The Arithmetic of Consecutive Polynomial Sequences over Finite Fields

TL;DR

This paper investigates the arithmetic properties of sequences of consecutive polynomials over finite fields, focusing on irreducibility, factorization bounds, and the existence of such sequences with all polynomials irreducible.

Contribution

It introduces new bounds and results on the irreducibility and factorization of consecutive polynomial sequences over finite fields, extending understanding of their arithmetic structure.

Findings

Bounds for the largest degree of irreducible factors

Number of irreducible factors in such sequences

Count of sequences with all polynomials irreducible

Abstract

Motivated by a question of van der Poorten about the existence of infinite chain of prime numbers (with respect to some base), in this paper we advance the study of sequences of consecutive polynomials whose coefficients are chosen consecutively from a sequence in a finite field of odd prime characteristic. We study the arithmetic of such sequences, including bounds for the largest degree of irreducible factors, the number of irreducible factors, as well as for the number of such sequences of fixed length in which all the polynomials are irreducible.