Patterns In The Coefficients Of Powers Of Polynomials Over A Finite Field

TL;DR

This paper studies the behavior of coefficients of polynomial powers over finite fields, deriving formulas, asymptotics, and conjectures related to their complexity, automata, and fractal dimensions.

Contribution

It introduces new formulas and asymptotic analysis for the sequence of string complexities of polynomial coefficients over finite fields, extending previous work.

Findings

Derived recursion relation for a(n) when f=1+x and general p

Computed asymptotics of a(n)/n^2 as n approaches infinity

Analyzed eigenvalues of matrix B related to fractal dimensions

Abstract

We examine the behavior of the coefficients of powers of polynomials over a finite field of prime order. Extending the work of Allouche-Berthe, 1997, we study a(n), the number of occurring strings of length n among coefficients of any power of a polynomial f reduced modulo a prime p. The sequence of line complexity a(n) is p-regular in the sense of Allouche-Shalit. For f=1+x and general p, we derive a recursion relation for a(n) then find a new formula for the generating function for a(n). We use the generating function to compute the asymptotics of a(n)/n^2 as n approaches infinity, which is an explicitly computable piecewise quadratic in x with n= [p^m/x] and x is a real number between 1/p and 1. Analyzing other cases, we form a conjecture about the generating function for general a(n). We examine the matrix B associated with f and p used to compute the count of a coefficient, which applies to the theory of linear cellular automata and fractals. For p=2 and polynomials of small degree we compute the largest positive eigenvalue, \lambda, of B, related to the fractal dimension d of the corresponding fractal by d= \log_2(\lambda). We find proofs and make a number of conjectures for some bounds on \lambda, and upper bounds on its degree.