Combinatorial Relationship Between Finite Fields and Fixed Points of Functions Going Up and Down

TL;DR

This paper uncovers a combinatorial connection between finite field roots and fixed points of specific piecewise linear functions, extending to Chebyshev polynomials and non-continuous functions, revealing structural insights.

Contribution

It establishes a bijection linking roots of irreducible polynomials over finite fields to fixed points of functions that go up and down, including extensions to Chebyshev polynomials.

Findings

Bijection between finite field elements and fixed points of functions

Extension of results to Chebyshev polynomials

Generalization to non-continuous piecewise linear functions

Abstract

We explore a combinatorial bijection between two seemingly unrelated topics: the roots of irreducible polynomials of degree $m$ over a finite field $F_p$ for a prime number $p$ and the number of points that are periodic of order $m$ for a continuous piece-wise linear function $g_p:[0,1]\rightarrow[0,1]$ that \emph{goes up and down $p$ times} with slope $\pm 1/p$. We provide a bijection between $F_{p^n}$ and the fixed points of $g^n_p$ that naturally relates some of the structure in both worlds. Also we extend our result to other families of continuous functions that goes up and down $p$ times, in particular to Chebyshev polynomials, where we get a better understanding of its fixed points. A generalization for other piece-wise linear functions that are not necessarily continuous is also provided.