Self-interacting polynomials

TL;DR

This paper introduces self-interacting polynomial dynamical systems, exploring their algebraic properties, invariants, and dynamics, especially focusing on quadratic cases and finite field experiments.

Contribution

It presents a novel class of algebraic dynamical systems based on polynomials acting on each other, providing new insights and analysis methods.

Findings

Identification of basic invariant sets

Detailed analysis of quadratic polynomial systems

Experimental results over finite fields

Abstract

We introduce a class of dynamical systems of algebraic origin, consisting of self-interacting irreducible polynomials over a field. A polynomial f is made to act on a polynomial g by mapping the roots of g. This action identifies a new polynomial h, as the minimal polynomial of the displaced roots. By allowing several polynomials to act on one another, we obtain a self-interacting system with a rich dynamics, which affords a fresh viewpoint on some algebraic dynamical constructs. We identify the basic invariant sets, and study in some detail the case of quadratic polynomials. We perform some experiments on self-interacting polynomials over finite fields.