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

2021/11/26 by Emerson León, León, Emerson, Julián Pulido +1
Computer Science · Mathematics · #Advanced Mathematical Identities #Coding theory and cryptography #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2111.13745

openalex publication_date 2021/11/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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]→[0,1] that goes up and down p times with slope ± 1/p. We provide a bijection between F_pⁿ and the fixed points of gⁿ_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.

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