Irreducible Polynomials with Varying Constraints on Coefficients

2017/12/11 by Moses, Eyal
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1712.04051

Abstract

We study the number of prime polynomials of degree n over \mathbbF_q in which the i^th coefficient is either preassigned to be a_i ∈ \mathbbF_q or outside a small set S_i ⊂ \mathbbF_q. This serves as a function field analogue of a recent work of Maynard, which counts integer primes that do not have specific digits in their base-q expansion. Our work relates to Pollack's and Ha's work, which count the amount of prime polynomials with ≪ √(n) and ≪ n preassigned coefficients, respectively. Our result demonstrates how one can prove asymptotics of the number of prime polynomials with different types of constraints to each coefficient.

Related

• Prime numbers and factorization of polynomials
• Primes represented by incomplete norm forms
• On the distribution of polynomials having a given number of irreducible factors over finite fields
• Irreducibility of Polynomials with Square Coefficients over Finite Fields
• Explicit upper bounds for the number of primes simultaneously representable by any set of irreducible polynomials