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Dynamical systems defined by polynomials with algebraic properties

Xiang Gao, Teturo Kamae

TL;DR

This work builds a dynamical-systems framework for polynomials with integer coefficients acting on two-sided streams over the torus via convolution, introducing the stream zeros $\Omega_P$ as the kernel of the $P(z)$-action and linking them to the roots of $P(z)=0$. It develops a symbolic representation $\Xi_P$ through Kaneko's formula, analyzes factorization through the resultant and dimensions of $\Omega_P$, and characterizes the automorphism group $Saut_P$ for low-degree polynomials, including Pell-type structures. The results extend to streams over Galois fields, where the automorphism group becomes cyclic of order $q^k-1$ and can be realized by GF($q$) matrices. Overall, the paper connects algebraic properties of $P$ with the dynamical and symbolic structure of the induced stream system, offering a novel viewpoint on roots and factorization via dynamical methods.

Abstract

Let (x_n; n\in Z) be a bisequence of elements x_n in the 1-dimensional torus R/Z, which is called a stream over R/Z. Let P(z)=a_k z^k+...+a_1 z+a_0 be a polynomial with integer coefficients. Define the set of streams over R/Z such that the convolution product P(z)\times(x_n; n\in Z)=(\sum_{i=0}^k a_i x_{n-i}; n\in Z)=(0; n\in Z), which is called the stream 0 of P. We study similarities between stream 0 of P and the roots of P(z)=0.

Dynamical systems defined by polynomials with algebraic properties

TL;DR

This work builds a dynamical-systems framework for polynomials with integer coefficients acting on two-sided streams over the torus via convolution, introducing the stream zeros as the kernel of the -action and linking them to the roots of . It develops a symbolic representation through Kaneko's formula, analyzes factorization through the resultant and dimensions of , and characterizes the automorphism group for low-degree polynomials, including Pell-type structures. The results extend to streams over Galois fields, where the automorphism group becomes cyclic of order and can be realized by GF() matrices. Overall, the paper connects algebraic properties of with the dynamical and symbolic structure of the induced stream system, offering a novel viewpoint on roots and factorization via dynamical methods.

Abstract

Let (x_n; n\in Z) be a bisequence of elements x_n in the 1-dimensional torus R/Z, which is called a stream over R/Z. Let P(z)=a_k z^k+...+a_1 z+a_0 be a polynomial with integer coefficients. Define the set of streams over R/Z such that the convolution product P(z)\times(x_n; n\in Z)=(\sum_{i=0}^k a_i x_{n-i}; n\in Z)=(0; n\in Z), which is called the stream 0 of P. We study similarities between stream 0 of P and the roots of P(z)=0.

Paper Structure

This paper contains 7 sections, 9 theorems, 100 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Convolution product and its inverse
  3. Stream zeros and dynamical systems
  4. Factorization of polynomials and stream zeros
  5. Stream zeros and the automorphism group
  6. $Saut_P$ in the case $\deg P=2,3$
  7. Streams over Galois fields

Key Result

Theorem 1

(i) Among $A,B,C\in St(\mathbb{C})$, two of them are absolutely summable and the remaining one is bounded, then the convolution products below exist and it holds that (ii) For any absolutely summable $(a_n;n\in\mathbb{Z})\in St(\mathbb{C})$, an absolutely summable $(b_n;n\in\mathbb{Z})$ satisfying $(1)$ is unique if it exists. We call $(b_n;n\in\mathbb{Z})$ as this convolution inverse of $(a_n;n

Figures (4)

  • Figure 1: Symbolic representation of $St([0,1))$: $\xi_{n+2}\in \{-2,-1,0,1\}.$
  • Figure 2: $\sigma$-orbits of $\Omega_P^3$
  • Figure 3: $B$-orbit of $\Omega_P^3$
  • Figure :

Theorems & Definitions (28)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 1
  • Example 1
  • Theorem 3
  • proof
  • Example 2
  • Corollary 1
  • ...and 18 more