Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Division properties of commuting polynomials

Kimiko Hasegawa, Rin Sugiyama

Abstract

Polynomials commute under composition are referred to as commuting polynomials. In this paper, we study division properties for commuting polynomials with rational (and integer) coefficients. As a consequence, we show an algebraic particularity of the commuting polynomials coming from weighted sums for cycle graphs with pendant edges (arXiv:2402.07209v1.). We also discuss a set of commuting polynomials over a field of positive characteristic.

Division properties of commuting polynomials

Abstract

Polynomials commute under composition are referred to as commuting polynomials. In this paper, we study division properties for commuting polynomials with rational (and integer) coefficients. As a consequence, we show an algebraic particularity of the commuting polynomials coming from weighted sums for cycle graphs with pendant edges (arXiv:2402.07209v1.). We also discuss a set of commuting polynomials over a field of positive characteristic.
Paper Structure (17 sections, 24 theorems, 87 equations)

This paper contains 17 sections, 24 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Commuting polynomials
  3. Results
  4. Chains of Chebyshev type
  5. Chebyshev polynomials
  6. Monic and integer coefficient conditions
  7. Euclidean division
  8. Greatest common divisor and Factorization
  9. Chains of monomial type
  10. Monic and integer coefficient conditions
  11. Euclidean Division
  12. Greatest common divisor and Factorization
  13. Chains over a field of positive characteristic
  14. Classification theorem
  15. Proof in case $p\neq 2$
  16. ...and 2 more sections

Key Result

Theorem 1.4

Assume that $K$ is a field of characteristic zero. Let $\{f_n(x)\}_{n \in \mathbb{N}}$ be a chain in $K[x]$. Then $\{f_n(x)\}_{n \in \mathbb{N}}$ is similar to only one of the two $\{x^n\}_{n \in \mathbb{N}}$ or $\{T_n(x)\}_{n \in \mathbb{N}}$.

Theorems & Definitions (53)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Theorem 1.4: BT, J
  • Definition 1.5
  • Definition 1.6
  • Corollary 1.7
  • Remark 1.8
  • Theorem 1.9: Theorem \ref{['thm-A']}
  • Corollary 1.10: Corollary \ref{['cor-A']}
  • ...and 43 more