Generating Ouroboros Polynomials and Ouroboros Matrices

TL;DR

This paper explores properties of Ouroboros functions, demonstrating how to generate associated polynomials and matrices, and analyzing their structure and trace characteristics using algebraic methods.

Contribution

It introduces a method to generate higher-order polynomials from Ouroboros functions and constructs matrices from these polynomials, extending previous theoretical work.

Findings

Second-order polynomials can be generated for any multivariable Ouroboros function.

Higher-order polynomials are constructed using properties of Ouroboros spaces.

A matrix derived from these polynomials has a trace degree formula and interesting structural properties.

Abstract

Ouroboros functions have shown some interesting properties when subjected to conventional operations. The aim of this paper is to continue our investigation and prove some additional properties of these functions. Using algebraic methods, we demonstrate that a collection of second-order polynomials can be generated for any multivariable Ouroboros function of the form we have mentioned in previous works ([1] [2]). We then generalize this observation to higher-order polynomials using the properties of Ouroboros spaces and the results of some of our previously proven theorems. After discussing the generation of these polynomials, we conclude by constructing a matrix from them and provide a few comments on its structure and aesthetic, culminating in the derivation of an intuitive formula for the degree of the trace of the square cases of these matrices and the discussion of some future research prospects.