TL;DR

The paper addresses the positivity of leading-coefficient terms arising from applying NRS(2) to a cubic polynomial by introducing radius-value trees as a combinatorial model. It builds an algebraic framework using the rings $\tilde{\mathrm{CH}}_2$ and $\mathrm{CH}_2$ to encode these coefficients and derives recurrence relations for the leading terms. The main contributions show that the leading coefficients can be expressed as a sum over radius-value trees with all positive contributions (via sign-reversing involutions and Ker$(L)$ considerations), and it provides a detailed combinatorial partitioning (central-path decomposition) that explains the positivity and symmetry of the coefficients. These results connect the dynamics of NRS(2) on cubics to explicit, positive combinatorial structures, enabling precise understanding of error growth in this setting.

Abstract

We prove that the leading coefficient of the "error" terms of NRS(2) applied to a cubic polynomial $f(z)$ with starting point $(-\frac{a_1}{a_2}, -\frac{a_1}{a_2})$ are positive-coefficient rational functions in the zeros of $f(z)$. We express these terms as a sum over combinatorial objects which we call radius-value trees.

On recurrence relations arising from NRS(2) applied to a cubic polynomial

TL;DR

The paper addresses the positivity of leading-coefficient terms arising from applying NRS(2) to a cubic polynomial by introducing radius-value trees as a combinatorial model. It builds an algebraic framework using the rings and to encode these coefficients and derives recurrence relations for the leading terms. The main contributions show that the leading coefficients can be expressed as a sum over radius-value trees with all positive contributions (via sign-reversing involutions and Ker considerations), and it provides a detailed combinatorial partitioning (central-path decomposition) that explains the positivity and symmetry of the coefficients. These results connect the dynamics of NRS(2) on cubics to explicit, positive combinatorial structures, enabling precise understanding of error growth in this setting.

Abstract

We prove that the leading coefficient of the "error" terms of NRS(2) applied to a cubic polynomial with starting point are positive-coefficient rational functions in the zeros of . We express these terms as a sum over combinatorial objects which we call radius-value trees.
Paper Structure (5 sections, 11 theorems, 147 equations)

This paper contains 5 sections, 11 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. NRS(2) applied to cubic polynomials
  3. The rings $\mathrm{CH}_2$ and $\tilde{\mathrm{CH}}_2$
  4. Radius-value trees
  5. Combinatorial expressions and positivity for recurrence relation

Key Result

Lemma 1

As polynomials in $u_3$, assuming $u_1$ and $u_2$ to be indeterminates, we have the degrees

Theorems & Definitions (33)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Definition 1
  • Lemma 3
  • proof
  • Definition 2
  • Definition 3
  • Remark 1
  • ...and 23 more