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

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.